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![Eisagwg
Mia isometrik embˆptish enìc poluptÔgmatoc Riemann se èna polÔptugma Rie-
mann lègetai elaqistik an to dianusmatikì pedÐo mèshc kampulìthtac eÐnai to mh-
denikì isodÔnama an to Ðqnoc thc deÔterhc jemeli¸douc morf c thc isometrik c
embˆptishc eÐnai tautotikˆ mhdèn. Apì ton tÔpo thc pr¸thc metabol c tou embadoÔ
gnwrÐzoume pwc oi isometrikèc elaqistikèc embaptÐseic eÐnai akrib¸c ta krÐsima shmeÐa
thc sunˆrthshc tou embadoÔ. Epomènwc, fusikì epakìloujo eÐnai pwc oi elaqistikèc
isometrikèc embaptÐseic apoteloÔn an¸terhc diˆstashc genÐkeush twn gewdaisiak¸n
kampul¸n kai apartÐzoun mia shmantik oikogèneia isometrik¸n embaptÐsewn.
Sthn paroÔsa ergasÐa asqoloÔmaste me elaqistikèc epifˆneiec sthn Sn, dhlad
isometrikèc elaqistikèc embaptÐseic apì èna prosanatolismèno, sunektikì, didiˆstato
polÔptugma Riemann sthn n-diˆstath monadiaÐa sfaÐra
Sn
= {(x1, x2, ..., xn+1) ∈ Rn+1
: x2
1 + x2
2 + ... + x2
n+1 = 1},
h opoÐa eÐnai efodiasmènh me th sun jh metrik Riemann , kai wc gnwstìn èqei
stajer kampulìthta tom c 1. Parìlh thn omorfiˆ pou èqei apì mình thc h melèth
elaqistik¸n epifanei¸n sth sfaÐra, èqei apodeiqjeÐ ìti sqetÐzetai me th melèth me-
monwmènwn idiazìntwn shmeÐwn elaqistik¸n embaptÐsewn ston EukleÐdeio q¸ro. O
E. Calabi sto ˆrjro [6] parat rhse ìti an èqoume èna tridiˆstato elaqistikì upo-
polÔptugma M3 ston EukleÐdeio q¸ro En+3, tìte to polÔptugma M3 ∩ Sn+2, gia
sfaÐra Sn+2 katˆllhlou kèntrou, eÐnai didiˆstato elaqistikì upopolÔptugma sthn
Sn+2. AntÐstrofa, an M2 eÐnai èna didiˆstato elaqistikì upopolÔptugma thc Sn+2,
tìte o k¸noc pou dhmiourgeÐtai apì tic hmieujeÐec me arq to kèntro thc sfaÐrac
Sn+2 kai dièrqontai apì ta shmeÐa tou M2, eÐnai tridiˆstato elaqistikì upopolÔptug-
ma ston EukleÐdeio q¸ro En+3 me memonwmèno idiˆzon shmeÐo to kèntro thc sfaÐrac an
to M2 den eÐnai olikˆ gewdaisiakì sthn Sn+2. Epomènwc, h melèth isometrik¸n ela-
qistik¸n embaptÐsewn tridiˆstatwn poluptugmˆtwn Riemann me memonwmèno idiˆzon
shmeÐo ston EukleÐdeio q¸ro, anˆgetai sth melèth isometrik¸n elaqistik¸n embaptÐ-
sewn didiˆstatwn poluptugmˆtwn Riemann sth sfaÐra. Thn idèa aut ulopoÐhse o
E. Calabi sto prwtoporiakì ˆrjro [6] melet¸ntac elaqistikèc epifˆneiec sth sfaÐra
me thn aploÔsterh dunat topologÐa, dhlad sumpageÐc elaqistikèc epifˆneiec gènouc
mhdèn isodÔnama omoiomorfikèc me thn S2. Autì to ˆrjro èdwse to ènausma gia th
melèth elaqistik¸n epifanei¸n sth sfaÐra. LÐgo metˆ thn emfˆnish tou ˆrjrou tou
E. Calabi, o S.S. Chern parousÐase mia pio gewmetrik prosèggish twn elaqistik¸n
epifanei¸n sth sfaÐra kˆnontac qr sh twn jemeliwd¸n morf¸n anwtèrac tˆxewc. H
melèth aut suneqÐsthke apì ton J.L.M. Barbosa.
iii](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-3-320.jpg)
![iv
O stìqoc thc ergasÐac eÐnai na apodeÐxoume ta apotelèsmata tou E. Calabi sto
[6], me ton trìpo pou ta anadiatÔpwse o S.S. Chern sto [8], kaj¸c kai tou J.L.M.
Barbosa sto [4].
GnwrÐzoume ìti ta olikˆ gewdaisiakˆ m-diˆstata upopoluptÔgmata thc Sn, ìpou
2 ≤ m ≤ n − 1, eÐnai oi mègistec m-sfaÐrec thc Sn, dhlad tomèc thc Sn me (m + 1)-
diˆstatouc upoq¸rouc tou Rn+1. Mia isometrik embˆptish f : (M, , ) −→ Sn, ìpou
(M, , ) eÐnai polÔptugma Riemann, kaleÐtai koresmènh (full) an h eikìna thc den
perièqetai se kanèna olikˆ gewdaisiakì upopolÔptugma thc Sn.
Sthn ergasÐa aut to pr¸to apotèlesma pou ja apodeÐxoume eÐnai ìti oi sumpageÐc,
prosanatolismènec elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra eÐnai koresmènec
mìno se ˆrtiac diˆstashc sfaÐra. Epiplèon ja d¸soume mia ektÐmhsh tou embadoÔ twn.
To sumpèrasma autì to apèdeixe o E. Calabi sto ˆrjro [6]. Argìtera, o S.S. Chern
sto [8] èdwse mia diaforetik prosèggish tou apotelèsmatoc to opoÐo eÐnai to ex c:
Je¸rhma I. 'Estw f : (M, , ) −→ Sn, n ≥ 3, sumpag c, prosanatolismènh, kore-
smènh, elaqistik epifˆneia gènouc mhdèn. Tìte:
(i) O arijmìc n eÐnai ˆrtioc (n = 2m).
(ii) To embadì A(M) thc epifˆneiac eÐnai akèraio pollaplˆsio tou 2π kai isqÔei
A(M) ≥ 2πm(m + 1).
Sth sunèqeia ja apodeÐxoume ìti oi sumpageÐc, prosanatolismènec, koresmènec,
elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra eÐnai ˆkamptec (rigid), èna apotèlesma
pou ofeÐletai ston J.L.M. Barbosa [4]:
Je¸rhma II. 'Estwsan f : (M, , ) −→ S2m, f : (M, , ) −→ S2m sumpageÐc,
prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn. Tìte upˆrqei
isometrÐa τ : S2m −→ S2m ¸ste f = τ ◦ f.
Tèloc, ja deÐxoume pwc oi sumpageÐc, prosanatolismènec, koresmènec, elaqistikèc
epifˆneiec gènouc mhdèn sth sfaÐra me stajer kampulìthta Gauss taxinomoÔntai
pl rwc. Gia thn akrÐbeia apodeiknÔoume to akìloujo apotèlesma tou E. Calabi [6]:
Je¸rhma III. 'Estw f : (M, , ) −→ S2m sumpag c, prosanatolismènh, kore-
smènh, elaqistik epifˆneia gènouc mhdèn. An to polÔptugma Riemann (M, , )
èqei stajer kampulìthta Gauss K, tìte K = 2
m(m+1) kai upˆrqoun isometrÐec
F : (M, , ) −→ S2 m(m+1)
2 kai τ : S2m −→ S2m ¸ste f ◦ F−1 = τ ◦ fm,
ìpou fm : S2 m(m+1)
2 −→ S2m eÐnai h epifˆneia Veronese sthn S2m.
Gia tic apodeÐxeic aut¸n twn apotelesmˆtwn qrhsimopoioÔme ergaleÐa kai mejìdouc
apì th jewrÐa kinoumènou plaisÐou, twn jemeliwd¸n morf¸n an¸terhc tˆxhc kai th
jewrÐa epifanei¸n Riemann (Je¸rhma Riemann-Roch).
Epiplèon, h prosèggish pou akoloujoÔme mac epitrèpei na d¸soume mia apìdeixh
thc eikasÐac tou U. Simon [15] gia tic peript¸seic pou eÐnai dh gnwstì ìti isqÔei.
Parajètoume thn apìdeixh aut¸n twn peript¸sewn sto tèloc thc diatrib c.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-4-320.jpg)
![Kefˆlaio 1
Prokatarktikˆ
1.1 Isometrikèc embaptÐseic
Sto parìn kefˆlaio ja anafèroume aparaÐthta stoiqeÐa apì th jewrÐa twn iso-
metrik¸n embaptÐsewn, tou kinoumènou plaisÐou kai twn jemeliwd¸n morf¸n an¸te-
rhc tˆxhc. Gia tic basikèc ènnoiec thc Diaforik c GewmetrÐac parapèmpoume sta
[10, 14, 13].
'Estw Mn diaforÐsimo polÔptugma diˆstashc n. Gia tuqìn p ∈ Mn sumbolÐzoume
me TpMn ton efaptìmeno q¸ro tou Mn sto p kai me TMn = {(p, v) : p ∈ Mn, v ∈
TpMn} thn efaptìmenh dèsmh tou Mn. To sÔnolo twn diaforÐsimwn dianusmatik¸n
pedÐwn tou Mn to sumbolÐzoume me ∆(Mn) Γ(TMn) kai to sÔnolo twn diaforÐsimwn
sunart sewn g : Mn −→ R to sumbolÐzoume me D(Mn) C∞(Mn, R).
Orismìc 1.1.1. 'Estw f : Mn −→ M
n+k
diaforÐsimh apeikìnish metaxÔ twn
diaforÐsimwn poluptugmˆtwn Mn kai M
n+k
. DiaforÐsimo dianusmatikì pedÐo katˆ
m koc thc f kaloÔme kˆje apeikìnish V h opoÐa se kˆje p ∈ Mn antistoiqeÐ èna
diˆnusma V |p ∈ Tf(p)M
n+k
kai eÐnai diaforÐsimh me thn ex c ènnoia: An (U, y)
eÐnai qˆrthc tou M
n+k
gÔrw apì to f(p) me suntetagmènec (y1, y2, ..., yn+k) kai
V |q = n+k
i=1 gi(q) ∂
∂yi
|f(q) gia kˆje q ∈ f−1(U), oi sunart seic gi, i = 1, 2, ..., n + k,
eÐnai diaforÐsimec.
SumbolÐzoume me ∆(f) to sÔnolo twn diaforÐsimwn dianusmatik¸n pedÐwn katˆ
m koc thc f. Profan¸c to ∆(f) eÐnai to sÔnolo twn pedÐwn (sections) thc epagìmenhc
dianusmatik c dèsmhc f∗(TM
n+k
) := {(p, v) : p ∈ Mn, v ∈ Tf(p)M
n+k
} bajmÐdac
(rank) n + k, dhlad ∆(f) = Γ f∗(TM
n+k
) . An X ∈ ∆(Mn), tìte h apeikìnish h
opoÐa se kˆje p ∈ Mn antistoiqeÐ to diˆnusma dfp(X|p), eÐnai diaforÐsimo dianusmatikì
pedÐo katˆ m koc thc f kai sumbolÐzetai me df(X). Epiplèon, an Y ∈ ∆(M
n+k
), tìte
to Y ◦ f eÐnai epÐshc diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f.
Orismìc 1.1.2. 'Estw f : Mn −→ M
n+k
diaforÐsimh apeikìnish kai X ∈ ∆(Mn),
X ∈ ∆(M
n+k
). Ta X, X lègontai f-susqetismèna (f-related) an X ◦ f = df(X).
1](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-9-320.jpg)
![2 Prokatarktikˆ
L mma 1.1.1. An f : Mn −→ M
n+k
eÐnai diaforÐsimh apeikìnish kai X, Y ∈
∆(Mn) eÐnai f-susqetismèna twn X, Y ∈ ∆(M
n+k
) antÐstoiqa, tìte ta ginìmena Lie
[X, Y ], [X, Y ] eÐnai f-susqetismèna.
Apìdeixh. 'Eqoume X ◦ f = df(X), ˆra gia kˆje p ∈ Mn, X|f(p) = dfp(X|p). Ja
deÐxoume ìti [X, Y ] ◦ f = df [X, Y ] , isodÔnama
X|f(p)(Y ) − Y |f(p)(X) = dfp [X, Y ]|p .
Gia kˆje ϕ ∈ D(M
n+k
) èqoume:
X|f(p)(Y ) − Y |f(p)(X) (ϕ) = X|f(p) Y (ϕ) − Y |f(p) X(ϕ)
= dfp(X|p) Y (ϕ) − dfp(Y |p) X(ϕ)
= X|p Y (ϕ) ◦ f − Y |p X(ϕ) ◦ f .
'Omwc Y (ϕ)◦f = Y (ϕ◦f), afoÔ gia kˆje p ∈ Mn isqÔei: Y |f(p)(ϕ) = dfp(Y |p)(ϕ) =
Y |p(ϕ ◦ f). Telikˆ,
X|f(p)(Y ) − Y |f(p)(X) (ϕ) = X|p Y (ϕ ◦ f) − Y |p X(ϕ ◦ f)
= (XY − Y X)|p(ϕ ◦ f)
= [X, Y ]|p(ϕ ◦ f)
= dfp [X, Y ]|p (ϕ).
Qreiazìmaste thn akìloujh prìtash, h apìdeixh thc opoÐac dÐnetai sto [13].
Prìtash 1.1.1. 'Estw Mn diaforÐsimo polÔptugma kai (M
n+k
, , ) diaforÐsimo po-
lÔptugma Riemann me sunoq Levi-Civita . An f : Mn −→ M
n+k
eÐnai diaforÐsimh
apeikìnish, tìte upˆrqei monadik apeikìnish
f
: ∆(Mn
) × ∆(f) −→ ∆(f),
(X, V ) −→ f
XV,
gia thn opoÐa isqÔoun:
(i) f
X1+X2
V = f
X1
V + f
X2
V,
(ii) f
gXV = g f
XV,
(iii) f
X(V1 + V2) = f
XV1 + f
XV2,
(iv) f
X(gV ) = X(g)V + g f
XV,
(v) f
X(Y ◦ f) = df(X)Y ,
(vi) X V1, V2 = f
XV1, V2 + V1, f
XV2 ,
(vii) f
Xdf(Y ) − f
Y df(X) = df [X, Y ] ,
ìpou X, X1, X2, Y ∈ ∆(Mn), Y ∈ ∆(M
n+k
), V, V1, V2 ∈ ∆(f) kai g ∈ D(Mn).](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-10-320.jpg)
![4 Prokatarktikˆ
ìpou {e1, ..., en} eÐnai topikì orjomonadiaÐo plaÐsio tou Mn. 'Estw p shmeÐo tou Mn.
H kampulìthta Ricci sto shmeÐo p kai sth monadiaÐa dieÔjunsh x ∈ TpMn eÐnai
Ric(x) = Q(x, x).
H arijmhtik kampulìthta tou Mn eÐnai Sc = n
j=1 Ric(ej).
Gia kˆje isometrik embˆptish f : (Mn, , ) −→ (M
n+k
, , ) orÐzetai o tanust c
kampulìthtac Rf thc epagìmenhc dèsmhc f∗(TM
n+k
) wc proc th sunoq
f , wc ex c:
Rf
: ∆(Mn
) × ∆(Mn
) × ∆(f) −→ ∆(f),
(X, Y, V ) −→ Rf
(X, Y )V := f
X
f
Y V − f
Y
f
XV − f
[X,Y ]V
kai o opoÐoc eÐnai D(Mn)-grammikìc wc proc kˆje metablht .
L mma 1.1.2. An f : (Mn, , ) −→ (M
n+k
, , ) eÐnai isometrik embˆptish kai
X, Y, Z ∈ ∆(Mn) eÐnai f-susqetismèna twn X, Y , Z ∈ ∆(M
n+k
) antÐstoiqa, tìte
isqÔei R(X, Y )Z ◦ f = Rf (X, Y )df(Z).
Apìdeixh. Lìgw thc Prìtashc 1.1.1 isqÔei
f
Y df(Z) = f
Y (Z ◦ f) = df(Y )Z = Y ◦f Z = ( Y Z) ◦ f,
f
X
f
Y df(Z) = f
X ( Y Z) ◦ f = df(X)( Y Z) = ( X Y Z) ◦ f.
EpÐshc, kˆnontac qr sh tou L mmatoc 1.1.1, èqoume
f
[X,Y ]df(Z) = f
[X,Y ](Z ◦ f) =
df [X,Y ]
Z = [X,Y ]◦f Z = ( [X,Y ]Z) ◦ f.
Epomènwc isqÔei
Rf
(X, Y )df(Z) = ( X Y Z) ◦ f − ( Y XZ) ◦ f − ( [X,Y ]Z) ◦ f
= R(X, Y )Z ◦ f.
Gia kˆje isometrik embˆptish f : (Mn, , ) −→ (M
n+k
, , ) kai tuqìn shmeÐo
p ∈ Mn, èqoume thn anˆlush tou efaptìmenou q¸rou Tf(p)M
n+k
sto shmeÐo f(p) ∈
M
n+k
sto ex c orjog¸nio eujÔ ˆjroisma wc proc to eswterikì ginìmeno pou orÐzei
h metrik Riemann tou M
n+k
Tf(p)M
n+k
= dfp(TpMn
) ⊕ dfp(TpMn
)
⊥
.
O efaptìmenoc q¸roc Tpf thc f sto p eÐnai o n-diˆstatoc dianusmatikìc upìqwroc
dfp(TpMn) tou (n + k)-diˆstatou dianusmatikoÔ q¸rou Tf(p)M
n+k
. H efaptìmenh
dianusmatik dèsmh Tf thc f eÐnai h Tf := {(p, v) : p ∈ Mn, v ∈ Tpf}, èqei bajmÐda
n kai to sÔnolo twn pedÐwn thc eÐnai Γ(Tf) =: ∆f (Mn). OrÐzoume ton kˆjeto q¸ro](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-12-320.jpg)
![6 Prokatarktikˆ
JewroÔme topikì orjomonadiaÐo plaÐsio {ξ1, ..., ξk} thc Nf. To dianusmatikì pedÐo
mèshc kampulìthtac thc isometrik c embˆptishc f orÐzetai na eÐnai to
H =
1
n
k
α=1
(traceAξα )ξα.
To H eÐnai kalˆ orismèno kˆjeto diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f.
Epiplèon isqÔei
H =
1
n
n
j=1
B(ej, ej),
ìpou {e1, ..., en} topikì orjomonadiaÐo plaÐsio tou Mn.
Orismìc 1.1.5. Mia isometrik embˆptish f lègetai elaqistik an H = 0.
O tanust c kˆjethc kampulìthtac thc isometrik c embˆptishc f orÐzetai na eÐnai
h apeikìnish
R⊥
: ∆(Mn
) × ∆(Mn
) × ∆⊥
(f) −→ ∆⊥
(f),
(X, Y, ξ) −→ R⊥
(X, Y )ξ := ⊥
X
⊥
Y ξ − ⊥
Y
⊥
Xξ − ⊥
[X,Y ]ξ.
ApodeiknÔetai ìti o R⊥ eÐnai D(Mn)-grammikìc wc proc kˆje metablht .
Tèloc, gia ξ ∈ ∆⊥(f) orÐzoume to (2,1)-tanustikì pedÐo tou Mn
( XAξ)Y := X(AξY ) − Aξ( XY ),
ìpou X, Y ∈ ∆(Mn).
Gia isometrikèc embaptÐseic isqÔoun:
(i) h exÐswsh Gauss
Rf
(X, Y )df(Z), df(W) = R(X, Y )Z, W + B(X, Z), B(Y, W)
− B(Y, Z), B(X, W) ,
(ii) h exÐswsh Codazzi
Rf
(X, Y )ξ = ( Y Aξ)X − A ⊥
Y ξX − ( XAξ)Y + A ⊥
X ξY,
(iii) h exÐswsh Ricci
Rf
(X, Y )ξ
⊥
= R⊥
(X, Y )ξ − B(X, AξY ) + B(Y, AξX),
ìpou X, Y, Z, W ∈ ∆(Mn) kai ξ ∈ ∆⊥(f).
JewroÔme topikì orjomonadiaÐo plaÐsio {e1, ..., en} tou Mn. To m koc thc deÔte-
rhc jemeli¸douc morf c orÐzetai wc h sunˆrthsh
B :=
n
j,l=1
|B(ej, el)|2.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-14-320.jpg)
![Jemeli¸deic morfèc an¸terhc tˆxhc 9
OrÐzoume tic 1-morfèc ωjα := f∗(ωjα), ωαj := f∗(ωαj) kai tic morfèc kˆjethc
sunoq c ωαβ := f∗(ωαβ). ApodeiknÔetai ìti ωjα(X) = Aeα X, ej , ìpou Aeα eÐnai h
apeikìnish Weingarten sth dieÔjunsh eα kai ωαβ(X) = ⊥
Xeα, eβ gia kˆje X ∈
∆(U). EpÐshc orÐzoume ωα := f∗(ωα). EÔkola prokÔptei ìti ωα = 0.
L mma 1.2.1. (Cartan) 'Estwsan oi grammikˆ anexˆrthtec 1-morfèc ϕ1, ..., ϕr (r ≤
n) tou Mn. An θ1, θ2, ..., θr eÐnai 1-morfèc tou Mn tètoiec ¸ste na isqÔei
r
j=1 ϕj ∧
θj = 0, tìte upˆrqoun sunart seic ajl ∈ D(Mn), ìpou j, l ∈ {1, ..., r} ¸ste na isqÔei
θj = r
l=1 ajlϕl kai ajl = alj.
Epeid ωα = 0 èqoume
r
j=1 ωjα∧ωj = 0 kai sÔmfwna me to L mma 1.2.1, upˆrqoun
sunart seic hα
jl ∈ D(U), me hα
jl = hα
lj, tètoiec ¸ste ωjα = n
l=1 hα
jlωl kai hα
jl =
Aeα ej, el = B(ej, el), eα .
Parat rhsh 1.2.1. H isometrik embˆptish f : (Mn, , ) → (M
n+k
, , ) eÐnai
elaqistik an gia kˆje α isqÔei
n
j=1 hα
jj = 0.
'Otan to polÔptugma M
n+k
èqei stajer kampulìthta tom c c, tìte oi exis¸seic
Gauss, Codazzi kai Ricci gÐnontai antÐstoiqa:
Ωjl =
α
ωjα ∧ ωαl − cωj ∧ ωl,
dωjα =
l
ωjl ∧ ωlα +
β
ωjβ ∧ ωβα,
dωαβ =
j
ωαj ∧ ωjβ +
γ
ωαγ ∧ ωγβ.
An ω eÐnai 1-morf tou Mn, tìte apodeiknÔetai sto [14] ìti isqÔei
dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]),
ìpou X, Y ∈ ∆(Mn).
'Estw M2 èna didiˆstato polÔptugma Riemann. Me th bo jeia kai thc parapˆnw
sqèshc apodeiknÔetai ìti
dω12 = −Kω1 ∧ ω2, (1.4)
ìpou K eÐnai h kampulìthta Gauss tou M2.
1.3 Jemeli¸deic morfèc an¸terhc tˆxhc
'Estw f : (Mn, , ) −→ (M
n+k
, , ) isometrik embˆptish, ìpou Mn, M
n+k
eÐnai
poluptÔgmata Riemann me sunoqèc Levi-Civita , antÐstoiqa. Gia p ∈ Mn kai
jetikì akèraio r, o dianusmatikìc q¸roc
Tr
p f := span dfp(Xj1 |p), f
Xl1
df(Xl2 ) |p, ..., f
Xs1
... f
Xsr−1
df(Xsr ) |p :
Xm ∈ ∆(Mn
), m ∈ {j1, l1, l2, ..., s1, ..., sr}](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-17-320.jpg)
![10 Prokatarktikˆ
kaleÐtai eggÔtatoc q¸roc r-tˆxhc (osculating space) thc f sto p kai eÐnai dianusma-
tikìc upìqwroc tou Tf(p)M
n+k
.
Profan¸c, T1
p f = dfp(TpMn) kai T1f = p∈Mn T1
p f eÐnai h efaptìmenh dianu-
smatik dèsmh thc f. An af soume to p na metabˆletai, tìte gia stajerì r > 1 oi
q¸roi Tr
p f endeqomènwc na èqoun diaforetik diˆstash. EpÐshc, o eggÔtatoc q¸roc
r-tˆxhc thc f sto p eÐnai upìqwroc tou eggutˆtou q¸rou (r + 1)-tˆxhc thc f sto
p. Epomènwc, mporoÔme na orÐsoume to orjog¸nio sumpl rwma Nr
p f tou Tr
p f ston
Tr+1
p f wc proc to eswterikì ginìmeno tou Tf(p)M
n+k
, dhlad
Tr+1
p f = Tr
p f ⊕ Nr
p f.
O Nr
p f lègetai kˆjetoc q¸roc r-tˆxhc thc f sto p. IsqÔei
Tr+1
p f = T1
p f ⊕ N1
p f ⊕ N2
p f ⊕ ... ⊕ Nr
p f.
Profan¸c, gia s = r èqoume Nr
p f ∩ Ns
p f = {0} kai v, w = 0 gia kˆje v ∈ Nr
p f,
w ∈ Ns
p f.
Orismìc 1.3.1. 'Estw f isometrik embˆptish metaxÔ twn poluptugmˆtwn Riemann
Mn kai M
n+k
c , ìpou to M
n+k
c èqei stajer kampulìthta tom c c. KaloÔme jemeli¸dh
morf (r + 1)-tˆxhc thc f sto p ∈ Mn thn apeikìnish
Br|p : TpMn
× TpMn
× ... × TpMn
r+1
−→ Nr
p f,
(x1, x2, ..., xr+1) → Br|p(x1, x2, ..., xr+1) = f
X1
f
X2
... f
Xr
df(Xr+1)|p
Nr
p f
,
ìpou me
f
X1
f
X2
... f
Xr
df(Xr+1)|p
Nr
p f
sumbolÐzoume thn probol tou dianÔsmatoc
f
X1
f
X2
... f
Xr
df(Xr+1)|p tou Tr+1
p f ston upìqwrì tou Nr
p f kai X1, ..., Xr+1 eÐnai to-
pikˆ diaforÐsima dianusmatikˆ pedÐa tou Mn pou epekteÐnoun ta x1, ..., xr+1 antÐstoiqa,
dhlad Xi|p = xi gia i = 1, ..., r + 1.
Epeid to diˆnusma
f
X1
f
X2
... f
Xr
df(Xr+1)|p an kei ston Tr+1
p f = Tr
p f ⊕ Nr
p f
kai isqÔei Tr
p f = T1
p f ⊕ N1
p f ⊕ ... ⊕ Nr−1
p f, ènac isodÔnamoc orismìc thc Br|p eÐnai
Br|p(x1, ..., xr+1) = f
X1
... f
Xs
df(Xr+1)|p
(T1
p f⊕N1
p f⊕...⊕Nr−1
p f)⊥
,
ìpou (T1
p f⊕N1
p f⊕...⊕Nr−1
p f)⊥ eÐnai to orjosumpl rwma tou T1
p f⊕N1
p f⊕...⊕Nr−1
p f
ston Tf(p)M
n+k
.
ApodeiknÔetai sto [18] ìti h Br|p eÐnai kalˆ orismènh kai summetrik , dhlad
Br|p(x1, x2, ..., xr+1) = Br|p(xσ(1), xσ(2), ..., xσ(r+1)) gia kˆje stoiqeÐo σ thc omˆdac
metajèsewn twn arijm¸n 1, 2, ..., r + 1. EpÐshc, h Br|p eÐnai R-grammik wc proc kˆje
metablht thc.
JewroÔme thn ènwsh Nrf ìlwn twn kajètwn q¸rwn r-tˆxhc thc f, dhlad
Nr
f :=
p∈Mn
Nr
p f.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-18-320.jpg)
![12 Prokatarktikˆ
Apìdeixh. JewroÔme X1, ..., Xr−1 ∈ ∆(M2) tètoia ¸ste Xi|p = xi, gia i = 1, ..., r−1.
Apì ton orismì thc Br èqoume:
Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) =
f
X1
... f
Xr−1
f
e1
df(e1) + f
e2
df(e2) |p
Nr
p f
.
AfoÔ h f eÐnai elaqistik isqÔei B(e1, e1) + B(e2, e2) = 0. Lìgw tou tÔpou tou
Gauss èqoume
f
e1
df(e1) + f
e2
df(e2) = df( e1 e1) + df( e2 e2) + B(e1, e1) + B(e2, e2)
= df( e1 e1 + e2 e2).
Epomènwc
f
e1 df(e1) + f
e2 df(e2) ∈ ∆f (M2), opìte
f
X1
... f
Xr−1
f
e1
df(e1) + f
e2
df(e2) |p ∈ Tr
p f,
to opoÐo shmaÐnei ìti Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) = 0.
L mma 1.3.3. 'Estw f : (M2, , ) −→ (M
2+k
c , , ) isometrik elaqistik embˆpti-
sh, ìpou M2 eÐnai polÔptugma Riemann kai M
2+k
c eÐnai polÔptugma Riemann me sta-
jer kampulìthta tom c c. Tìte gia kˆje shmeÐo p tou M2 h diˆstash tou kˆjetou
q¸rou r-tˆxhc Nr
p f eÐnai to polÔ 2 gia kˆje r.
Apìdeixh. 'Estw p èna shmeÐo tou M2. JewroÔme orjomonadiaÐa bˆsh {e1, e2} tou
TpM2. Lìgw tou L mmatoc 1.3.1, gia tuqìnta dianÔsmata x1, ..., xr+1 tou TpM2
to diˆnusma Br(x1, ..., xr+1) eÐnai grammikìc sunduasmìc twn Br(ei1 , ..., eir+1 ), ìpou
i1, ..., ir+1 ∈ {1, 2}. Kˆnontac qr sh thc summetrÐac thc Br kai tou L mmatoc 1.3.2
èqoume,
Br(ei1 , ..., eir+1 ) = ±Br(e1, ..., e1) ±Br(e1, ..., e1, e2).
Apì ta parapˆnw paÐrnoume
Nr
p f = spanImBr
= span{Br(x1, x2, ..., xr+1) : x1, x2, ..., xr+1 ∈ TpMn
}
= span{Br(e1, ..., e1), Br(e1, ..., e1, e2)}.
Epomènwc se kˆje shmeÐo p h diˆstash tou Nr
p f eÐnai to polÔ 2 gia kˆje r.
Ja apodeÐxoume thn parakˆtw prìtash [3, 7].
Prìtash 1.3.1. Upojètoume ìti f : (M2, , ) −→ (M
2+k
c , , ) eÐnai isometrik
elaqistik embˆptish, ìpou M2 eÐnai polÔptugma Riemann kai M
2+k
c eÐnai polÔptugma
Riemann me stajer kampulìthta tom c c. Gia kˆje shmeÐo p tou M2 h eikìna tou
monadiaÐou kÔklou tou efaptìmenou q¸rou TpM2 me kèntro to 0 mèsw thc Br, dhlad](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-20-320.jpg)
![14 Prokatarktikˆ
Telikˆ, èqoume
Br(x, ..., x) = cos (r + 1)θ Br(e1, ..., e1) + sin r + 1)θ Br(e1, ..., e1, e2).
Katˆ sunèpeia to sÔnolo
Er(p) = {cos (r + 1)θ Br(e1, ..., e1) + sin (r + 1)θ Br(e1, ..., e1, e2) : θ ∈ R}
eÐnai èlleiyh, endeqomènwc ekfulismènh kai gÐnetai kÔkloc ìtan ta dÔo dianÔsmata
Br(e1, ..., e1), Br(e1, ..., e1, e2) èqoun Ðdio m koc kai eÐnai kˆjeta metaxÔ touc.
H èlleiyh Er(p) ja anafèretai kai wc èlleiyh r-tˆxhc thc f sto p. SumbolÐzoume
me κr, µr ta m kh twn hmiaxìnwn thc èlleiyhc r-tˆxhc ¸ste κr µr 0. OrÐzoume th
sunˆrthsh K⊥
r : M2 −→ R, p −→ K⊥
r (p) := 2κr(p)µr(p) kai thn onomˆzoume kˆjeth
kampulìthta r-tˆxhc thc f. EÐnai fanerì ìti an K⊥
r (p) = 0, tìte dimNr
p f < 2 kai h
èlleiyh Er(p) ekfulÐzetai se eujÔgrammo tm ma shmeÐo. En¸ ìtan K⊥
r (p) > 0, tìte
dimNr
p f = 2 kai h èlleiyh den eÐnai ekfulismènh.
H parakˆtw prìtash ofeÐletai ston Otsuki [17].
Prìtash 1.3.2. Upojètoume ìti f : (M2, , ) −→ (M
2+k
c , , ) eÐnai isometrik
elaqistik embˆptish, ìpou M2 eÐnai sunektikì polÔptugma Riemann kai M
2+k
c eÐnai
polÔptugma Riemann me stajer kampulìthta tom c c.
(i) An upˆrqei fusikìc r me 0 < r < [k+1
2 ], ìpou [k+1
2 ] eÐnai to akèraio mèroc tou
k+1
2 , tètoioc ¸ste gia kˆje shmeÐo p tou M2 na èqoume Nr
p f = {0} kai K⊥
s (p) > 0 gia
kˆje s ∈ {1, ..., r − 1}, tìte upˆrqei olikˆ gewdaisiakì upopolÔptugma Q2r tou M
2+k
c
¸ste f(M2) ⊂ Q2r.
(ii) An upˆrqei fusikìc r me 0 < r < [k+1
2 ] tètoioc ¸ste gia kˆje shmeÐo p tou
M2 na èqoume dimNr
p f = 1 kai K⊥
s (p) > 0 gia kˆje s ∈ {1, ..., r − 1}, tìte upˆrqei
olikˆ gewdaisiakì upopolÔptugma Q2r+1 tou M
2+k
c ¸ste f(M2) ⊂ Q2r+1.
Mia isometrik elaqistik embˆptish f : M2 −→ Sn enìc poluptÔgmatoc Riemann
M2 sthn Sn kaleÐtai koresmènh (full) an h eikìna thc f(M2) den perièqetai se kanèna
olikˆ gewdaisiakì upopolÔptugma thc Sn.
O Lawson [16] apèdeixe to ex c apotèlesma:
Prìtash 1.3.3. An f : M2 −→ Sn eÐnai isometrik elaqistik embˆptish, ìpou M2
prosanatolismèno polÔptugma Riemann, tìte h f eÐnai analutik , dhlad an (U, ϕ)
eÐnai qˆrthc tou M2, tìte h apeikìnish f ◦ϕ−1 : ϕ(U) ⊂ R2 −→ Rn+1 eÐnai analutik .
Parat rhsh 1.3.1. 'Amesh sunèpeia twn Protˆsewn 1.3.2 kai 1.3.3, eÐnai pwc
an f : M2 −→ Sn eÐnai isometrik elaqistik embˆptish enìc prosanatolismènou
poluptÔgmatoc Riemann M2 sthn Sn me Br|U = 0, ìpou U eÐnai anoiktì sÔnolo kai
r ∈ {1, ..., [n−1
2 ]−1}, tìte upˆrqei mègisth sfaÐra S2r thc Sn ¸ste f(U) ⊂ S2r. Lìgw
analutikìthtac isqÔei f(M2) ⊂ S2r. Epomènwc, an upˆrqei r kai anoiktì sÔnolo U
tou M2 ¸ste Br|U = 0, tìte h f den eÐnai koresmènh.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-22-320.jpg)
![Kefˆlaio 2
Elaqistikèc epifˆneiec sthn Sn
omoiomorfikèc me thn S2
2.1 Epifˆneiec Riemann
Sthn enìthta aut parajètoume stoiqeÐa apì th jewrÐa epifanei¸n Riemann, ta
opoÐa eÐnai aparaÐthta stic apodeÐxeic twn kurÐwn apotelesmˆtwn.
Orismìc 2.1.1. KaloÔme epifˆneia Riemann kˆje topologikì q¸ro Hausdorff M
me arijm simh bˆsh gia thn topologÐa tou, o opoÐoc eÐnai efodiasmènoc me ènan ˆtlanta
{(Uα, ϕα)}α∈I pou plhroÐ ta parakˆtw:
(i) H oikogèneia {Uα}α∈I eÐnai anoikt kˆluyh tou M kai oi apeikonÐseic ϕα : Uα ⊂
M −→ ϕα(Uα) ⊂ C eÐnai omoiomorfismoÐ. To zeÔgoc (Uα, ϕα) onomˆzetai migadikìc
qˆrthc sÔsthma suntetagmènwn tou M.
(ii) Gia kˆje α, β ∈ I, me Uα ∩ Uβ = ∅ h migadik sunˆrthsh ϕα ◦ ϕ−1
β : ϕβ(Uα ∩
Uβ) ⊂ C −→ ϕα(Uα ∩ Uβ) ⊂ C eÐnai olìmorfh.
(iii) H oikogèneia migadik¸n qart¸n {(Uα, ϕα)}α∈I eÐnai mègisth, dhlad an (U, ϕ)
migadikìc qˆrthc me ϕα ◦ϕ−1, ϕ◦ϕ−1
α , α ∈ I, olìmorfec sunart seic, tìte o migadikìc
qˆrthc (U, ϕ) an kei sthn oikogèneia {(Uα, ϕα)}α∈I.
An (U, ϕ) eÐnai migadikìc qˆrthc, tìte gia kˆje p ∈ U, o migadikìc arijmìc
z(p) := ϕ(p) = x(p) + iy(p) dÐdei tic suntetagmènec tou p wc proc ton en lìgw
qˆrth. AxÐzei na shmei¸soume ìti kˆje epifˆneia Riemann eÐnai prosanatolismèno
didiˆstato diaforÐsimo polÔptugma.
Orismìc 2.1.2. 'Estw M epifˆneia Riemann. Mia suneq c migadik sunˆrthsh
f : M −→ C lègetai olìmorfh an gia kˆje migadikì qˆrth (U, ϕ) tou M h migadik
sunˆrthsh f ◦ ϕ−1 : ϕ(U) −→ C eÐnai olìmorfh.
'Estw (M2, , ) polÔptugma Riemann kai (U, ϕ) qˆrthc gÔrw apì to p ∈ M2 me
suntetagmènec (x, y). An isqÔei , = Edx2 + Edy2, E ∈ D(U) kai E > 0, tìte to
sÔsthma suntetagmènwn (x, y) kaleÐtai isìjermo. O Chern [9] apèdeixe to parakˆtw
apotèlesma: 'Estw (M2, , ) diaforÐsimo polÔptugma Riemann. Tìte gÔrw apì kˆje
shmeÐo p ∈ M2 upˆrqei isìjermo sÔsthma suntetagmènwn.
15](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-23-320.jpg)
![SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn 19
ìpou
W(r,0)
dzr
:= W(
∂
∂z
, ...,
∂
∂z
r
) dz ⊗ ... ⊗ dz
r
,
W(r−1,1)
dzr−1
dz := W(
∂
∂z
, ...,
∂
∂z
r−1
,
∂
∂z
) dz ⊗ ... ⊗ dz
r−1
⊗dz
+ W(
∂
∂z
, ...,
∂
∂z
r−2
,
∂
∂z
∂
∂z
) dz ⊗ ... ⊗ dz
r−2
⊗dz ⊗ dz
+ ... + W(
∂
∂z
∂
∂z
, ...,
∂
∂z
r−1
)dz ⊗ dz ⊗ ... ⊗ dz
r−1
,
.
.
.
W(0,r)
dzr
:= W(
∂
∂z
, ...,
∂
∂z
r
) dz ⊗ ... ⊗ dz
r
.
Kˆje migadikì (r, 0)-tanustikì pedÐo T thc morf c
T = f(z) dz ⊗ ... ⊗ dz
r
= f(z)dzr
,
ìpou (U, ϕ) eÐnai migadikìc qˆrthc me migadik suntetagmènh z kai f ∈ C∞(U, C) lège-
tai r-diaforikì. EÐnai fanerì ìti to r-diaforikì T eÐnai migadikì (r, 0)-tanustikì pedÐo
tètoio ¸ste an kˆpoio apì ta v1, ..., vr an kei sth dèsmh T M2, tìte T(v1, ..., vr) = 0.
To r-diaforikì T = f(z)dzr kaleÐtai olìmorfo an h f eÐnai olìmorfh sunˆrthsh.
Gia ta r-diaforikˆ isqÔei to akìloujo shmantikì apotèlesma, gnwstì kai wc Je-
¸rhma Riemann-Roch [12].
Je¸rhma 2.1.1. 'Estw M epifˆneia Riemann omoiomorfik me thn S2. An Φ eÐnai
olìmorfo r-diaforikì orismèno sthn epifˆneia M, tìte Φ = 0.
2.2 SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn
To kleidÐ gia thn apìdeixh twn jewrhmˆtwn, pou anafèrontai sthn eisagwg , eÐnai
h diapÐstwsh ìti gia sumpageÐc elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra, oi el-
leÐyeic kˆje tˆxhc eÐnai kÔkloi sqedìn pantoÔ. Stìqoc thc paroÔshc paragrˆfou eÐnai
h apìdeixh aut c thc diapÐstwshc. Gia to skopì autì, orÐzoume katˆllhla r-diaforikˆ
ta opoÐa apodeiknÔoume ìti eÐnai olìmorfa kai lìgw tou Jewr matoc Riemann-Roch,
eÐnai ek tautìthtoc mhdèn.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-27-320.jpg)
![SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn 27
Stìqoc mac eÐnai na deÐxoume ìti to Φ2 eÐnai tautotikˆ mhdèn. Gia autì ja deÐxoume
ìti to Mc
1 eÐnai peperasmèno sÔnolo kai ìti upˆrqei upodèsmh thc kˆjethc dèsmhc me
bajmÐda 2 orismènh sto M2, h opoÐa sto M1 tautÐzetai me th dèsmh N1f|M1 . Gia to
skopì autì qreiazìmaste to epìmeno l mma to opoÐo ofeÐletai ston Chern [8] (blèpe
epÐshc [5] [11]).
L mma 2.2.3. An f1, ..., fm : U ⊂ C −→ C eÐnai migadikèc sunart seic oi opoÐec se
mia perioq U tou mhdenìc ikanopoioÔn to diaforikì sÔsthma
∂fi
∂z = j aijfj, ìpou
aij : U −→ C eÐnai diaforÐsimec sunart seic kai i, j = 1, ..., m, tìte eÐte f1 = ... =
fm = 0, eÐte oi koinèc rÐzec twn f1, ..., fm eÐnai memonwmènec kai mˆlista upˆrqei
jetikìc akèraioc l kai diaforÐsimec sunart seic f∗
1 , ..., f∗
m : U ⊂ C −→ C ¸ste na
isqÔei fi(z) = zlf∗
i (z) me f∗
1 (0), ..., f∗
m(0) = (0, ..., 0).
L mma 2.2.4. An f : M2 −→ Sn, n ≥ 5, sumpag c, koresmènh, elaqistik epi-
fˆneia gènouc mhdèn, tìte to sÔnolo Mc
1 eÐnai peperasmèno. Epiplèon, upˆrqei dia-
nusmatik upodèsmh N∗1f thc kˆjethc dèsmhc Nf me bajmÐda 2 ¸ste N∗1f|M1 =
N1f|M1 .
Apìdeixh. 'Estw p ∈ Mc
1. JewroÔme gÔrw apì to p migadikì qˆrth (U, z) me z(p) = 0,
z = x + iy, , = E|dz|2 kai orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno
mèroc {e1, e2}, ìpou e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y , kai kˆjeto mèroc {eα}. Sto U èqoume
to migadikì (1,0)-tanustikì pedÐo φ = ω1 + iω2 kai tic migadikèc sunart seic Hα
1 =
hα
11 + ihα
12 gia α = 3, ..., n. Me th bo jeia twn sqèsewn
dH
α
1 =
∂H
α
1
∂z
dz +
∂H
α
1
∂z
dz,
ωαβ = ωαβ(
∂
∂z
)dz + ωαβ(
∂
∂z
)dz
kai twn sqèsewn (2.4), (2.5), apì to L mma 2.2.1(i) lambˆnoume
∂H
α
1
∂z
=
β
gαβH
β
1 ,
ìpou gαβ := −ωβα( ∂
∂z ) − δαβd log E( ∂
∂z ) eÐnai diaforÐsimec sunart seic kai δαβ eÐnai
to dèlta tou Kronecker.
SÔmfwna me to L mma 2.2.3 eÐte H
α
1 = 0 gia kˆje α = 3, ..., n, eÐte oi koinèc rÐzec
twn H
α
1 eÐnai memonwmènec kai upˆrqei jetikìc akèraioc l1 kai diaforÐsimec sunart seic
H
∗α
1 : U −→ C ¸ste na isqÔei
H
α
1 = zl1
H
∗α
1 (2.6)
gia kˆje α kai H
∗3
1 (0), ..., H
∗n
1 (0) = (0, ..., 0).
An tan H
α
1 = 0 gia kˆje α = 3, ..., n, tìte B|U = 0. 'Atopo, afoÔ sÔmfwna me
thn Parat rhsh 1.3.1 h f den ja tan koresmènh. 'Ara oi koinèc rÐzec twn H
α
1 eÐnai
memonwmènec. Oi koinèc rÐzec twn H
α
1 , α = 3, ..., n, eÐnai akrib¸c ta shmeÐa ìpou o
pr¸toc kˆjetoc q¸roc gÐnetai mhdenikìc. Autì shmaÐnei ìti to sÔnolo Mc
1 apoteleÐtai
apì memonwmèna shmeÐa kai afoÔ to M2 eÐnai sumpagèc, eÐnai peperasmèno sÔnolo.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-35-320.jpg)
![SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn 31
To Φ∗
2 eÐnai kalˆ orismèno se ìlo to M2 kai lìgw tou L mmatoc 2.2.5(iii), isqÔei
Φ∗
2|p =
Φ2|p, p ∈ M1,
0, p ∈ Mc
1.
JewroÔme èna shmeÐo p ∈ Mc
1 kai èstw (U, z) migadikìc qˆrthc gÔrw apì to p
me z(p) = 0 kai U ∩ Mc
1 = {p}. Apì to L mma 2.2.2(iv) to Φ∗
2 eÐnai olìmorfo sto
U −{p}. Epeid eÐnai kai suneqèc sto shmeÐo p, sumperaÐnoume ìti to Φ∗
2 eÐnai olìmorfo
sto U [1]. Epanalambˆnontac thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou
Mc
1, èqoume ìti to Φ∗
2 eÐnai olìmorfo 6-diaforikì se ìlo to M2. Apì to Je¸rhma
Riemann-Roch èqoume Φ∗
2 = 0. Sunep¸c deÐxame ìti Φ2 = 0. Katˆ sunèpeia gia kˆje
p ∈ M1 isqÔei Φ2|p = 0 isodÔnama
B2(e1, e1, e1)
2
(p) − B2(e1, e1, e2)
2
(p) − 2i B2(e1, e1, e1), B2(e1, e1, e2) (p) = 0.
Apì thn teleutaÐa sqèsh, lìgw thc Prìtashc 1.3.1 prokÔptei ìti se kˆje shmeÐo
p ∈ M1 h èlleiyh E2(p) eÐnai kÔkloc me aktÐna
κ2(p) = B2(e1, e1, e1) (p) = B2(e1, e1, e2) (p).
Epomènwc, h diˆstash tou kˆjetou q¸rou deÔterhc tˆxhc sto tuqìn p ∈ M1 eÐnai
0 2. An dimN2
p f = 0 gia kˆje p ∈ M1, tìte sÔmfwna me thn Parat rhsh 1.3.1,
f(M2) ⊂ S4, ìpou S4 eÐnai mia mègisth 4-sfaÐra thc Sn. 'Atopo, afoÔ èqoume dh
anafèrei pwc exetˆzoume koresmènh elaqistik epifˆneia f : M2 −→ Sn, n ≥ 5.
Sunep¸c, maxp∈M2 dimN2
p f = 2 to opoÐo shmaÐnei ìti n ≥ 6. An n = 6, tìte h
diadikasÐa stamatˆ ed¸ kai an n ≥ 7, tìte suneqÐzei.
EÐnai t¸ra eÔlogo h diadikasÐa aut na genikeÔetai epagwgikˆ. Gia lìgouc plh-
rìthtac thc apìdeixhc twn apotelesmˆtwn, ja perigrˆyoume leptomer¸c to epagwgikì
b ma.
'Estw f : M2 −→ Sn, n ≥ 7, sumpag c, koresmènh, elaqistik epifˆneia gènouc
mhdèn kai jetikìc akèraioc r me 2 ≤ r ≤ [n−1
2 ] − 1. Upojètoume ìti gia kˆje s ∈
{2, ..., r} isqÔoun ta akìlouja:
(I) Upˆrqoun dianusmatikèc upodèsmec N∗1f, ..., N∗r−1f thc kˆjethc dèsmhc Nf
me bajmÐda 2 uperˆnw tou M2 ètsi ¸ste N∗s−1f|Ms−1 = Ns−1f|Ms−1 , ìpou
Ms−1 := {p ∈ M2
: B∗
s−1 (p) > 0}
kai
B∗
s :=
2
j1,...,js+1=1
|B∗
s (ej1 , ..., ejs+1 )|2
eÐnai to m koc tou summetrikoÔ (s + 1, 1)-tanustikoÔ pedÐou
B∗
s : ∆(M2
) × ... × ∆(M2
)
s+1
−→ Γ (T1
f ⊕ N∗1
f ⊕ ... ⊕ N∗s−1
f)⊥
,
(X1, ..., Xs+1) −→ B∗
s (X1, ..., Xs+1) = f
X1
... f
Xs
df(Xs+1)
(T1f⊕N∗1f⊕...⊕N∗s−1f)⊥
.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-39-320.jpg)
![46 Elaqistikèc epifˆneiec sthn Sn omoiomorfikèc me thn S2
JewroÔme èna shmeÐo p ∈ Mc
r kai èstw (U, z) migadikìc qˆrthc gÔrw apì to p
me z(p) = 0 kai U ∩ Mc
r = {p}. To Φ∗
r+1 eÐnai olìmorfo sto U − {p} kai suneqèc
sto shmeÐo p, ˆra Φ∗
r+1 eÐnai olìmorfo sto U [1]. Epanalambˆnontac thn parapˆnw
diadikasÐa gÔrw apì ìla ta peperasmèna shmeÐa tou Mc
r , èqoume to Φ∗
r+1 olìmorfo
(2r + 4)-diaforikì se ìlo to M2 kai apì to Je¸rhma Riemann-Roch Φ∗
r+1 = 0.
2.3 Bohjhtikˆ apotelèsmata
Ja sunoyÐsoume ìsa mèqri stigm c èqoume apodeÐxei kai ja qreiastoÔme sth su-
nèqeia gia tic sumpageÐc, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra.
Gia to skopì autì qreiazìmaste ton orismì kai to l mma pou akoloujoÔn.
Orismìc 2.3.1. Se prosanatolismèno polÔptugma Riemann M2 jewroÔme topikì
orjomonadiaÐo plaÐsio {e1, e2} tou prosanatolismoÔ, to sumplaÐsiì tou {ω1, ω2} kai to
sÔnolo twn 1-morf¸n Λ1(M2). O telest c tou Hodge orÐzetai na eÐnai h apeikìnish
∗ : Λ1
(M2
) −→ Λ1
(M2
),
ω = ω(e1)ω1 + ω(e2)ω2 −→ ∗ω := −ω(e2)ω1 + ω(e1)ω2.
ApodeiknÔetai ìti o anwtèrw orismìc eÐnai kalìc, dhlad anexˆrthtoc tou plaisÐou
{e1, e2}. Profan¸c isqÔei ∗ω1 = ω2 kai ∗ω2 = −ω1.
JumÐzoume ìti gia mia sunˆrthsh g ∈ D(M2) h klÐsh thc gradg kai h Laplasian
thc ∆g, ìpou ∆ eÐnai o telest c Laplace tou M2, dÐnontai wc gradg = e1(g)e1+e2(g)e2
kai ∆g = e1 e1(g) + e2 e2(g) − e1 e1 (g) − e2 e2 (g).
L mma 2.3.1. 'Estw {e1, e2} topikì orjomonadiaÐo plaÐsio tou M2 kai {ω1, ω2} to
sumplaÐsiì tou.
(i) Gia tuqoÔsa sunˆrthsh g ∈ D(M2) isqÔei h sqèsh
d(∗dg) = ∆gω1 ∧ ω2.
(ii) An η, ω ∈ Λ1(M2), tìte
η = ∗ω ⇔ η ∧ φ = iω ∧ φ,
ìpou φ = ω1 + iω2.
Apìdeixh. (i) ArkeÐ na deÐxoume ìti d(∗dg)(e1, e2) = ∆g. Prˆgmati, epeid
dg = dg(e1)ω1 + dg(e2)ω2 = e1(g)ω1 + e2(g)ω2,
èqoume
∗dg = −e2(g)ω1 + e1(g)ω2.
UpologÐzoume
d(∗dg)(e1, e2) =
= −d e2(g) ∧ ω1(e1, e2) + d e1(g) ∧ ω2(e1, e2)](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-54-320.jpg)
![Bohjhtikˆ apotelèsmata 47
− e2(g)ω12 ∧ ω2(e1, e2) + e1(g)ω21 ∧ ω1(e1, e2)
= d e2(g) (e2) + d e1(g) (e1) − e2(g)ω12(e1) − e1(g)ω21(e2)
= e2 e2(g) + e1 e1(g) − e2(g) e1 e1, e2 − e1(g) e2 e2, e1
= ∆g.
(ii) H apìdeixh eÐnai aplìc upologismìc.
Prìtash 2.3.1. 'Estw f : M2 −→ Sn, n ≥ 3, sumpag c, koresmènh, elaqistik
epifˆneia gènouc mhdèn. Upˆrqoun anoiktˆ uposÔnola M1, ..., Mm−1 tou M2 me M1 ⊃
M2 ⊃ ... ⊃ Mm−1, ìpou m := 1 + [n−1
2 ], tètoia ¸ste ta sumplhr¸matˆ touc Mc
s =
M2 − Ms gia kˆje s ∈ {1, ..., m − 1} na eÐnai peperasmèna sÔnola kai isqÔoun:
(i) H èlleiyh Es s-tˆxhc eÐnai kÔkloc aktÐnac κs > 0 se kˆje shmeÐo tou Mm−1
gia kˆje s ∈ {1, ..., m − 1}. Epiplèon, o arijmìc n eÐnai ˆrtioc kai n = 2m.
(ii) Upˆrqoun diaforÐsimec sunart seic gs : M2 −→ [0, +∞) me gs|Mm−1 =
κ2
s|Mm−1 , gs|Mc
m−1
= 0 gia kˆje s ∈ {1, ..., m − 1}. Epiplèon, gia kˆje p ∈ Mc
m−1,
upˆrqei migadikìc qˆrthc (U, z) me z(p) = 0 kai gs = |z|2ls us sto U, ìpou ls jetikìc
akèraioc kai us ∈ D(U) jetik sunˆrthsh.
(iii) Gia kˆje migadikì qˆrth (U, z) me z = x + iy, , = E|dz|2 upˆrqei topikì
orjomonadiaÐo plaÐsio {eA} katˆ m koc thc f|U∩Mm−1 me efaptìmeno mèroc {e1, e2},
ìpou e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y kai kˆjeto mèroc {eα} tètoio ¸ste
e2s+1 =
1
κs
Bs(e1, ..., e1), e2s+2 =
1
κs
Bs(e1, ..., e1, e2), s ∈ {1, ..., m − 1},
gia to opoÐo isqÔoun ta akìlouja
ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs,
ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) =
κs
κs−1
,
ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1,
ω2s,2s+2(e1) = ω2s−1,2s+2(e2) =
κs
κs−1
,
ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2,
ωrα = 0, 1 ≤ r ≤ 2s, α ≥ 2s + 3.
Apìdeixh. (i) 'Eqei apodeiqjeÐ sto epagwgikì b ma ìti se kˆje shmeÐo p ∈ Ms h
èlleiyh mèqri kai s-tˆxhc sto p eÐnai kÔkloc jetik c aktÐnac, gia kˆje s ∈ {1, ..., m −
1}. Epeid M1 ⊃ M2 ⊃ ... ⊃ Mm−1, se kˆje shmeÐo tou Mm−1 h èlleiyh s-tˆxhc Es
eÐnai kÔkloc aktÐnac κs > 0 gia ìla ta s ∈ {1, ..., m − 1}.
Epeid h f eÐnai koresmènh, gia tuqìn shmeÐo p tou Mm−1 èqoume
Tf(p)Sn
= T1
p f ⊕ N1
p f ⊕ ... ⊕ Nm−1
p f.
'Olec oi elleÐyeic kˆje tˆxhc eÐnai kÔkloi me jetik aktÐna, ˆra ìloi oi kˆjetoi q¸roi
thc f sto tuqìn p ∈ Mm−1 eÐnai didiˆstatoi. Epomènwc, h parapˆnw anˆlush mac dÐnei
n = 2m, dhlad o arijmìc n eÐnai ˆrtioc.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-55-320.jpg)
![48 Elaqistikèc epifˆneiec sthn Sn omoiomorfikèc me thn S2
(ii) Gia kˆje s jètoume
gs =
1
2s+1
B∗
s
2
.
Epeid M1 ⊃ M2 ⊃ ... ⊃ Mm−1, isqÔei B∗
s |Mm−1 = Bs|Mm−1 kai B∗
s |Mc
m−1
= 0.
Epomènwc èqoume
gs(p) =
κ2
s(p), p ∈ Mm−1,
0, p ∈ Mc
m−1.
Sto epagwgikì b ma apodeÐxame ìti Φ∗
1 = ... = Φ∗
m−1 = 0 apì ìpou sumperaÐnoume
ìti |B∗
s (e1, ..., e1)| = |B∗
s (e1, ..., e1, e2)|. Lambˆnontac upìyh tic sqèseic (2.6) kai (2.8)
upologÐzoume
gs =
1
2
|B∗
s (e1, ..., e1)|2
+
1
2
|B∗
s (e1, ..., e1, e2)|2
=
1
2
n
α=2s+1
(h∗α
(s),1)2
+ (h∗α
(s),2)2
=
1
2
n
α=2s+1
|H
∗α
s |2
= |z|2ls
us,
ìpou us := 1
2
n
α=2s+1 |G
∗α
s |2.
(iii) Efìson Φ1|Mm−1 = ... = Φm−1|Mm−1 = 0, ta Bs(e1, ..., e1), Bs(e1, ..., e1, e2)
eÐnai isom kh kai kˆjeta metaxÔ touc gia kˆje s ∈ {1, ..., m−1}. Epomènwc, h epilog
autoÔ tou plaisÐou eÐnai efikt .
GnwrÐzoume ìti sto Mm−1 gia kˆje s ∈ {1, ..., m − 1} isqÔei h sqèsh
ω2s+1,2s+2 − (s + 1)ω12 ∧ ϕ = id log κs ∧ ϕ,
epomènwc apì to L mma 2.3.1(ii) èqoume
ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs.
Oi upìloipec sqèseic èqoun apodeiqjeÐ sto epagwgikì b ma (2.11), (2.12), (2.13),
(2.14), (2.15) .
Sto anoiktì kai puknì uposÔnolo Mm−1 tou M2 èqoume tic dianusmatikèc u-
podèsmec N1f|Mm−1 , ..., Nm−1f|Mm−1 thc kˆjethc dèsmhc me bajmÐda 2. Gia kˆje
s ∈ {1, ..., m − 1}, h dèsmh Nsf|Mm−1 èqei metrik , s th metrik thc Sn periori-
smènh se kˆje èna apì autˆ kai sunoq
s thn kˆjeth sunoq thc embˆptishc f,
epÐshc periorismènh se kˆje èna apì autˆ. Epiplèon, h dèsmh Nsf|Mm−1 èqei tanust
kampulìthtac
Rs
: ∆(Mm−1) × ∆(Mm−1) × Γ(Ns
f|Mm−1 ) −→ Γ(Ns
f|Mm−1 ),
(X, Y, V ) −→ Rs
(X, Y )V = s
X
s
Y V − s
Y
s
XV − s
[X,Y ]V.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-56-320.jpg)
![Bohjhtikˆ apotelèsmata 49
An {e1, e2} eÐnai topikì orjomonadiaÐo plaÐsio tou prosanatolismoÔ tou M2 kai
{vs
1, vs
2} orjomonadiaÐo plaÐsio tou prosanatolismoÔ thc Nsf|Mm−1 pou epˆgei h Bs,
dhlad o prosanatolismìc pou orÐzei sthn Nsf|Mm−1 to plaÐsiì thc
{Bs(e1, ..., e1), Bs(e1, ..., e1, e2)},
tìte h kampulìthta Ks thc Nsf|Mm−1 orÐzetai wc
Ks = − Rs
(e1, e2)vs
1, vs
2 s.
'Eqoume epilèxei topikì orjomonadiaÐo plaÐsio {e2s+1, e2s+2} thc Nsf|Mm−1 ètsi ¸ste
Bs(e1, ..., e1) = κse2s+1, Bs(e1, ..., e1, e2) = κse2s+2
gia kˆje s ∈ {1, ..., m − 1}.
H morf sunoq c ω2s+1,2s+2 thc dèsmhc Nsf|Mm−1 eÐnai
ω2s+1,2s+2(X) = s
Xe2s+1, e2s+2 s
gia X ∈ ∆(M2) kai isqÔei
dω2s+1,2s+2 = −Ksω1 ∧ ω2. (2.18)
Prìtash 2.3.2. 'Estw f : M2 −→ Sn sumpag c, koresmènh, elaqistik epifˆneia
gènouc mhdèn. Tìte sto Mm−1, ìpou m = 1 + [n−1
2 ], isqÔoun:
(i) K = 1 − 2κ2
1.
(ii) Gia s ∈ {1, ..., m − 1} h kampulìthta Ks thc dèsmhc Nsf|Mm−1 eÐnai
Ks =
2κ2
1 − 2
κ2
2
κ2
1
, s = 1,
2
κ2
s
κ2
s−1
− 2
κ2
s+1
κ2
s
, 1 < s < m − 1,
κ2
m−1
κ2
m−2
, s = m − 1.
(iii) ∆ log κs = (s+1)K−Ks, 1 ≤ s ≤ m−1, kai ∆ log(κ1...κm−1) = m(m+1)
2 K−1.
Apìdeixh. SÔmfwna me thn Prìtash 2.3.1(i) èqoume n = 2m. Epilègoume orjomona-
diaÐo plaÐsio ìpwc sthn Prìtash 2.3.1(iii).
(i) Se kˆje shmeÐo tou M2 èqoume |B(e1, e1)| = |B(e1, e2)| = κ1, epomènwc apì
th sqèsh (1.3) èqoume K = 1 − 2κ2
1 sto M2, ˆra kai sto Mm−1.
(ii) H kampulìthta thc N1f|Mm−1 dÐnetai apì th sqèsh dω34 = −K1ω1 ∧ ω2. Apì
thn exÐswsh Ricci èqoume dω34 = ω31 ∧ ω14 + ω32 ∧ ω24 + α≥3 ω3α ∧ ωα4. 'Omwc
ω31 ∧ ω14(e1, e2) = −κ2
1 = ω32 ∧ ω24(e1, e2) kai apì tic sqèseic
ω33 = ω44 = 0,
ω35(e1) = ω54(e2) =
κ2
κ1
,](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-57-320.jpg)
![Kefˆlaio 3
KÔria apotelèsmata
3.1 ApodeÐxeic twn kurÐwn apotelesmˆtwn
EÐmaste plèon ètoimoi na d¸soume tic apodeÐxeic twn jewrhmˆtwn, pou èqoun a-
naferjeÐ sthn eisagwg . Stic apodeÐxeic ja gÐnei qr sh, ektìc apì ta apotelèsmata
tou KefalaÐou 2, apotelesmˆtwn thc Olik c Diaforik c GewmetrÐac, ìpwc tou Jew-
r matoc Gauss-Bonnet, tou Jewr matoc Gauss-Green, thc Arq c MegÐstou kai tou
Jewr matoc Monadikìthtac twn isometrik¸n embaptÐsewn.
To apotèlesma pou akoloujeÐ ofeÐletai ston Calabi [6].
Je¸rhma 3.1.1. 'Estw f : (M, , ) −→ Sn, n ≥ 3, sumpag c, prosanatolismènh,
koresmènh, elaqistik epifˆneia gènouc mhdèn. Tìte:
(i) O arijmìc n eÐnai ˆrtioc (n = 2m).
(ii) To embadì A(M) thc epifˆneiac eÐnai akèraio pollaplˆsio tou 2π kai isqÔei
A(M) ≥ 2πm(m + 1).
Apìdeixh. (i) 'Eqei apodeiqjeÐ sthn Prìtash 2.3.1(i).
(ii) Lambˆnontac upìyh thn Prìtash 2.3.1(ii) kai thn Prìtash 2.3.2(iii) èqoume
∆ log u = m(m + 1)K − 2 (3.1)
sto Mm−1, ìpou u := g2
1...g2
m−1.
Apì thn Prìtash 2.3.1, to Mc
m−1 apoteleÐtai apì peperasmèno to pl joc shmeÐwn.
JewroÔme tuqìn shmeÐo tou p. GÔrw apì to p jewroÔme migadikì qˆrth (U, z) tou
M ¸ste z(p) = 0, z = x + iy, , = E|dz|2 kai U ∩ Mc
m−1 = {p}. Epilègoume
orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno mèroc {e1, e2}, ìpou e1 =
1√
E
∂
∂x , e2 = 1√
E
∂
∂y kai kˆjeto mèroc {eα} ¸ste sto U − {p} na isqÔei e2s+1 =
1
κs
Bs(e1, ..., e1), e2s+2 = 1
κs
Bs(e1, ..., e1, e2) gia kˆje s ∈ {1, ..., m − 1}.
Lìgw thc Prìtashc 2.3.1(ii) isqÔei u = |z|2lu0 sto U me l := l1 + ... + lm−1,
ìpou u0 eÐnai diaforÐsimh jetik sunˆrthsh orismènh kontˆ sto p ∈ Mc
m−1. 'Estw
Mc
m−1 = {p1, ..., pt}. Gia kˆje s ∈ {1, ..., t} jewroÔme gewdaisiak mpˆla Bε(ps) me
kèntro to ps kai aktÐna ε > 0 ètsi ¸ste Bε(ps) ∩ Mc
m−1 = {ps} kai jètoume
Mε := M −
t
s=1
Bε(ps).
53](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-61-320.jpg)
![ApodeÐxeic twn kurÐwn apotelesmˆtwn 55
epeid h posìthta
∂
∂ρ , grad log u0 eÐnai fragmènh. Oloklhr¸noume th sqèsh (3.1)
sto Mε, paÐrnoume to ìrio kai lambˆnontac upìyh thn (3.2) èqoume diadoqikˆ
lim
ε→0 Mε
∆ log udM = m(m + 1) lim
ε→0 Mε
KdM − 2 lim
ε→0 Mε
dM,
−4πlt = m(m + 1)
M
KdM − 2
M
dM,
A(M) = 2πlt +
m(m + 1)
2 M
KdM.
Epeid to M eÐnai sumpagèc kai gènouc mhdèn, èqoume
M KdM = 4π, lìgw tou
Jewr matoc Gauss-Bonnet. Epomènwc isqÔei
A(M) = 2π m(m + 1) + lt ,
dhlad to embadì tou M eÐnai akèraio pollaplˆsio tou 2π kai epiplèon A(M) ≥
2πm(m + 1).
Parat rhsh 3.1.1. AxÐzei na shmeiwjeÐ ìti gia to embadì tou M isqÔei A(M) =
2πm(m + 1) ìtan Mm−1 = M.
'Amesh apìrroia tou Jewr matoc 3.1.1 eÐnai to akìloujo [2]
Pìrisma 3.1.1. 'Estw f : (M, , ) −→ S3 sumpag c, prosanatolismènh, elaqisti-
k epifˆneia gènouc mhdèn. Tìte h f eÐnai olikˆ gewdaisiak .
Sth sunèqeia ja apodeÐxoume to shmantikì apotèlesma akamyÐac pou ofeÐletai
ston Barbosa [4], gia thn apìdeixh tou opoÐou qreiazìmaste to Je¸rhma Monadikìth-
tac twn isometrik¸n embaptÐsewn sth sfaÐra [10].
Je¸rhma 3.1.2. 'Estwsan f, f : Mn −→ Sn+k isometrikèc embaptÐseic tou sune-
ktikoÔ poluptÔgmatoc Riemann Mn sthn Sn+k. SumbolÐzoume me Nf, B, ⊥ kai
Nf, B, ⊥ thn kˆjeth dèsmh, th deÔterh jemeli¸dh morf kai thn kˆjeth su-
noq twn f kai f antÐstoiqa. Upojètoume ìti upˆrqei isomorfismìc T metaxÔ twn
dianusmatik¸n desm¸n Nf, Nf, ξp = (p, ξ) −→ Tξp = (p, ξ) tètoioc ¸ste:
(i) Tξp, Tηp = ξp, ηp gia kˆje ξp, ηp ∈ Nf.
(ii) Oi deÔterec jemeli¸deic morfèc sundèontai mèsw tou isomorfismoÔ me th sqèsh
T ◦ B = B.
(iii) Gia X ∈ ∆(Mn) kai ξ ∈ ∆⊥(f) isqÔei T( ⊥
Xξ) = ⊥
X(Tξ).
Tìte upˆrqei isometrÐa τ thc Sn+k ètsi ¸ste f = τ ◦ f.
Je¸rhma 3.1.3. 'Estwsan f : (M, , ) −→ S2m, f : (M, , ) −→ S2m sumpageÐc,
prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn. Tìte upˆrqei
isometrÐa τ thc S2m tètoia ¸ste f = τ ◦ f.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-63-320.jpg)
![ApodeÐxeic twn kurÐwn apotelesmˆtwn 57
kai
∆ log κs = (s + 1)K − Ks,
ìpou Ks eÐnai h kampulìthta thc dianusmatik c dèsmhc Nsf|U kai κs eÐnai to m koc
tou Bs(e1, ..., e1) gia kˆje s ∈ {1, ..., m − 1}.
Sth sunèqeia ergazìmaste sto U. Epeid κ1 = κ1, apì tic proanaferjèntec
sqèseic eÔkola paÐrnoume ìti κs = κs kai Ks = Ks gia 1 ≤ s ≤ m − 1. Apì thn
Prìtash 2.3.1(iii) gia thn f gnwrÐzoume ìti ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs kai
omoÐwc gia thn f gnwrÐzoume ìti ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs. Apì autèc su-
nˆgetai ìti
ω2s+1,2s+2 = ω2s+1,2s+2.
Epiplèon apì thn Prìtash 2.3.1(iii) gia thn f|U èqoume
ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) =
κs
κs−1
,
ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1,
ω2s,2s+2(e1) = ω2s−1,2s+2(e2) =
κs
κs−1
,
ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2
kai gia thn f|U èqoume
ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) =
κs
κs−1
,
ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1,
ω2s,2s+2(e1) = ω2s−1,2s+2(e2) =
κs
κs−1
,
ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2,
ìpou 1 ≤ s ≤ m − 1. Apì autèc tic sqèseic prokÔptei ìti ωαβ = ωαβ, afoÔ κs = κs.
'Estw X ∈ ∆(U) kai ξ ∈ Γ(Nf|U ). To ξ èqei thn anˆlush ξ = 2m
α=3 ξ, eα eα.
Epeid gia kˆje p ∈ U isqÔei
T( ⊥
Xeα|p) = T
2m
β=3
ωαβ(X)eβ |p =
2m
β=3
ωαβ(X)eβ|p = ⊥
Xeα|p,
paÐrnoume T( ⊥
Xξ) = ⊥
X(Tξ).
SÔmfwna me to Je¸rhma 3.1.2 upˆrqei isometrÐa τ : S2m −→ S2m tètoia ¸ste
f|U = τ ◦ f|U . Jètoume F := f − τ ◦ f : M −→ R2m+1. GnwrÐzoume ìti oi f, f
eÐnai analutikèc (Prìtash 1.3.3). Sunep¸c h F eÐnai analutik sto M, wc diaforˆ
analutik¸n kai afoÔ F|U = 0 sumperaÐnoume ìti F = 0 sto M, dhlad f = τ ◦ f.
To teleutaÐo apotèlesma eÐnai h taxinìmhsh twn sumpag¸n, prosanatolismènwn,
koresmènwn, elaqistik¸n epifanei¸n gènouc mhdèn sth sfaÐra me stajer kampulìthta
Gauss, to opoÐo ofeÐletai ston Calabi [6]. Gia to skopì autì ja anafèroume tic
epifˆneiec Veronese, oi opoÐec eÐnai sumpageÐc kai èqoun stajer kampulìthta Gauss.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-65-320.jpg)
![58 KÔria apotelèsmata
SÔmfwna me gnwstì Je¸rhma tou Takahashi [19], mia isometrik embˆptish g =
(g1, ..., gn+k+1) apì èna n-diˆstato polÔptugma Riemann Mn sthn Sn+k eÐnai elaqi-
stik an kai mìno an oi sunart seic suntetagmènwn thc g eÐnai idiosunart seic tou
telest Laplace ∆ tou Mn me antÐstoiqh idiotim n, dhlad an ∆gi = −ngi, i ∈
{1, ..., n + k + 1}.
JewroÔme th didiˆstath sfaÐra S2(R) aktÐnac R > 0, efodiasmènh me th sun jh
metrik Riemann. EÐnai gnwstì ìti oi idiotimèc tou telest Laplace thc S2(R) eÐnai
λκ = κ(κ+1)
R2 , ìpou κ eÐnai mh arnhtikìc akèraioc. Epiprìsjeta, o idioq¸roc Vλκ pou
antistoiqeÐ sthn idiotim λκ, parˆgetai apì ta armonikˆ omogen polu¸numa bajmoÔ κ
tou R3 periorismèna sthn S2(R) kai èqei diˆstash 2κ + 1.
H epifˆneia Veronese sthn S2κ, ìpou κ eÐnai jetikìc akèraioc, eÐnai h isometrik
elaqistik embˆptish
fκ : S2
(R) −→ S2κ
, R =
κ(κ + 1)
2
,
me fκ = (g1, ..., g2κ+1) kai g2
1 + ... + g2
2κ+1 = 1, ìpou {g1, ..., g2κ+1} eÐnai bˆsh tou
idioq¸rou Vλκ apoteloÔmenh apì armonikˆ omogen polu¸numa bajmoÔ κ tou R3 pe-
riorismèna sthn S2(R) kai h opoÐa eÐnai orjomonadiaÐa wc proc to eswterikì ginìmeno
tou dianusmatikoÔ q¸rou D S2(R)
, : D S2
(R) × D S2
(R) −→ R, (g, h) −→ g, h :=
S2(R)
ghdS2
(R),
ìpou dS2(R) eÐnai to stoiqeÐo embadoÔ thc S2(R).
EÐnai t¸ra fanerì ìti oi sunart seic suntetagmènwn thc fκ eÐnai idiosunart seic
tou telest Laplace thc S2(R). Sunep¸c, apì to Je¸rhma Takahashi gia kˆje jetikì
akèraio κ h fκ eÐnai elaqistik epifˆneia. Profan¸c oi epifˆneiec Veronese eÐnai
gènouc mhdèn kai èqoun stajer kampulìthta Gauss K = 2
κ(κ+1) .
Gia parˆdeigma, h epifˆneia Veronese sthn S4 eÐnai h
f2 : S2
(
√
3) −→ S4
, f2(x, y, z) =
xy
√
3
,
xz
√
3
,
yz
√
3
,
x2 − y2
2
√
3
,
x2 + y2 − 2z2
6
.
To akìloujo je¸rhma dhl¸nei pwc ousiastikˆ oi epifˆneiec Veronese eÐnai oi mìnec
elaqistikèc epifˆneiec me autèc tic idiìthtec.
Je¸rhma 3.1.4. 'Estw f : (M, , ) −→ S2m sumpag c, prosanatolismènh, kore-
smènh, elaqistik epifˆneia gènouc mhdèn. An to M èqei stajer kampulìthta Gauss
K, tìte K = 2
m(m+1) kai upˆrqoun isometrÐec F : (M, , ) −→ S2 m(m+1)
2 kai
τ : S2m −→ S2m ¸ste f ◦ F−1 = τ ◦ fm, ìpou fm : S2 m(m+1)
2 −→ S2m eÐnai h
epifˆneia Veronese sthn S2m.
Apìdeixh. Apì thn Prìtash 2.3.2 èqoume tic sqèseic
K = 1 − 2κ2
1,
∆ log(κ1...κm−1)2
= m(m + 1)K − 2](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-66-320.jpg)
![EikasÐa tou U. Simon 59
kai
∆ log κs = (s + 1)K − Ks
sto Mm−1 gia kˆje s ∈ {1, ..., m − 1}. AfoÔ h kampulìthta Gauss K eÐnai stajer ,
to κ1 eÐnai stajerì kai sunep¸c ∆ log κ1 = 0. Lìgw thc ∆ log κ1 = 2K − K1 h
teleutaÐa sqèsh èqei wc sunèpeia to K1 stajerì kai epeid sthn Prìtash 2.3.2 eÐdame
ìti K1 = 2κ2
1 − 2
κ2
2
κ2
1
, èqoume κ2 stajerì kai ∆ log κ2 = 0, to opoÐo me th seirˆ tou
mac dÐnei to K2 stajerì. Epagwgikˆ ftˆnoume sto sumpèrasma ìti κ1, ..., κm−1 eÐnai
stajerèc kai epomènwc èqoume ∆ log(κ1...κm−1)2 = 0, dhlad K = 2
m(m+1).
To M eÐnai pl rec kai aplˆ sunektikì. Apì to Je¸rhma Taxinìmhshc Aplˆ Su-
nektik¸n Q¸rwn Morf c [14] upˆrqei isometrÐa F : M −→ S2 m(m+1)
2 . EÐnai
fanerì ìti h f ◦ F−1 : S2 m(m+1)
2 −→ S2m eÐnai sumpag c koresmènh elaqistik
epifˆneia gènouc mhdèn. 'Omwc kai h epifˆneia Veronese fm : S2 m(m+1)
2 −→ S2m
eÐnai sumpag c koresmènh elaqistik epifˆneia gènouc mhdèn. Apì to Je¸rhma 3.1.3
upˆrqei isometrÐa τ thc S2m ¸ste τ ◦ fm = f ◦ F−1.
3.2 EikasÐa tou U. Simon
H eikasÐa pou akoloujeÐ diatup¸jhke sto [15] kai èqei epalhjeujeÐ mìno se eidikèc
peript¸seic.
EikasÐa 3.2.1. 'Estw f : (M, , ) −→ Sn sumpag c, prosanatolismènh, elaqistik
epifˆneia me kampulìthta Gauss K. An h K plhroÐ thn anisìthta K(s + 1) ≤ K ≤
K(s), ìpou K(s) := 2
s(s+1) , s jetikìc akèraioc, tìte eÐte K = K(s) kai upˆrqoun
isometrÐec F : (M, , ) −→ S2 s(s+1)
2 kai τ : S2s −→ S2s ¸ste f ◦ F−1 = τ ◦ fs,
ìpou fs : S2 s(s+1)
2 −→ S2s eÐnai h epifˆneia Veronese sthn S2s, K = K(s + 1)
kai upˆrqoun isometrÐec F : (M, , ) −→ S2 (s+1)(s+2)
2 kai τ : S2(s+1) −→ S2(s+1)
¸ste f ◦ F−1 = τ ◦ fs+1, ìpou fs+1 : S2 (s+1)(s+2)
2 −→ S2(s+1) eÐnai h epifˆneia
Veronese sthn S2(s+1).
H eikasÐa èqei apodeiqjeÐ sto [15] stic peript¸seic ìpou s = 1 kai s = 2. Ja
d¸soume mia apìdeixh gia autèc tic peript¸seic me th mejodologÐa pou anaptÔqjhke
sthn paroÔsa ergasÐa, dhlad ja apodeÐxoume to akìloujo
Je¸rhma 3.2.1. H eikasÐa tou Simon eÐnai alhj c gia s = 1 kai s = 2.
Apìdeixh. Lìgw thc upìjeshc K(s+1) ≤ K ≤ K(s), s = 1, 2, èqoume
M KdM > 0,
ìpou dM eÐnai to stoiqeÐo embadoÔ tou M. Apì to Je¸rhma twn Gauss-Bonnet pro-
kÔptei ìti to M eÐnai omoiomorfikì me thn S2 kai epomènwc
M KdM = 4π. SÔmfwna
me to Je¸rhma 3.1.1 h f eÐnai koresmènh se sfaÐra S2m ˆrtiac diˆstashc. DiakrÐnoume
dÔo peript¸seic:
(i) PerÐptwsh s = 1.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-67-320.jpg)
![BibliografÐa
[1] L.V. Ahlfors, Complex Analysis, McGraw-Hill, 1979.
[2] F.J.Jr. Almgren, Some interior regularity theorems for minimal surfaces and
an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277-292.
[3] A.C. Asperti, Generic minimal surfaces, Math. Z. 200 (1989), 181-186.
[4] J.L.M. Barbosa, On minimal immersions of S2 into S2m, Trans. Amer. Math.
Soc. 210 (1975), 75-106.
[5] J.L.M. Barbosa, An extrinsic rigidity theorem for minimal immersions from S2
into Sn, J. Differential Geom. 14 (1979), 355-368.
[6] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential
Geom. 1 (1967), 111-125.
[7] C.C. Chen, The generalized curvature ellipses and minimal surfaces, Bull. Inst.
Math. Acad. Sin. 11 (1983), 329-336.
[8] S.S. Chern, On the minimal immersions of the two-sphere in a space of constant
curvature, Problems in Analysis, University Press, Princeton, N. J. (1970), 27-
40.
[9] S.S. Chern, W.H. Chen and K.S. Lam, Lectures on Differential Geometry, Wo-
rld Scientific, 2000.
[10] M. Dajczer, M. Antonucci, G. Oliveira, P. Lima-Filho and R. Tojeiro, Subma-
nifolds and isometric immersions, Publish or Perish, 1990.
[11] J. Eschenburg and R. Tribuzy, Branch points of conformal mappings of surfaces,
Math. Ann. 279 (1988), 621-633.
[12] H. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, 1992.
[13] D. Ferus, Differentialgeometrie, I-II, Technische Universitaet Berlin, 2000.
[14] D. Koutroufi¸thc, Diaforik GewmetrÐa, Panepist mio IwannÐnwn, 1994.
[15] M. Kozlowski and U. Simon, Minimal immersions of 2-manifolds into spheres,
Math. Z. 186 (1984), 377-382.
65](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-73-320.jpg)
![66
[16] H.B. Lawson, The global behavior of minimal surfaces in Sn, Ann. of Math. (2)
92 (1970), 224-237.
[17] T. Otsuki, Minimal submanifolds with m-index 2 and generalized Veronese sur-
faces, J. Math. Soc. Japan 24 (1972), 89-122.
[18] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. IV.
Berkeley: Publish or Perish, 1979.
[19] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc.
Japan 18 (1966), 380-385.](https://image.slidesharecdn.com/n-150114045005-conversion-gate01/85/n-74-320.jpg)
Uploaded byLamprini Zourka
Μεταπτυχιακή διατριβή στη Διαφορική Γεωμετρία: Συμπαγεις Ελαχιστικες Επιφανειες Γενους μηδεν Στην n-Διαστατη Σφαιρα
Μεταπτυχιακή διατριβή στη Διαφορική Γεωμετρία: Συμπαγεις Ελαχιστικες Επιφανειες Γενους μηδεν Στην n-Διαστατη Σφαιρα
- 1. PANEPISTHMIO IWANNINWN TMHMA MAJHMATIKWN TOMEASALGEBRAS KAI GEWMETRIAS METAPTUQIAKH DIATRIBH LAMPRINH ZOURKA SUMPAGEIS ELAQISTIKES EPIFANEIES GENOUS MHDEN STHN n-DIASTATH SFAIRA IWANNINA 2008
- 2. ii H paroÔsa Metaptuqiak Diatrib ekpon jhke sto plaÐsio twn spoud¸n gia thn apìkthsh tou MetaptuqiakoÔ Dipl¸matoc EidÐkeushc sta MAJHMATIKA pou aponèmei to Tm ma Majhmatik¸n tou PanepisthmÐou IwannÐnwn. EgkrÐjhke th Deutèra, 30-06-2008, apì thn exetastik epitrop : ONOMATEPWNUMO BAJMIDA UPOGRAFH Jeìdwroc Blˆqoc Anaplhrwt c Kajhght c (Epiblèpwn Kajhght c) tou Tm matoc Majhmatik¸n tou PanepisthmÐou IwannÐnwn Qr stoc MpaðkoÔshc Kajhght c tou Tm matoc Majhmatik¸n tou PanepisthmÐou IwannÐnwn Jwmˆc Qasˆnhc Kajhght c tou Tm matoc Majhmatik¸n tou PanepisthmÐou IwannÐnwn
- 3. Eisagwg Mia isometrik embˆptishenìc poluptÔgmatoc Riemann se èna polÔptugma Rie- mann lègetai elaqistik an to dianusmatikì pedÐo mèshc kampulìthtac eÐnai to mh- denikì isodÔnama an to Ðqnoc thc deÔterhc jemeli¸douc morf c thc isometrik c embˆptishc eÐnai tautotikˆ mhdèn. Apì ton tÔpo thc pr¸thc metabol c tou embadoÔ gnwrÐzoume pwc oi isometrikèc elaqistikèc embaptÐseic eÐnai akrib¸c ta krÐsima shmeÐa thc sunˆrthshc tou embadoÔ. Epomènwc, fusikì epakìloujo eÐnai pwc oi elaqistikèc isometrikèc embaptÐseic apoteloÔn an¸terhc diˆstashc genÐkeush twn gewdaisiak¸n kampul¸n kai apartÐzoun mia shmantik oikogèneia isometrik¸n embaptÐsewn. Sthn paroÔsa ergasÐa asqoloÔmaste me elaqistikèc epifˆneiec sthn Sn, dhlad isometrikèc elaqistikèc embaptÐseic apì èna prosanatolismèno, sunektikì, didiˆstato polÔptugma Riemann sthn n-diˆstath monadiaÐa sfaÐra Sn = {(x1, x2, ..., xn+1) ∈ Rn+1 : x2 1 + x2 2 + ... + x2 n+1 = 1}, h opoÐa eÐnai efodiasmènh me th sun jh metrik Riemann , kai wc gnwstìn èqei stajer kampulìthta tom c 1. Parìlh thn omorfiˆ pou èqei apì mình thc h melèth elaqistik¸n epifanei¸n sth sfaÐra, èqei apodeiqjeÐ ìti sqetÐzetai me th melèth me- monwmènwn idiazìntwn shmeÐwn elaqistik¸n embaptÐsewn ston EukleÐdeio q¸ro. O E. Calabi sto ˆrjro [6] parat rhse ìti an èqoume èna tridiˆstato elaqistikì upo- polÔptugma M3 ston EukleÐdeio q¸ro En+3, tìte to polÔptugma M3 ∩ Sn+2, gia sfaÐra Sn+2 katˆllhlou kèntrou, eÐnai didiˆstato elaqistikì upopolÔptugma sthn Sn+2. AntÐstrofa, an M2 eÐnai èna didiˆstato elaqistikì upopolÔptugma thc Sn+2, tìte o k¸noc pou dhmiourgeÐtai apì tic hmieujeÐec me arq to kèntro thc sfaÐrac Sn+2 kai dièrqontai apì ta shmeÐa tou M2, eÐnai tridiˆstato elaqistikì upopolÔptug- ma ston EukleÐdeio q¸ro En+3 me memonwmèno idiˆzon shmeÐo to kèntro thc sfaÐrac an to M2 den eÐnai olikˆ gewdaisiakì sthn Sn+2. Epomènwc, h melèth isometrik¸n ela- qistik¸n embaptÐsewn tridiˆstatwn poluptugmˆtwn Riemann me memonwmèno idiˆzon shmeÐo ston EukleÐdeio q¸ro, anˆgetai sth melèth isometrik¸n elaqistik¸n embaptÐ- sewn didiˆstatwn poluptugmˆtwn Riemann sth sfaÐra. Thn idèa aut ulopoÐhse o E. Calabi sto prwtoporiakì ˆrjro [6] melet¸ntac elaqistikèc epifˆneiec sth sfaÐra me thn aploÔsterh dunat topologÐa, dhlad sumpageÐc elaqistikèc epifˆneiec gènouc mhdèn isodÔnama omoiomorfikèc me thn S2. Autì to ˆrjro èdwse to ènausma gia th melèth elaqistik¸n epifanei¸n sth sfaÐra. LÐgo metˆ thn emfˆnish tou ˆrjrou tou E. Calabi, o S.S. Chern parousÐase mia pio gewmetrik prosèggish twn elaqistik¸n epifanei¸n sth sfaÐra kˆnontac qr sh twn jemeliwd¸n morf¸n anwtèrac tˆxewc. H melèth aut suneqÐsthke apì ton J.L.M. Barbosa. iii
- 4. iv O stìqoc thcergasÐac eÐnai na apodeÐxoume ta apotelèsmata tou E. Calabi sto [6], me ton trìpo pou ta anadiatÔpwse o S.S. Chern sto [8], kaj¸c kai tou J.L.M. Barbosa sto [4]. GnwrÐzoume ìti ta olikˆ gewdaisiakˆ m-diˆstata upopoluptÔgmata thc Sn, ìpou 2 ≤ m ≤ n − 1, eÐnai oi mègistec m-sfaÐrec thc Sn, dhlad tomèc thc Sn me (m + 1)- diˆstatouc upoq¸rouc tou Rn+1. Mia isometrik embˆptish f : (M, , ) −→ Sn, ìpou (M, , ) eÐnai polÔptugma Riemann, kaleÐtai koresmènh (full) an h eikìna thc den perièqetai se kanèna olikˆ gewdaisiakì upopolÔptugma thc Sn. Sthn ergasÐa aut to pr¸to apotèlesma pou ja apodeÐxoume eÐnai ìti oi sumpageÐc, prosanatolismènec elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra eÐnai koresmènec mìno se ˆrtiac diˆstashc sfaÐra. Epiplèon ja d¸soume mia ektÐmhsh tou embadoÔ twn. To sumpèrasma autì to apèdeixe o E. Calabi sto ˆrjro [6]. Argìtera, o S.S. Chern sto [8] èdwse mia diaforetik prosèggish tou apotelèsmatoc to opoÐo eÐnai to ex c: Je¸rhma I. 'Estw f : (M, , ) −→ Sn, n ≥ 3, sumpag c, prosanatolismènh, kore- smènh, elaqistik epifˆneia gènouc mhdèn. Tìte: (i) O arijmìc n eÐnai ˆrtioc (n = 2m). (ii) To embadì A(M) thc epifˆneiac eÐnai akèraio pollaplˆsio tou 2π kai isqÔei A(M) ≥ 2πm(m + 1). Sth sunèqeia ja apodeÐxoume ìti oi sumpageÐc, prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra eÐnai ˆkamptec (rigid), èna apotèlesma pou ofeÐletai ston J.L.M. Barbosa [4]: Je¸rhma II. 'Estwsan f : (M, , ) −→ S2m, f : (M, , ) −→ S2m sumpageÐc, prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn. Tìte upˆrqei isometrÐa τ : S2m −→ S2m ¸ste f = τ ◦ f. Tèloc, ja deÐxoume pwc oi sumpageÐc, prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra me stajer kampulìthta Gauss taxinomoÔntai pl rwc. Gia thn akrÐbeia apodeiknÔoume to akìloujo apotèlesma tou E. Calabi [6]: Je¸rhma III. 'Estw f : (M, , ) −→ S2m sumpag c, prosanatolismènh, kore- smènh, elaqistik epifˆneia gènouc mhdèn. An to polÔptugma Riemann (M, , ) èqei stajer kampulìthta Gauss K, tìte K = 2 m(m+1) kai upˆrqoun isometrÐec F : (M, , ) −→ S2 m(m+1) 2 kai τ : S2m −→ S2m ¸ste f ◦ F−1 = τ ◦ fm, ìpou fm : S2 m(m+1) 2 −→ S2m eÐnai h epifˆneia Veronese sthn S2m. Gia tic apodeÐxeic aut¸n twn apotelesmˆtwn qrhsimopoioÔme ergaleÐa kai mejìdouc apì th jewrÐa kinoumènou plaisÐou, twn jemeliwd¸n morf¸n an¸terhc tˆxhc kai th jewrÐa epifanei¸n Riemann (Je¸rhma Riemann-Roch). Epiplèon, h prosèggish pou akoloujoÔme mac epitrèpei na d¸soume mia apìdeixh thc eikasÐac tou U. Simon [15] gia tic peript¸seic pou eÐnai dh gnwstì ìti isqÔei. Parajètoume thn apìdeixh aut¸n twn peript¸sewn sto tèloc thc diatrib c.
- 5. v EuqaristÐec Ekfrˆzw tic jermècmou euqaristÐec ston epiblèponta kajhght mou, k. Jeìdwro Blˆqo gia th suneq epÐbleyh kai kajod ghs tou. EuqaristÐec ofeÐlw kai sta ˆlla dÔo mèlh thc exetastik c epitrop c, ton k. Jwmˆ Qasˆnh kai ton k. Qr sto MpaðkoÔsh gia tic qr simec parathr seic touc sthn ergasÐa mou. EpÐshc, euqarist¸ touc upoy fiouc didˆktorec Qr sto Tatˆkh kai Qrusìstomo Yaroudˆkh gia th bo jeiˆ touc. Tèloc, idiaÐterec euqaristÐec ofeÐlw sth metaptuqiak foit tria AshmÐna MpoÔs- mpoura gia thn polÔtimh bo jeia, upost rixh kai filÐa thc katˆ th diˆrkeia twn meta- ptuqiak¸n spoud¸n mou.
- 6.
- 7. Perieqìmena 1 Prokatarktikˆ 1 1.1Isometrikèc embaptÐseic . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Exis¸seic dom c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Jemeli¸deic morfèc an¸terhc tˆxhc . . . . . . . . . . . . . . . . . . . 9 2 Elaqistikèc epifˆneiec sthn Sn omoiomorfikèc me thn S2 15 2.1 Epifˆneiec Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn . . . . . . . . . . . . . 19 2.3 Bohjhtikˆ apotelèsmata . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 KÔria apotelèsmata 53 3.1 ApodeÐxeic twn kurÐwn apotelesmˆtwn . . . . . . . . . . . . . . . . . . 53 3.2 EikasÐa tou U. Simon . . . . . . . . . . . . . . . . . . . . . . . . . . 59 vii
- 8.
- 9. Kefˆlaio 1 Prokatarktikˆ 1.1 IsometrikècembaptÐseic Sto parìn kefˆlaio ja anafèroume aparaÐthta stoiqeÐa apì th jewrÐa twn iso- metrik¸n embaptÐsewn, tou kinoumènou plaisÐou kai twn jemeliwd¸n morf¸n an¸te- rhc tˆxhc. Gia tic basikèc ènnoiec thc Diaforik c GewmetrÐac parapèmpoume sta [10, 14, 13]. 'Estw Mn diaforÐsimo polÔptugma diˆstashc n. Gia tuqìn p ∈ Mn sumbolÐzoume me TpMn ton efaptìmeno q¸ro tou Mn sto p kai me TMn = {(p, v) : p ∈ Mn, v ∈ TpMn} thn efaptìmenh dèsmh tou Mn. To sÔnolo twn diaforÐsimwn dianusmatik¸n pedÐwn tou Mn to sumbolÐzoume me ∆(Mn) Γ(TMn) kai to sÔnolo twn diaforÐsimwn sunart sewn g : Mn −→ R to sumbolÐzoume me D(Mn) C∞(Mn, R). Orismìc 1.1.1. 'Estw f : Mn −→ M n+k diaforÐsimh apeikìnish metaxÔ twn diaforÐsimwn poluptugmˆtwn Mn kai M n+k . DiaforÐsimo dianusmatikì pedÐo katˆ m koc thc f kaloÔme kˆje apeikìnish V h opoÐa se kˆje p ∈ Mn antistoiqeÐ èna diˆnusma V |p ∈ Tf(p)M n+k kai eÐnai diaforÐsimh me thn ex c ènnoia: An (U, y) eÐnai qˆrthc tou M n+k gÔrw apì to f(p) me suntetagmènec (y1, y2, ..., yn+k) kai V |q = n+k i=1 gi(q) ∂ ∂yi |f(q) gia kˆje q ∈ f−1(U), oi sunart seic gi, i = 1, 2, ..., n + k, eÐnai diaforÐsimec. SumbolÐzoume me ∆(f) to sÔnolo twn diaforÐsimwn dianusmatik¸n pedÐwn katˆ m koc thc f. Profan¸c to ∆(f) eÐnai to sÔnolo twn pedÐwn (sections) thc epagìmenhc dianusmatik c dèsmhc f∗(TM n+k ) := {(p, v) : p ∈ Mn, v ∈ Tf(p)M n+k } bajmÐdac (rank) n + k, dhlad ∆(f) = Γ f∗(TM n+k ) . An X ∈ ∆(Mn), tìte h apeikìnish h opoÐa se kˆje p ∈ Mn antistoiqeÐ to diˆnusma dfp(X|p), eÐnai diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f kai sumbolÐzetai me df(X). Epiplèon, an Y ∈ ∆(M n+k ), tìte to Y ◦ f eÐnai epÐshc diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f. Orismìc 1.1.2. 'Estw f : Mn −→ M n+k diaforÐsimh apeikìnish kai X ∈ ∆(Mn), X ∈ ∆(M n+k ). Ta X, X lègontai f-susqetismèna (f-related) an X ◦ f = df(X). 1
- 10. 2 Prokatarktikˆ L mma 1.1.1.An f : Mn −→ M n+k eÐnai diaforÐsimh apeikìnish kai X, Y ∈ ∆(Mn) eÐnai f-susqetismèna twn X, Y ∈ ∆(M n+k ) antÐstoiqa, tìte ta ginìmena Lie [X, Y ], [X, Y ] eÐnai f-susqetismèna. Apìdeixh. 'Eqoume X ◦ f = df(X), ˆra gia kˆje p ∈ Mn, X|f(p) = dfp(X|p). Ja deÐxoume ìti [X, Y ] ◦ f = df [X, Y ] , isodÔnama X|f(p)(Y ) − Y |f(p)(X) = dfp [X, Y ]|p . Gia kˆje ϕ ∈ D(M n+k ) èqoume: X|f(p)(Y ) − Y |f(p)(X) (ϕ) = X|f(p) Y (ϕ) − Y |f(p) X(ϕ) = dfp(X|p) Y (ϕ) − dfp(Y |p) X(ϕ) = X|p Y (ϕ) ◦ f − Y |p X(ϕ) ◦ f . 'Omwc Y (ϕ)◦f = Y (ϕ◦f), afoÔ gia kˆje p ∈ Mn isqÔei: Y |f(p)(ϕ) = dfp(Y |p)(ϕ) = Y |p(ϕ ◦ f). Telikˆ, X|f(p)(Y ) − Y |f(p)(X) (ϕ) = X|p Y (ϕ ◦ f) − Y |p X(ϕ ◦ f) = (XY − Y X)|p(ϕ ◦ f) = [X, Y ]|p(ϕ ◦ f) = dfp [X, Y ]|p (ϕ). Qreiazìmaste thn akìloujh prìtash, h apìdeixh thc opoÐac dÐnetai sto [13]. Prìtash 1.1.1. 'Estw Mn diaforÐsimo polÔptugma kai (M n+k , , ) diaforÐsimo po- lÔptugma Riemann me sunoq Levi-Civita . An f : Mn −→ M n+k eÐnai diaforÐsimh apeikìnish, tìte upˆrqei monadik apeikìnish f : ∆(Mn ) × ∆(f) −→ ∆(f), (X, V ) −→ f XV, gia thn opoÐa isqÔoun: (i) f X1+X2 V = f X1 V + f X2 V, (ii) f gXV = g f XV, (iii) f X(V1 + V2) = f XV1 + f XV2, (iv) f X(gV ) = X(g)V + g f XV, (v) f X(Y ◦ f) = df(X)Y , (vi) X V1, V2 = f XV1, V2 + V1, f XV2 , (vii) f Xdf(Y ) − f Y df(X) = df [X, Y ] , ìpou X, X1, X2, Y ∈ ∆(Mn), Y ∈ ∆(M n+k ), V, V1, V2 ∈ ∆(f) kai g ∈ D(Mn).
- 11. Isometrikèc embaptÐseic 3 Hapeikìnish f eÐnai h sunoq pou epˆgei h sunoq Levi-Civita tou M n+k sthn epagìmenh dèsmh f∗(TM n+k ). Orismìc 1.1.3. 'Estwsan (Mn, , ) kai (M n+k , , ) diaforÐsima poluptÔgmata Rie- mann. Mia diaforÐsimh apeikìnish f : (Mn, , ) −→ (M n+k , , ) kaleÐtai isometrik embˆptish an gia kˆje p ∈ Mn isqÔoun ta akìlouja: (i) to diaforikì dfp : TpMn −→ Tf(p)M n+k eÐnai ènesh kai (ii) dfp(v), dfp(w) f(p) = v, w p, gia kˆje v, w ∈ TpMn. SumbolÐzoume th sunoq Levi-Civita tou Mn me kai ton tanust kampulìthtˆc tou me R. Me sumbolÐzoume th sunoq Levi-Civita tou M n+k kai me R ton tanust kampulìthtˆc tou. An {e1, ..., en} eÐnai topikì orjomonadiaÐo plaÐsio tou Mn, tìte R(ei, ej)ek = n l=1 Rijklel, ìpou Rijkl := R(ei, ej)ek, el . Orismìc 1.1.4. 'Estw Mn diaforÐsimo polÔptugma. Mia apeikìnish T : ∆(Mn ) × ... × ∆(Mn ) r −→ D(Mn ) kaleÐtai (r, 0)-tanustikì pedÐo an eÐnai D(Mn)-grammik wc proc kˆje metablht thc. Epiplèon, an E eÐnai dianusmatik dèsmh (vector bundle) uperˆnw tou Mn, tìte mia apeikìnish T : ∆(Mn ) × ... × ∆(Mn ) r −→ Γ(E), ìpou Γ(E) eÐnai to sÔnolo twn pedÐwn (sections) thc dianusmatik c dèsmhc, kaleÐtai (r, 1)-tanustikì pedÐo an eÐnai D(Mn)-grammik wc proc kˆje metablht thc. EÐnai gnwstì ìti an T eÐnai (r, 0)-tanustikì pedÐo (r, 1)-tanustikì pedÐo kai X1, ..., Xr, Y1, ..., Yr ∈ ∆(Mn) me Xi|p = Yi|p gia kˆje i ∈ {1, ..., r} se kˆpoio shmeÐo p tou Mn, tìte T(X1, ..., Xr)|p = T(Y1, ..., Yr)|p. Autì mac epitrèpei na blèpoume to tanustikì pedÐo T se kˆje shmeÐo p ∈ Mn wc pleiogrammik apeikìnish T|p : TpMn × ... × TpMn −→ R T|p : TpMn × ... × TpMn −→ Ep, ìpou Ep eÐnai to n ma (fiber) thc dèsmhc E uperˆnw tou p. 'Estw σ ènac didiˆstatoc upìqwroc tou TpMn. H kampulìthta tom c tou Mn sto shmeÐo p gia to epÐpedo σ eÐnai o arijmìc K(p, σ) = R(e1, e2)e2, e1 , ìpou {e1, e2} eÐnai tuqaÐa orjomonadiaÐa bˆsh tou σ. O tanust c Ricci tou Mn eÐnai to summetrikì (2,0)-tanustikì pedÐo Q : ∆(Mn ) × ∆(Mn ) −→ D(Mn ), (X, Y ) −→ Q(X, Y ) := n j=1 R(ej, X)Y, ej ,
- 12. 4 Prokatarktikˆ ìpou {e1,..., en} eÐnai topikì orjomonadiaÐo plaÐsio tou Mn. 'Estw p shmeÐo tou Mn. H kampulìthta Ricci sto shmeÐo p kai sth monadiaÐa dieÔjunsh x ∈ TpMn eÐnai Ric(x) = Q(x, x). H arijmhtik kampulìthta tou Mn eÐnai Sc = n j=1 Ric(ej). Gia kˆje isometrik embˆptish f : (Mn, , ) −→ (M n+k , , ) orÐzetai o tanust c kampulìthtac Rf thc epagìmenhc dèsmhc f∗(TM n+k ) wc proc th sunoq f , wc ex c: Rf : ∆(Mn ) × ∆(Mn ) × ∆(f) −→ ∆(f), (X, Y, V ) −→ Rf (X, Y )V := f X f Y V − f Y f XV − f [X,Y ]V kai o opoÐoc eÐnai D(Mn)-grammikìc wc proc kˆje metablht . L mma 1.1.2. An f : (Mn, , ) −→ (M n+k , , ) eÐnai isometrik embˆptish kai X, Y, Z ∈ ∆(Mn) eÐnai f-susqetismèna twn X, Y , Z ∈ ∆(M n+k ) antÐstoiqa, tìte isqÔei R(X, Y )Z ◦ f = Rf (X, Y )df(Z). Apìdeixh. Lìgw thc Prìtashc 1.1.1 isqÔei f Y df(Z) = f Y (Z ◦ f) = df(Y )Z = Y ◦f Z = ( Y Z) ◦ f, f X f Y df(Z) = f X ( Y Z) ◦ f = df(X)( Y Z) = ( X Y Z) ◦ f. EpÐshc, kˆnontac qr sh tou L mmatoc 1.1.1, èqoume f [X,Y ]df(Z) = f [X,Y ](Z ◦ f) = df [X,Y ] Z = [X,Y ]◦f Z = ( [X,Y ]Z) ◦ f. Epomènwc isqÔei Rf (X, Y )df(Z) = ( X Y Z) ◦ f − ( Y XZ) ◦ f − ( [X,Y ]Z) ◦ f = R(X, Y )Z ◦ f. Gia kˆje isometrik embˆptish f : (Mn, , ) −→ (M n+k , , ) kai tuqìn shmeÐo p ∈ Mn, èqoume thn anˆlush tou efaptìmenou q¸rou Tf(p)M n+k sto shmeÐo f(p) ∈ M n+k sto ex c orjog¸nio eujÔ ˆjroisma wc proc to eswterikì ginìmeno pou orÐzei h metrik Riemann tou M n+k Tf(p)M n+k = dfp(TpMn ) ⊕ dfp(TpMn ) ⊥ . O efaptìmenoc q¸roc Tpf thc f sto p eÐnai o n-diˆstatoc dianusmatikìc upìqwroc dfp(TpMn) tou (n + k)-diˆstatou dianusmatikoÔ q¸rou Tf(p)M n+k . H efaptìmenh dianusmatik dèsmh Tf thc f eÐnai h Tf := {(p, v) : p ∈ Mn, v ∈ Tpf}, èqei bajmÐda n kai to sÔnolo twn pedÐwn thc eÐnai Γ(Tf) =: ∆f (Mn). OrÐzoume ton kˆjeto q¸ro
- 13. Isometrikèc embaptÐseic 5 thcf sto p na eÐnai o k-diˆstatoc dianusmatikìc q¸roc Npf := {ξ : ξ ∈ (Tpf)⊥}. H kˆjeth dianusmatik dèsmh Nf thc f eÐnai h ènwsh ìlwn twn kajètwn q¸rwn, dhlad Nf := {(p, ξ) : p ∈ Mn, ξ ∈ Npf}, èqei bajmÐda k kai to sÔnolo twn pedÐwn thc eÐnai Γ(Nf) =: ∆⊥(f). Ta pedÐa thc kˆjethc dianusmatik c dèsmhc Nf ta kaloÔme kˆjeta dianusmatikˆ pedÐa katˆ m koc thc f. Lìgw thc parapˆnw anˆlushc, gia tuqìn v ∈ Tf(p)M n+k upˆrqoun monadikˆ dia- nÔsmata v ∈ TpMn kai v⊥ ∈ Npf ètsi ¸ste v = dfp(v ) + v⊥. Katˆ sunèpeia, gia kˆje V ∈ ∆(f) upˆrqoun monadikˆ dianusmatikˆ pedÐa V ∈ ∆(Mn) kai V ⊥ ∈ ∆⊥(f) ¸ste V = df(V ) + V ⊥ . Gia X, Y ∈ ∆(Mn) èqoume epomènwc thn anˆlush: f Xdf(Y ) = df f Xdf(Y ) + f Xdf(Y ) ⊥ , ìpou f Xdf(Y ) ∈ ∆(Mn) kai f Xdf(Y ) ⊥ ∈ ∆⊥(f). ApodeiknÔetai ìti f Xdf(Y ) = XY. OrÐzoume thn apeikìnish B : ∆(Mn ) × ∆(Mn ) −→ ∆⊥ (f), (X, Y ) −→ B(X, Y ) := f Xdf(Y ) ⊥ . ApodeiknÔetai ìti h B eÐnai summetrikì (2,1)-tanustikì pedÐo kai kaleÐtai deÔterh jemeli¸dhc morf thc f. SÔmfwna me ta parapˆnw, h teleutaÐa sqèsh gÐnetai f Xdf(Y ) = df( XY ) + B(X, Y ), o legìmenoc tÔpoc tou Gauss. Gia X ∈ ∆(Mn), ξ ∈ ∆⊥(f) èqoume thn anˆlush: f Xξ = df ( f Xξ) + ( f Xξ)⊥ , ìpou ( f Xξ) ∈ ∆(Mn) kai ( f Xξ)⊥ ∈ ∆⊥(f). H apeikìnish Weingarten sth dieÔjunsh ξ ∈ ∆⊥(f) eÐnai Aξ : ∆(Mn ) −→ ∆(Mn ), X −→ AξX := −( f Xξ) . H apeikìnish Weingarten eÐnai D(Mn)-grammik wc proc ξ kai autoproshrthmèno (1,1)-tanustikì pedÐo. ApodeiknÔetai ìti AξX, Y = B(X, Y ), ξ . EpÐshc, orÐzoume thn apeikìnish ⊥ : ∆(Mn ) × ∆⊥ (f) −→ ∆⊥ (f), (X, ξ) −→ ⊥ Xξ := ( f Xξ)⊥ , h opoÐa eÐnai h sunoq thc kˆjethc dianusmatik c dèsmhc Nf. Epomènwc, apì ta parapˆnw paÐrnoume ton tÔpo tou Weingarten : f Xξ = −df(AξX) + ⊥ Xξ.
- 14. 6 Prokatarktikˆ JewroÔme topikìorjomonadiaÐo plaÐsio {ξ1, ..., ξk} thc Nf. To dianusmatikì pedÐo mèshc kampulìthtac thc isometrik c embˆptishc f orÐzetai na eÐnai to H = 1 n k α=1 (traceAξα )ξα. To H eÐnai kalˆ orismèno kˆjeto diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f. Epiplèon isqÔei H = 1 n n j=1 B(ej, ej), ìpou {e1, ..., en} topikì orjomonadiaÐo plaÐsio tou Mn. Orismìc 1.1.5. Mia isometrik embˆptish f lègetai elaqistik an H = 0. O tanust c kˆjethc kampulìthtac thc isometrik c embˆptishc f orÐzetai na eÐnai h apeikìnish R⊥ : ∆(Mn ) × ∆(Mn ) × ∆⊥ (f) −→ ∆⊥ (f), (X, Y, ξ) −→ R⊥ (X, Y )ξ := ⊥ X ⊥ Y ξ − ⊥ Y ⊥ Xξ − ⊥ [X,Y ]ξ. ApodeiknÔetai ìti o R⊥ eÐnai D(Mn)-grammikìc wc proc kˆje metablht . Tèloc, gia ξ ∈ ∆⊥(f) orÐzoume to (2,1)-tanustikì pedÐo tou Mn ( XAξ)Y := X(AξY ) − Aξ( XY ), ìpou X, Y ∈ ∆(Mn). Gia isometrikèc embaptÐseic isqÔoun: (i) h exÐswsh Gauss Rf (X, Y )df(Z), df(W) = R(X, Y )Z, W + B(X, Z), B(Y, W) − B(Y, Z), B(X, W) , (ii) h exÐswsh Codazzi Rf (X, Y )ξ = ( Y Aξ)X − A ⊥ Y ξX − ( XAξ)Y + A ⊥ X ξY, (iii) h exÐswsh Ricci Rf (X, Y )ξ ⊥ = R⊥ (X, Y )ξ − B(X, AξY ) + B(Y, AξX), ìpou X, Y, Z, W ∈ ∆(Mn) kai ξ ∈ ∆⊥(f). JewroÔme topikì orjomonadiaÐo plaÐsio {e1, ..., en} tou Mn. To m koc thc deÔte- rhc jemeli¸douc morf c orÐzetai wc h sunˆrthsh B := n j,l=1 |B(ej, el)|2.
- 15. Exis¸seic dom c 7 ApodeiknÔetaiìti eÐnai kalˆ orismènh, dhlad anexˆrthth tou plaisÐou kai ìti B 2 = k α=1 trace(Aξα ◦ Aξα ). 'Estw f : (Mn, , ) −→ (M n+k c , , ) isometrik embˆptish ìpou Mn eÐnai dia- forÐsimo polÔptugma Riemann kai M n+k c eÐnai diaforÐsimo polÔptugma Riemann me stajer kampulìthta tom c c. Gia p ∈ Mn, jewroÔme dianÔsmata x, y ∈ TpMn kai orjomonadiaÐa bˆsh {ξ1, ..., ξk} tou Npf. Apì thn exÐswsh Gauss apodeiknÔetai ìti gia ton tanust Ricci isqÔei Q(x, y) = k α=1 (traceAξα ) Aξα x, y − k α=1 Aξα x, Aξα y + (n − 1)c x, y . An x eÐnai monadiaÐo, tìte h kampulìthta Ricci sth dieÔjunsh x dÐnetai apì th sqèsh Ric(x) = k α=1 (traceAξα ) Aξα x, x − k α=1 |Aξα x|2 + (n − 1)c. Epiplèon, gia thn arijmhtik kampulìthta isqÔei Sc = n(n − 1)c + n2 |H|2 − B 2 . (1.1) 'Estw f : M2 −→ Sn isometrik embˆptish enìc polÔptugmatoc Riemann M2 sth monadiaÐa sfaÐra Sn. JewroÔme topikì orjomonadiaÐo plaÐsio {e1, e2} tou M2. Tìte h f eÐnai elaqistik an kai mìno an B(e1, e1) + B(e2, e2) = 0. Gia thn arijmhtik kampulìthta èqoume Sc = 2K, ìpou K eÐnai h kampulìthta Gauss tou M2. Epomènwc, h sqèsh (1.1) gÐnetai sthn perÐptwsh pou h f eÐnai elaqistik 2K = 2 − B 2 (1.2) isodÔnama K = 1 − |B(e1, e1)|2 − |B(e1, e2)|2 . (1.3) Katˆ sunèpeia, ìtan h f eÐnai elaqistik isqÔei K ≤ 1 kai èqoume K = 1 pantoÔ an kai mìno an h elaqistik embˆptish f : M2 −→ Sn eÐnai olikˆ gewdaisiak . 1.2 Exis¸seic dom c 'Estw f : (Mn, , ) −→ (M n+k , , ) isometrik embˆptish, ìpou Mn kai M n+k eÐnai poluptÔgmata Riemann me sunoqèc Levi-Civita , kai tanustèc kampulìthtac R, R antÐstoiqa. JewroÔme anoiktì uposÔnolo U tou Mn, ¸ste h f|U na eÐnai emfÔteush, kai anoiktì uposÔnolo U tou M n+k me f(U) = U ∩f(Mn). Gia ta epìmena ja qrhsimopoi soume thn ex c sÔmbash deikt¸n: 1 ≤ j, l, s, t, ... ≤ n, n + 1 ≤ α, β, γ, ... ≤ n + k, 1 ≤ A, B, C, ... ≤ n + k,
- 16. 8 Prokatarktikˆ ektìc ananafèretai diaforetikˆ. 'Estw {eA} orjomonadiaÐo plaÐsio sto U ètsi ¸ste gia kˆje shmeÐo q = f(p) tou f(U), na isqÔei ej|q ∈ Tpf gia kˆje j. SumbolÐzoume me {ωA} to sumplaÐsio tou {eA}. Gia V ∈ ∆(U) èqoume ωA(V ) = V, eA . OrÐzoume orjomonadiaÐo plaÐsio {ej} sto U me ej := df−1(ej ◦ f|U ) kai èstw {ωj} to sumplaÐsiì tou. Gia X ∈ ∆(U) èqoume ωj(X) = X, ej . An jèsoume eα := eα ◦ f|U , tìte apoktoÔme orjomonadiaÐo plaÐsio {eA} katˆ m koc thc f me efaptìmeno mèroc {ej} kai kˆjeto mèroc {eα}. Orismìc 1.2.1. KaloÔme r-morf epÐ enìc diaforÐsimou poluptÔgmatoc kˆje anti- summetrikì (r, 0)-tanustikì pedÐo. SumbolÐzoume me Λr(M n+k ) kai me Λr(Mn) to sÔnolo twn r-morf¸n (r ≥ 0) tou M n+k kai tou Mn antÐstoiqa. Gia r = 0 èqoume ta sÔnola twn diaforÐsimwn sunart sewn D(M n+k ) kai D(Mn) antÐstoiqa. H f epˆgei gia kˆje akèraio r ≥ 0 mia apeikìnish f∗ : Λr(M n+k ) −→ Λr(Mn), thn anˆsursh (pullback) thc f, pou orÐzetai wc ex c: Gia r = 0 kai g ∈ D(M n+k ), f∗(g) = g ◦ f. Gia r > 0 kai w ∈ Λr(M n+k ) h f∗(w) ∈ Λr(Mn) eÐnai h r-morf , pou sto p ∈ Mn orÐzetai wc f∗(w)|p(v1, v2, ..., vr) := w|f(p) dfp(v1), dfp(v2), ..., dfp(vr) , ìpou v1, ..., vr ∈ TpMn. Oi morfèc sunoq c tou Mn gia to plaÐsio {ej}, eÐnai oi 1-morfèc ωjl : ∆(U) −→ D(U), X −→ ωjl(X) := Xej, el . EpÐshc, oi morfèc sunoq c tou M n+k gia to plaÐsio {eA}, eÐnai oi 1-morfèc ωAB : ∆(U) −→ D(U), X −→ ωAB(X) := XeA, eB . IsqÔei ωjl = −ωlj, ωAB = −ωBA kai apodeiknÔetai ìti ωj = f∗(ωj), ωjl = f∗(ωjl). Oi exis¸seic dom c tou Mn eÐnai: dωj = l ωjl ∧ ωl, dωjl = s ωjs ∧ ωsl + Ωjl, ìpou Ωjl ∈ Λ2(U), Ωjl = s<t Rjlstωs ∧ ωt, Ωjl = −Ωlj kai ∧ eÐnai to exwterikì ginìmeno morf¸n. Oi 2-morfèc Ωjl kaloÔntai morfèc kampulìthtac tou Mn. Oi exis¸seic dom c tou M n+k eÐnai: dωA = B ωAB ∧ ωB, dωAB = C ωAC ∧ ωCB + ΩAB, ìpou ΩAB ∈ Λ2(U), ΩAB = C<D RABCDωC ∧ ωD, ΩAB = −ΩBA.
- 17. Jemeli¸deic morfèc an¸terhctˆxhc 9 OrÐzoume tic 1-morfèc ωjα := f∗(ωjα), ωαj := f∗(ωαj) kai tic morfèc kˆjethc sunoq c ωαβ := f∗(ωαβ). ApodeiknÔetai ìti ωjα(X) = Aeα X, ej , ìpou Aeα eÐnai h apeikìnish Weingarten sth dieÔjunsh eα kai ωαβ(X) = ⊥ Xeα, eβ gia kˆje X ∈ ∆(U). EpÐshc orÐzoume ωα := f∗(ωα). EÔkola prokÔptei ìti ωα = 0. L mma 1.2.1. (Cartan) 'Estwsan oi grammikˆ anexˆrthtec 1-morfèc ϕ1, ..., ϕr (r ≤ n) tou Mn. An θ1, θ2, ..., θr eÐnai 1-morfèc tou Mn tètoiec ¸ste na isqÔei r j=1 ϕj ∧ θj = 0, tìte upˆrqoun sunart seic ajl ∈ D(Mn), ìpou j, l ∈ {1, ..., r} ¸ste na isqÔei θj = r l=1 ajlϕl kai ajl = alj. Epeid ωα = 0 èqoume r j=1 ωjα∧ωj = 0 kai sÔmfwna me to L mma 1.2.1, upˆrqoun sunart seic hα jl ∈ D(U), me hα jl = hα lj, tètoiec ¸ste ωjα = n l=1 hα jlωl kai hα jl = Aeα ej, el = B(ej, el), eα . Parat rhsh 1.2.1. H isometrik embˆptish f : (Mn, , ) → (M n+k , , ) eÐnai elaqistik an gia kˆje α isqÔei n j=1 hα jj = 0. 'Otan to polÔptugma M n+k èqei stajer kampulìthta tom c c, tìte oi exis¸seic Gauss, Codazzi kai Ricci gÐnontai antÐstoiqa: Ωjl = α ωjα ∧ ωαl − cωj ∧ ωl, dωjα = l ωjl ∧ ωlα + β ωjβ ∧ ωβα, dωαβ = j ωαj ∧ ωjβ + γ ωαγ ∧ ωγβ. An ω eÐnai 1-morf tou Mn, tìte apodeiknÔetai sto [14] ìti isqÔei dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]), ìpou X, Y ∈ ∆(Mn). 'Estw M2 èna didiˆstato polÔptugma Riemann. Me th bo jeia kai thc parapˆnw sqèshc apodeiknÔetai ìti dω12 = −Kω1 ∧ ω2, (1.4) ìpou K eÐnai h kampulìthta Gauss tou M2. 1.3 Jemeli¸deic morfèc an¸terhc tˆxhc 'Estw f : (Mn, , ) −→ (M n+k , , ) isometrik embˆptish, ìpou Mn, M n+k eÐnai poluptÔgmata Riemann me sunoqèc Levi-Civita , antÐstoiqa. Gia p ∈ Mn kai jetikì akèraio r, o dianusmatikìc q¸roc Tr p f := span dfp(Xj1 |p), f Xl1 df(Xl2 ) |p, ..., f Xs1 ... f Xsr−1 df(Xsr ) |p : Xm ∈ ∆(Mn ), m ∈ {j1, l1, l2, ..., s1, ..., sr}
- 18. 10 Prokatarktikˆ kaleÐtai eggÔtatocq¸roc r-tˆxhc (osculating space) thc f sto p kai eÐnai dianusma- tikìc upìqwroc tou Tf(p)M n+k . Profan¸c, T1 p f = dfp(TpMn) kai T1f = p∈Mn T1 p f eÐnai h efaptìmenh dianu- smatik dèsmh thc f. An af soume to p na metabˆletai, tìte gia stajerì r > 1 oi q¸roi Tr p f endeqomènwc na èqoun diaforetik diˆstash. EpÐshc, o eggÔtatoc q¸roc r-tˆxhc thc f sto p eÐnai upìqwroc tou eggutˆtou q¸rou (r + 1)-tˆxhc thc f sto p. Epomènwc, mporoÔme na orÐsoume to orjog¸nio sumpl rwma Nr p f tou Tr p f ston Tr+1 p f wc proc to eswterikì ginìmeno tou Tf(p)M n+k , dhlad Tr+1 p f = Tr p f ⊕ Nr p f. O Nr p f lègetai kˆjetoc q¸roc r-tˆxhc thc f sto p. IsqÔei Tr+1 p f = T1 p f ⊕ N1 p f ⊕ N2 p f ⊕ ... ⊕ Nr p f. Profan¸c, gia s = r èqoume Nr p f ∩ Ns p f = {0} kai v, w = 0 gia kˆje v ∈ Nr p f, w ∈ Ns p f. Orismìc 1.3.1. 'Estw f isometrik embˆptish metaxÔ twn poluptugmˆtwn Riemann Mn kai M n+k c , ìpou to M n+k c èqei stajer kampulìthta tom c c. KaloÔme jemeli¸dh morf (r + 1)-tˆxhc thc f sto p ∈ Mn thn apeikìnish Br|p : TpMn × TpMn × ... × TpMn r+1 −→ Nr p f, (x1, x2, ..., xr+1) → Br|p(x1, x2, ..., xr+1) = f X1 f X2 ... f Xr df(Xr+1)|p Nr p f , ìpou me f X1 f X2 ... f Xr df(Xr+1)|p Nr p f sumbolÐzoume thn probol tou dianÔsmatoc f X1 f X2 ... f Xr df(Xr+1)|p tou Tr+1 p f ston upìqwrì tou Nr p f kai X1, ..., Xr+1 eÐnai to- pikˆ diaforÐsima dianusmatikˆ pedÐa tou Mn pou epekteÐnoun ta x1, ..., xr+1 antÐstoiqa, dhlad Xi|p = xi gia i = 1, ..., r + 1. Epeid to diˆnusma f X1 f X2 ... f Xr df(Xr+1)|p an kei ston Tr+1 p f = Tr p f ⊕ Nr p f kai isqÔei Tr p f = T1 p f ⊕ N1 p f ⊕ ... ⊕ Nr−1 p f, ènac isodÔnamoc orismìc thc Br|p eÐnai Br|p(x1, ..., xr+1) = f X1 ... f Xs df(Xr+1)|p (T1 p f⊕N1 p f⊕...⊕Nr−1 p f)⊥ , ìpou (T1 p f⊕N1 p f⊕...⊕Nr−1 p f)⊥ eÐnai to orjosumpl rwma tou T1 p f⊕N1 p f⊕...⊕Nr−1 p f ston Tf(p)M n+k . ApodeiknÔetai sto [18] ìti h Br|p eÐnai kalˆ orismènh kai summetrik , dhlad Br|p(x1, x2, ..., xr+1) = Br|p(xσ(1), xσ(2), ..., xσ(r+1)) gia kˆje stoiqeÐo σ thc omˆdac metajèsewn twn arijm¸n 1, 2, ..., r + 1. EpÐshc, h Br|p eÐnai R-grammik wc proc kˆje metablht thc. JewroÔme thn ènwsh Nrf ìlwn twn kajètwn q¸rwn r-tˆxhc thc f, dhlad Nr f := p∈Mn Nr p f.
- 19. Jemeli¸deic morfèc an¸terhctˆxhc 11 An af soume to p na metabˆletai ston Orismì 1.3.1, tìte èqoume th jemeli¸dh morf (r + 1)-tˆxhc thc f : Br : TMn ⊕ TMn ⊕ ... ⊕ TMn r+1 −→ Nr f, isodÔnama Br : TMn ⊕ TMn ⊕ ... ⊕ TMn r+1 −→ (T1 f ⊕ N1 f ⊕ ... ⊕ Nr−1 f)⊥ , (p, x1), ..., (p, xr+1) −→ Br|p(x1, ..., xr+1). Profan¸c h B1 eÐnai h deÔterh jemeli¸dhc morf thc f, dhlad èqoume B1 = B. Gia r > 1, h Br den eÐnai en gènei (r+1,1)-tanustikì pedÐo, afoÔ to sÔnolo T1f ⊕ N1f ⊕ ... ⊕ Nr−1f de gnwrÐzoume an eÐnai dianusmatik dèsmh. Ac shmeiwjeÐ ìti sto ex c ja grˆfoume Br eÐte gia th jemeli¸dh morf (r + 1)- tˆxhc thc f sto p, eÐte gia th jemeli¸dh morf (r + 1)-tˆxhc thc f. L mma 1.3.1. 'Estw f : (Mn, , ) −→ (M n+k c , , ) isometrik embˆptish, ìpou Mn eÐnai polÔptugma Riemann kai M n+k c eÐnai polÔptugma Riemann me stajer kampu- lìthta tom c c. Tìte gia kˆje p ∈ Mn isqÔei Nr p f = spanImBr = span{Br(x1, x2, ..., xr+1) : x1, x2, ..., xr+1 ∈ TpMn }. Apìdeixh. Profan¸c spanImBr ⊂ Nr p f, afoÔ ImBr ⊂ Nr p f. 'Estw ξ ∈ Nr p f. Gnw- rÐzoume ìti Tr+1 p f = Tr p f ⊕ Nr p f, epomènwc ξ ∈ Tr+1 p f kai ja grˆfetai wc ξ = j αjdfp(Xj|p) + l,m βlm f Xl df(Xm)|p + ... + + s1,...,sr γs1...sr f Xs1 ... f Xsr−1 df(Xsr )|p + t1,...,tr+1 δt1...tr+1 f Xt1 ... f Xtr df(Xtr+1 )|p Nr p f , gia katˆllhlouc suntelestèc, ìpou Xj, ..., Xtr+1 ∈ ∆(Mn). Epeid h probol eÐnai grammik apeikìnish, Ts p f ⊂ Tr p f gia kˆje s < r kai Tr+1 p f = Tr p f ⊕ Nr p f, èqoume ξ = t1,...,tr+1 δt1...tr+1 f Xt1 ... f Xtr df(Xtr+1 )|p Nr p f = t1,...,tr+1 δt1...tr+1 Br(Xt1 |p, Xt2 |p, ..., Xtr+1 |p). Epomènwc, ξ ∈ spanImBr. L mma 1.3.2. 'Estw f : (M2, , ) −→ (M 2+k c , , ) isometrik elaqistik embˆpti- sh, ìpou M2 eÐnai polÔptugma Riemann kai M 2+k c eÐnai polÔptugma Riemann me stajer kampulìthta tom c c. An {e1, e2} eÐnai topikì orjomonadiaÐo plaÐsio gÔrw apì to tuqìn shmeÐo p ∈ M2, tìte gia kˆje x1, ..., xr−1 ∈ TpM2 isqÔei Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) = 0.
- 20. 12 Prokatarktikˆ Apìdeixh. JewroÔmeX1, ..., Xr−1 ∈ ∆(M2) tètoia ¸ste Xi|p = xi, gia i = 1, ..., r−1. Apì ton orismì thc Br èqoume: Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) = f X1 ... f Xr−1 f e1 df(e1) + f e2 df(e2) |p Nr p f . AfoÔ h f eÐnai elaqistik isqÔei B(e1, e1) + B(e2, e2) = 0. Lìgw tou tÔpou tou Gauss èqoume f e1 df(e1) + f e2 df(e2) = df( e1 e1) + df( e2 e2) + B(e1, e1) + B(e2, e2) = df( e1 e1 + e2 e2). Epomènwc f e1 df(e1) + f e2 df(e2) ∈ ∆f (M2), opìte f X1 ... f Xr−1 f e1 df(e1) + f e2 df(e2) |p ∈ Tr p f, to opoÐo shmaÐnei ìti Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) = 0. L mma 1.3.3. 'Estw f : (M2, , ) −→ (M 2+k c , , ) isometrik elaqistik embˆpti- sh, ìpou M2 eÐnai polÔptugma Riemann kai M 2+k c eÐnai polÔptugma Riemann me sta- jer kampulìthta tom c c. Tìte gia kˆje shmeÐo p tou M2 h diˆstash tou kˆjetou q¸rou r-tˆxhc Nr p f eÐnai to polÔ 2 gia kˆje r. Apìdeixh. 'Estw p èna shmeÐo tou M2. JewroÔme orjomonadiaÐa bˆsh {e1, e2} tou TpM2. Lìgw tou L mmatoc 1.3.1, gia tuqìnta dianÔsmata x1, ..., xr+1 tou TpM2 to diˆnusma Br(x1, ..., xr+1) eÐnai grammikìc sunduasmìc twn Br(ei1 , ..., eir+1 ), ìpou i1, ..., ir+1 ∈ {1, 2}. Kˆnontac qr sh thc summetrÐac thc Br kai tou L mmatoc 1.3.2 èqoume, Br(ei1 , ..., eir+1 ) = ±Br(e1, ..., e1) ±Br(e1, ..., e1, e2). Apì ta parapˆnw paÐrnoume Nr p f = spanImBr = span{Br(x1, x2, ..., xr+1) : x1, x2, ..., xr+1 ∈ TpMn } = span{Br(e1, ..., e1), Br(e1, ..., e1, e2)}. Epomènwc se kˆje shmeÐo p h diˆstash tou Nr p f eÐnai to polÔ 2 gia kˆje r. Ja apodeÐxoume thn parakˆtw prìtash [3, 7]. Prìtash 1.3.1. Upojètoume ìti f : (M2, , ) −→ (M 2+k c , , ) eÐnai isometrik elaqistik embˆptish, ìpou M2 eÐnai polÔptugma Riemann kai M 2+k c eÐnai polÔptugma Riemann me stajer kampulìthta tom c c. Gia kˆje shmeÐo p tou M2 h eikìna tou monadiaÐou kÔklou tou efaptìmenou q¸rou TpM2 me kèntro to 0 mèsw thc Br, dhlad
- 21. Jemeli¸deic morfèc an¸terhctˆxhc 13 to sÔnolo Er(p) := {Br(x, ..., x) : x ∈ TpM2, |x| = 1}, eÐnai èlleiyh, endeqomènwc ekfulismènh. Epiplèon, to sÔnolo Er(p) eÐnai kÔkloc aktÐnac ρ ≥ 0 an gia tuqoÔsa orjomonadiaÐa bˆsh {e1, e2} tou TpM2 isqÔei |Br(e1, ..., e1)| = |Br(e1, ..., e1, e2)| = ρ kai Br(e1, ..., e1), Br(e1, ..., e1, e2) = 0. Apìdeixh. JewroÔme to monadiaÐo kÔklo tou efaptìmenou q¸rou sto shmeÐo p, dhlad to sÔnolo Sp = {x ∈ TpM2 : |x| = 1}. 'Estw {e1, e2} orjomonadiaÐa bˆsh tou TpM2. Gia kˆje x ∈ Sp, èqoume x = cos θe1 + sin θe2, ìpou θ ∈ R. Kˆnontac qr sh thc summetrÐac thc Br, èqoume Br(x, ..., x) = = Br(cos θe1 + sin θe2, ..., cos θe1 + sin θe2) = r+1 m=0 r + 1 m (cos θ)r+1−m (sin θ)m Br(e1, ..., e1 r+1−m , e2, ..., e2 m ). SumbolÐzoume me I to sÔnolo twn ˆrtiwn arijm¸n tou sunìlou {1, ..., r + 1} kai me J to sÔnolo twn peritt¸n arijm¸n tou. Br(x, ..., x) = = m∈I r + 1 m (cos θ)r+1−m (sin θ)m Br(e1, ..., e1 r+1−m , e2, ..., e2 m ) + r+1 m∈J r + 1 m (cos θ)r+1−m (sin θ)m Br(e1, ..., e1 r+1−m , e2, ..., e2 m ). 'Omwc lìgw tou L mmatoc 1.3.2 eÐnai Br(e1, ..., e1, e2, ..., e2 m ) = (−1)lBr(e1, ..., e1), m = 2l, (−1)lBr(e1, ..., e1, e2), m = 2l + 1. Epomènwc, èqoume Br(x, ..., x) = m∈I,m=2l r + 1 m (cos θ)r+1−m (sin θ)m (−1)l Br(e1, ..., e1) + m∈J,m=2l+1 r + 1 m (cos θ)r+1−m (sin θ)m (−1)l Br(e1, ..., e1, e2). EÐnai gnwstì ìti isqÔoun oi sqèseic cos (r + 1)θ = m∈I,m=2l (−1)l r + 1 m (cos θ)r+1−m (sin θ)m , sin (r + 1)θ = m∈J,m=2l+1 (−1)l r + 1 m (cos θ)r+1−m (sin θ)m .
- 22. 14 Prokatarktikˆ Telikˆ, èqoume Br(x,..., x) = cos (r + 1)θ Br(e1, ..., e1) + sin r + 1)θ Br(e1, ..., e1, e2). Katˆ sunèpeia to sÔnolo Er(p) = {cos (r + 1)θ Br(e1, ..., e1) + sin (r + 1)θ Br(e1, ..., e1, e2) : θ ∈ R} eÐnai èlleiyh, endeqomènwc ekfulismènh kai gÐnetai kÔkloc ìtan ta dÔo dianÔsmata Br(e1, ..., e1), Br(e1, ..., e1, e2) èqoun Ðdio m koc kai eÐnai kˆjeta metaxÔ touc. H èlleiyh Er(p) ja anafèretai kai wc èlleiyh r-tˆxhc thc f sto p. SumbolÐzoume me κr, µr ta m kh twn hmiaxìnwn thc èlleiyhc r-tˆxhc ¸ste κr µr 0. OrÐzoume th sunˆrthsh K⊥ r : M2 −→ R, p −→ K⊥ r (p) := 2κr(p)µr(p) kai thn onomˆzoume kˆjeth kampulìthta r-tˆxhc thc f. EÐnai fanerì ìti an K⊥ r (p) = 0, tìte dimNr p f < 2 kai h èlleiyh Er(p) ekfulÐzetai se eujÔgrammo tm ma shmeÐo. En¸ ìtan K⊥ r (p) > 0, tìte dimNr p f = 2 kai h èlleiyh den eÐnai ekfulismènh. H parakˆtw prìtash ofeÐletai ston Otsuki [17]. Prìtash 1.3.2. Upojètoume ìti f : (M2, , ) −→ (M 2+k c , , ) eÐnai isometrik elaqistik embˆptish, ìpou M2 eÐnai sunektikì polÔptugma Riemann kai M 2+k c eÐnai polÔptugma Riemann me stajer kampulìthta tom c c. (i) An upˆrqei fusikìc r me 0 < r < [k+1 2 ], ìpou [k+1 2 ] eÐnai to akèraio mèroc tou k+1 2 , tètoioc ¸ste gia kˆje shmeÐo p tou M2 na èqoume Nr p f = {0} kai K⊥ s (p) > 0 gia kˆje s ∈ {1, ..., r − 1}, tìte upˆrqei olikˆ gewdaisiakì upopolÔptugma Q2r tou M 2+k c ¸ste f(M2) ⊂ Q2r. (ii) An upˆrqei fusikìc r me 0 < r < [k+1 2 ] tètoioc ¸ste gia kˆje shmeÐo p tou M2 na èqoume dimNr p f = 1 kai K⊥ s (p) > 0 gia kˆje s ∈ {1, ..., r − 1}, tìte upˆrqei olikˆ gewdaisiakì upopolÔptugma Q2r+1 tou M 2+k c ¸ste f(M2) ⊂ Q2r+1. Mia isometrik elaqistik embˆptish f : M2 −→ Sn enìc poluptÔgmatoc Riemann M2 sthn Sn kaleÐtai koresmènh (full) an h eikìna thc f(M2) den perièqetai se kanèna olikˆ gewdaisiakì upopolÔptugma thc Sn. O Lawson [16] apèdeixe to ex c apotèlesma: Prìtash 1.3.3. An f : M2 −→ Sn eÐnai isometrik elaqistik embˆptish, ìpou M2 prosanatolismèno polÔptugma Riemann, tìte h f eÐnai analutik , dhlad an (U, ϕ) eÐnai qˆrthc tou M2, tìte h apeikìnish f ◦ϕ−1 : ϕ(U) ⊂ R2 −→ Rn+1 eÐnai analutik . Parat rhsh 1.3.1. 'Amesh sunèpeia twn Protˆsewn 1.3.2 kai 1.3.3, eÐnai pwc an f : M2 −→ Sn eÐnai isometrik elaqistik embˆptish enìc prosanatolismènou poluptÔgmatoc Riemann M2 sthn Sn me Br|U = 0, ìpou U eÐnai anoiktì sÔnolo kai r ∈ {1, ..., [n−1 2 ]−1}, tìte upˆrqei mègisth sfaÐra S2r thc Sn ¸ste f(U) ⊂ S2r. Lìgw analutikìthtac isqÔei f(M2) ⊂ S2r. Epomènwc, an upˆrqei r kai anoiktì sÔnolo U tou M2 ¸ste Br|U = 0, tìte h f den eÐnai koresmènh.
- 23. Kefˆlaio 2 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 2.1 Epifˆneiec Riemann Sthn enìthta aut parajètoume stoiqeÐa apì th jewrÐa epifanei¸n Riemann, ta opoÐa eÐnai aparaÐthta stic apodeÐxeic twn kurÐwn apotelesmˆtwn. Orismìc 2.1.1. KaloÔme epifˆneia Riemann kˆje topologikì q¸ro Hausdorff M me arijm simh bˆsh gia thn topologÐa tou, o opoÐoc eÐnai efodiasmènoc me ènan ˆtlanta {(Uα, ϕα)}α∈I pou plhroÐ ta parakˆtw: (i) H oikogèneia {Uα}α∈I eÐnai anoikt kˆluyh tou M kai oi apeikonÐseic ϕα : Uα ⊂ M −→ ϕα(Uα) ⊂ C eÐnai omoiomorfismoÐ. To zeÔgoc (Uα, ϕα) onomˆzetai migadikìc qˆrthc sÔsthma suntetagmènwn tou M. (ii) Gia kˆje α, β ∈ I, me Uα ∩ Uβ = ∅ h migadik sunˆrthsh ϕα ◦ ϕ−1 β : ϕβ(Uα ∩ Uβ) ⊂ C −→ ϕα(Uα ∩ Uβ) ⊂ C eÐnai olìmorfh. (iii) H oikogèneia migadik¸n qart¸n {(Uα, ϕα)}α∈I eÐnai mègisth, dhlad an (U, ϕ) migadikìc qˆrthc me ϕα ◦ϕ−1, ϕ◦ϕ−1 α , α ∈ I, olìmorfec sunart seic, tìte o migadikìc qˆrthc (U, ϕ) an kei sthn oikogèneia {(Uα, ϕα)}α∈I. An (U, ϕ) eÐnai migadikìc qˆrthc, tìte gia kˆje p ∈ U, o migadikìc arijmìc z(p) := ϕ(p) = x(p) + iy(p) dÐdei tic suntetagmènec tou p wc proc ton en lìgw qˆrth. AxÐzei na shmei¸soume ìti kˆje epifˆneia Riemann eÐnai prosanatolismèno didiˆstato diaforÐsimo polÔptugma. Orismìc 2.1.2. 'Estw M epifˆneia Riemann. Mia suneq c migadik sunˆrthsh f : M −→ C lègetai olìmorfh an gia kˆje migadikì qˆrth (U, ϕ) tou M h migadik sunˆrthsh f ◦ ϕ−1 : ϕ(U) −→ C eÐnai olìmorfh. 'Estw (M2, , ) polÔptugma Riemann kai (U, ϕ) qˆrthc gÔrw apì to p ∈ M2 me suntetagmènec (x, y). An isqÔei , = Edx2 + Edy2, E ∈ D(U) kai E > 0, tìte to sÔsthma suntetagmènwn (x, y) kaleÐtai isìjermo. O Chern [9] apèdeixe to parakˆtw apotèlesma: 'Estw (M2, , ) diaforÐsimo polÔptugma Riemann. Tìte gÔrw apì kˆje shmeÐo p ∈ M2 upˆrqei isìjermo sÔsthma suntetagmènwn. 15
- 24. 16 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 Prìtash 2.1.1. Kˆje prosanatolismèno didiˆstato diaforÐsimo polÔptugma gÐnetai katˆ fusikì trìpo epifˆneia Riemann. Apìdeixh. 'Estw M2 prosanatolismèno didiˆstato diaforÐsimo polÔptugma. Gnw- rÐzoume ìti kˆje polÔptugma dèqetai metrik Riemann. Epomènwc mporoÔme na efo- diˆsoume to M2 me metrik Riemann , . Lìgw tou proanaferjèntoc apotelèsmatoc tou Chern, gÔrw apì kˆje shmeÐo tou M2 upˆrqei sÔsthma isìjermwn suntetagmènwn. JewroÔme èna shmeÐo p tou M2. GÔrw apì to p, jewroÔme qˆrtec (U, ϕ) kai (V, ψ) tou prosanatolismoÔ me suntetagmènec (x, y) kai (u, v) antistoÐqwc, ètsi ¸ste ∂ ∂x , ∂ ∂y = 0, ∂ ∂x , ∂ ∂x = ∂ ∂y , ∂ ∂y = E, (2.1) ∂ ∂u , ∂ ∂v = 0, ∂ ∂u , ∂ ∂u = ∂ ∂v , ∂ ∂v = E. (2.2) Sto anoiktì sÔnolo W = U ∩ V èqoume ∂ ∂x = ∂u ∂x ∂ ∂u + ∂v ∂x ∂ ∂v , ∂ ∂y = ∂u ∂y ∂ ∂u + ∂v ∂y ∂ ∂v . Apì tic sqèseic (2.1) kai (2.2) paÐrnoume tic sqèseic ∂u ∂x ∂u ∂y + ∂v ∂x ∂v ∂y = 0, ( ∂u ∂x )2 + ( ∂v ∂x )2 = E E , ( ∂u ∂y )2 + ( ∂v ∂y )2 = E E . EpÐshc, h apeikìnish ψ ◦ ϕ−1 : ϕ(W) −→ ψ(W) èqei jetik Iakwbian orÐzousa afoÔ oi qˆrtec an koun ston Ðdio prosanatolismì, dhlad ∂u ∂x ∂v ∂y − ∂v ∂x ∂u ∂y > 0. Apì tic teleutaÐec tèsseric sqèseic paÐrnoume ∂u ∂x = ∂v ∂y , ∂u ∂y = − ∂v ∂x , dhlad h apeikìnish ψ ◦ ϕ−1 eÐnai olìmorfh kai epomènwc to M2 kajÐstatai epifˆneia Riemann. H Prìtash 2.1.1 mac epitrèpei na jewroÔme kˆje prosanatolismèno didiˆstato diaforÐsimo polÔptugma Riemann M2 wc epifˆneia Riemann. Sto ex c ìtan ja lème ìti jewroÔme migadikì qˆrth (U, z), ìpou z = x + iy, ja ennooÔme qˆrth (U, ϕ)
- 25. Epifˆneiec Riemann 17 touprosanatolismoÔ tou M2 me suntetagmènec (x, y), ètsi ¸ste ∂ ∂x , ∂ ∂y = 0 kai ∂ ∂x , ∂ ∂x = E = ∂ ∂y , ∂ ∂y isodÔnama , = E|dz|2. JewroÔme èna prosanatolismèno didiˆstato diaforÐsimo polÔptugma Riemann M2 kai orjomonadiaÐo plaÐsio {e1, e2} tou prosanatolismoÔ. OrÐzetai èna (1,1)-tanustikì pedÐo J : ∆(M2) −→ ∆(M2), ¸ste gia kˆje shmeÐo p tou M2, J|p : TpM2 −→ TpM2 eÐnai h strof katˆ gwnÐa +π 2 . To (1,1)-tanustikì pedÐo J kaleÐtai migadik dom tou M2. MigadikopoioÔme ton efaptìmeno q¸ro sto p kai epekteÐnoume C-grammikˆ to J sto TpM2 ⊗ C wc ex c: J(v + iw) = J(v) + iJ(w). Epeid J ◦ J = −Id, oi mìnec idiotimèc tou J eÐnai i, −i. O idioq¸roc pou antistoiqeÐ sthn idiotim i eÐnai TpM2 := {v ∈ TpM2 ⊗ C : J(v) = iv} = {x − iJ(x) : x ∈ TpM2 } kai o idioq¸roc pou antistoiqeÐ sthn idiotim −i eÐnai Tp M2 := {v ∈ TpM2 ⊗ C : J(v) = −iv} = {x + iJ(x) : x ∈ TpM2 }. L mma 2.1.1. Se kˆje shmeÐo p tou M2 isqÔei TpM2 ⊗ C = TpM2 ⊕ Tp M2. Apìdeixh. 'Estw v ∈ TpM2 ⊗ C, tìte v = u + iw me u, w ∈ TpM2 kai èqoume v = u + J(w) 2 − iJ( u + J(w) 2 ) ∈TpM2 + u − J(w) 2 + iJ( u − J(w) 2 ) ∈Tp M2 , dhlad TpM2 ⊗ C = TpM2 + Tp M2. Apomènei na deÐxoume ìti to ˆjroisma eÐnai eujÔ. 'Estw x = a + ib ∈ TpM2 ∩ Tp M2, a, b ∈ TpM2. Tìte x = y − iJ(y) gia kˆpoio y ∈ TpM2, afoÔ x ∈ TpM2 kai x = h + iJ(h) gia kˆpoio h ∈ TpM2, afoÔ x ∈ Tp M2. Epomènwc a = y = h kai b = −J(y) = J(h), dhlad −J(y) = J(y), kai ˆra y = 0. Katˆ sunèpeia a = b = 0 kai to ˆjroisma eÐnai eujÔ. Lìgw tou L mmatoc 2.1.1, eÐnai fanerì ìti h migadikopoihmènh efaptìmenh dianu- smatik dèsmh TM2 ⊗ C diaspˆtai wc ex c: TM2 ⊗ C = T M2 ⊕ T M2 , ìpou T M2 := {(p, v) : p ∈ M2, v ∈ TpM2} kai T M2 := {(p, v) : p ∈ M2, v ∈ Tp M2}. SumbolÐzoume ta pedÐa thc migadikopoihmènhc efaptìmenhc dianusmatik c dèsmhc me Γ(TM2 ⊗ C) kai jewroÔme to sÔnolo C∞(M2, C) twn diaforÐsimwn sunart sewn g : M2 −→ C. Orismìc 2.1.3. KaloÔme migadikì (r, 0)-tanustikì pedÐo kˆje apeikìnish T : Γ(TM2 ⊗ C) × ... × Γ(TM2 ⊗ C) r −→ C∞ (M2 , C) h opoÐa eÐnai C∞(M2, C)-grammik wc proc kˆje metablht thc. EpÐshc, an E eÐnai dianusmatik dèsmh uperˆnw tou M2, kaloÔme migadikì (r, 1)- tanustikì pedÐo kˆje apeikìnish T : Γ(TM2 ⊗ C) × ... × Γ(TM2 ⊗ C) r −→ Γ(E ⊗ C)
- 26. 18 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 h opoÐa eÐnai C∞(M2, C)-grammik wc proc kˆje metablht thc. 'Estw τ èna migadikì (r, 0)-tanustikì pedÐo kai σ èna migadikì (s, 0)-tanustikì pedÐo, tìte orÐzetai to tanustikì ginìmeno τ ⊗σ twn τ, σ wc to ex c migadikì (r+s, 0)- tanustikì pedÐo τ ⊗ σ : Γ(TM2 ⊗ C) × ... × Γ(TM2 ⊗ C) r+s −→ C∞ (M2 , C), (X1 + iY1, ..., Xr+s + iYr+s) −→ τ ⊗ σ(X1 + iY1, ..., Xr+s + iYr+s) = τ(X1 + iY1, ..., Xr + iYr)σ(Xr+1 + iYr+1, ..., Xr+s + iYr+s). 'Ena ˆllo ginìmeno migadik¸n tanustik¸n pedÐwn eÐnai to exwterikì ginìmeno, to opoÐo gia migadikˆ (1,0)-tanustikˆ pedÐa τ kai σ eÐnai to migadikì (2,0)-tanustikì pedÐo τ ∧ σ := τ ⊗ σ − σ ⊗ τ. 'Estw M2 prosanatolismèno diaforÐsimo polÔptugma kai (U, z) migadikìc qˆrthc tou me z = x + iy. Sto U èqoume ta (1,0)-tanustikˆ pedÐa dx, dy : ∆(U) −→ D(U), isodÔnama dx, dy : Γ(TU) −→ C∞(U, R) ta opoÐa epekteÐnoume C-grammikˆ sth migadikopoihmènh efaptìmenh dianusmatik dèsmh TU ⊗C. OrÐzoume to migadikì (1,0)- tanustikì pedÐo dz : Γ(TU ⊗ C) −→ C∞(U, C), dz := dx + idy kai to suzugèc tou dz : Γ(TU ⊗ C) −→ C∞(U, C), dz := dx − idy. EpÐshc orÐzoume touc telestèc Wirtinger ∂ ∂z := 1 2 ∂ ∂x − i ∂ ∂y kai ∂ ∂z := 1 2 ∂ ∂x + i ∂ ∂y gia touc opoÐouc isqÔei dz( ∂ ∂z ) = 1, dz( ∂ ∂z ) = 0, dz( ∂ ∂z ) = 0 kai dz( ∂ ∂z ) = 1. Ta { ∂ ∂z |p, ∂ ∂z |p} apoteloÔn bˆsh tou TpM2⊗C kai mˆlista TpM2 = span{ ∂ ∂z |p}, Tp M2 = span{ ∂ ∂z |p}. To sÔnolo twn migadik¸n (1,0)-tanustik¸n pedÐwn eÐnai C∞(U, C)-mìdio me prˆxeic thn prìsjesh twn (1,0)-tanustik¸n pedÐwn kai to bajmwtì pollalasiasmì kai bˆsh tou apoteloÔn ta migadikˆ (1,0)-tanustikˆ pedÐa dz, dz. Epiplèon, an W eÐnai migadikì (1,0)-tanustikì pedÐo, tìte W = W( ∂ ∂z )dz + W( ∂ ∂z )dz. To sÔnolo twn migadik¸n (2,0)-tanustik¸n pedÐwn eÐnai C∞(M2, C)-mìdio, ìpwc kai to sÔnolo twn migadik¸n (1,0)-tanustik¸n pedÐwn, kai èqei wc bˆsh ta migadikˆ (2,0)-tanustikˆ pedÐa dz⊗dz, dz⊗dz, dz⊗dz kai dz⊗dz. 'Ena migadikì (2,0)-tanustikì pedÐo W grˆfetai wc W = W( ∂ ∂z , ∂ ∂z )dz ⊗ dz + W( ∂ ∂z , ∂ ∂z )dz ⊗ dz + W( ∂ ∂z , ∂ ∂z )dz ⊗ dz + W( ∂ ∂z , ∂ ∂z )dz ⊗ dz. Genikìtera, to sÔnolo twn migadik¸n (r, 0)-tanustik¸n pedÐwn eÐnai C∞(M2, C)- mìdio me bˆsh ta migadikˆ (r, 0)-tanustikˆ pedÐa dw1 ⊗ ... ⊗ dwr, ìpou wj = z z gia kˆje j = 1, ..., r. Kˆje migadikì (r,0)-tanustikì pedÐo W grˆfetai wc W = p+q=r W(p,q) dzp dzq ,
- 27. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 19 ìpou W(r,0) dzr := W( ∂ ∂z , ..., ∂ ∂z r ) dz ⊗ ... ⊗ dz r , W(r−1,1) dzr−1 dz := W( ∂ ∂z , ..., ∂ ∂z r−1 , ∂ ∂z ) dz ⊗ ... ⊗ dz r−1 ⊗dz + W( ∂ ∂z , ..., ∂ ∂z r−2 , ∂ ∂z ∂ ∂z ) dz ⊗ ... ⊗ dz r−2 ⊗dz ⊗ dz + ... + W( ∂ ∂z ∂ ∂z , ..., ∂ ∂z r−1 )dz ⊗ dz ⊗ ... ⊗ dz r−1 , . . . W(0,r) dzr := W( ∂ ∂z , ..., ∂ ∂z r ) dz ⊗ ... ⊗ dz r . Kˆje migadikì (r, 0)-tanustikì pedÐo T thc morf c T = f(z) dz ⊗ ... ⊗ dz r = f(z)dzr , ìpou (U, ϕ) eÐnai migadikìc qˆrthc me migadik suntetagmènh z kai f ∈ C∞(U, C) lège- tai r-diaforikì. EÐnai fanerì ìti to r-diaforikì T eÐnai migadikì (r, 0)-tanustikì pedÐo tètoio ¸ste an kˆpoio apì ta v1, ..., vr an kei sth dèsmh T M2, tìte T(v1, ..., vr) = 0. To r-diaforikì T = f(z)dzr kaleÐtai olìmorfo an h f eÐnai olìmorfh sunˆrthsh. Gia ta r-diaforikˆ isqÔei to akìloujo shmantikì apotèlesma, gnwstì kai wc Je- ¸rhma Riemann-Roch [12]. Je¸rhma 2.1.1. 'Estw M epifˆneia Riemann omoiomorfik me thn S2. An Φ eÐnai olìmorfo r-diaforikì orismèno sthn epifˆneia M, tìte Φ = 0. 2.2 SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn To kleidÐ gia thn apìdeixh twn jewrhmˆtwn, pou anafèrontai sthn eisagwg , eÐnai h diapÐstwsh ìti gia sumpageÐc elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra, oi el- leÐyeic kˆje tˆxhc eÐnai kÔkloi sqedìn pantoÔ. Stìqoc thc paroÔshc paragrˆfou eÐnai h apìdeixh aut c thc diapÐstwshc. Gia to skopì autì, orÐzoume katˆllhla r-diaforikˆ ta opoÐa apodeiknÔoume ìti eÐnai olìmorfa kai lìgw tou Jewr matoc Riemann-Roch, eÐnai ek tautìthtoc mhdèn.
- 28. 20 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 'Estw f : M2 −→ Sn, n ≥ 3, elaqistik epifˆneia, dhlad isometrik elaqistik embˆptish enìc prosanatolismènou, sunektikoÔ, didiˆstatou poluptÔgmatoc Riemann (M2, , ) sthn Sn me deÔterh jemeli¸dh morf B. MigadikopoioÔme thn efaptìmenh dèsmh TM2 kai thn kˆjeth dèsmh Nf, kaj¸c kai kˆje upodèsmh thc. EpÐshc, epekteÐnoume C-grammikˆ kˆje tanustikì pedÐo pou ja oristeÐ sth sunèqeia. Apì ed¸ kai sto ex c jewroÔme migadikì qˆrth (U, z) tou M2 me z = x+iy kai , = E|dz|2. Me , sumbolÐzoume th C-grammik epèktash thc metrik c tou M2 kaj¸c kai th sun jh metrik thc Sn. Epilègoume topikì orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno mèroc {e1, e2} ¸ste e1 = 1√ E ∂ ∂x, e2 = 1√ E ∂ ∂y kai kˆjeto mèroc {eα}. Oi telestèc Wirtinger tìte eÐnai ∂ ∂z = 1 2 √ E(e1 − ie2) kai ∂ ∂z = 1 2 √ E(e1 + ie2). SumbolÐzoume me {ω1, ω2} to sumplaÐsio tou {e1, e2}. Profan¸c ω1 = √ Edx kai ω2 = √ Edy. Sto U orÐzoume to migadikì (1,0)-tanustikì pedÐo φ := ω1 + iω2 gia to opoÐo isqÔoun oi sqèseic φ = √ Edz, (2.3) dφ = 1 2 √ E dE ∧ dz. (2.4) Apì tic exis¸seic dom c tou M2 èqoume dφ = −iω12 ∧ φ. (2.5) Gia α = 3, ..., n orÐzoume tic sunart seic Hα 1 : U −→ C, Hα 1 := hα 11 + ihα 12, ìpou hα 11 = B(e1, e1), eα kai hα 12 = B(e1, e2), eα . EÐnai fanerì ìti Hα 1 ∈ C∞(U, C). Gia kˆje fusikì s ≥ 2 kai gia kˆje α = 3, ..., n orÐzoume tic sunart seic Hα s : U −→ C, Hα s := hα (s),1 + ihα (s),2, ìpou hα (s),1 := Bs(e1, ..., e1), eα kai hα (s),2 := Bs(e1, ..., e1, e2), eα . Tèloc, orÐzoume to m koc thc (r + 1)-jemeli¸douc morf c na eÐnai h sunˆrthsh Br := 2 j1,...,jr+1=1 |Br(ej1 , ..., ejr+1 )|2. EpekteÐnontac C-grammikˆ th deÔterh jemeli¸dh morf B, èqoume to migadikì (2,1)-tanustikì pedÐo B : Γ(TM2 ⊗ C) × Γ(TM2 ⊗ C) −→ Γ(Nf ⊗ C). Kˆje mÐa apì tic n + 1 sunist¸sec tou tanustikoÔ pedÐou B ston EukleÐdeio q¸ro Rn+1 eÐnai migadikì (2,0)-tanustikì pedÐo. Epomènwc, gia kˆje migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2, analÔontac kˆje sunist¸sa tou B, ìpwc sthn parˆgrafo 2.1, èqoume thn akìloujh anˆlush gia to B sto U B = B(2,0) dz2 + B(1,1) dzdz + B(0,2) dz2 ,
- 29. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 21 ìpou B(2,0) dz2 := B( ∂ ∂z , ∂ ∂z )dz ⊗ dz, B(1,1) dzdz := B( ∂ ∂z , ∂ ∂z )dz ⊗ dz + B( ∂ ∂z , ∂ ∂z )dz ⊗ dz, B(0,2) dz2 := B( ∂ ∂z , ∂ ∂z )dz ⊗ dz kai isqÔei B(0,2) = B(2,0). Epeid h f eÐnai elaqistik , isqÔei B(e1, e1)+B(e2, e2) = 0. 'Omwc apì thn epilog tou plaisÐou èqoume e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y . Epiplèon, eÐnai ∂ ∂x = ∂ ∂z + ∂ ∂z kai ∂ ∂y = i( ∂ ∂z − ∂ ∂z ). Katˆ sunèpeia isqÔei B( ∂ ∂z , ∂ ∂z ) = 0, isodÔnama B(1,1)dzdz = 0. Epomènwc to B dèqetai thn anˆlush B = B(2,0) dz2 + B(2,0)dz2 . OrÐzoume to 4-diaforikì Φ1 := B(2,0), B(2,0) dz4. Profan¸c B(2,0), B(2,0) ∈ C∞(U, C). Ja deÐxoume ìti to Φ1 eÐnai kalˆ orismèno, dhlad anexˆrthto thc epilog c tou qˆrth. Gia to skopì autì, jewroÔme migadikoÔc qˆrtec (U, ϕ) me migadik suntetagmènh z kai (V, ψ) me migadik suntetagmènh ζ ¸ste U ∩ V = ∅. To M2 eÐnai epifˆneia Riemann ˆra ψ ◦ ϕ−1 : ϕ(U ∩ V ) −→ ψ(U ∩ V ) kai ϕ ◦ ψ−1 : ψ(U ∩ V ) −→ ϕ(U ∩ V ) eÐnai olìmorfec, dhlad ∂ζ ∂z = 0 kai ∂z ∂ζ = 0. Sto U ∩ V èqoume loipìn tic sqèseic ∂ ∂z = ∂ζ ∂z ∂ ∂ζ , dζ = ∂ζ ∂z dz, kai epomènwc isqÔei B( ∂ ∂ζ , ∂ ∂ζ ), B( ∂ ∂ζ , ∂ ∂ζ ) dζ4 = ( ∂ζ ∂z )2 B( ∂ ∂ζ , ∂ ∂ζ ), ( ∂ζ ∂z )2 B( ∂ ∂ζ , ∂ ∂ζ ) dz4 = B( ∂ ∂z , ∂ ∂z ), B( ∂ ∂z , ∂ ∂z ) dz4 . Sunep¸c, to 4-diaforikì Φ1 eÐnai kalˆ orismèno. Sth sunèqeia ja doÔme ìti to Φ1 eÐnai olìmorfo 4-diaforikì. L mma 2.2.1. 'Estw f : M2 −→ Sn, n ≥ 3, elaqistik epifˆneia. Tìte isqÔoun: (i) (dH α 1 − 2iH α 1 ω12 + β H β 1 ωβα) ∧ φ = 0 gia kˆje α ≥ 3. (ii) To 4-diaforikì Φ1 eÐnai olìmorfo. Eidikˆ, an h epifˆneia Riemann M2 eÐnai omoiomorfik me th didiˆstath sfaÐra, tìte Φ1 = 0. Apìdeixh. (i) JumÐzoume ìti isqÔoun oi sqèseic ω1α = hα 11ω1 + hα 12ω2, ω2α = hα 12ω1 − hα 11ω2, dω1 = ω12 ∧ ω2, dω2 = −ω12 ∧ ω1. Qrhsimopoi¸ntac tic parapˆnw sqèseic, oi exis¸seic Codazzi dωjα = l ωjl ∧ ωlα + β ωjβ ∧ ωβα, j = 1, 2,
- 30. 22 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 gÐnontai β hβ 11ωβα ∧ ω2 + β hβ 12ωβα ∧ ω1 = −dhα 11 ∧ ω1 − dhα 12 ∧ ω2 − 2hα 11dω1 − 2hα 12dω2, β hβ 12ωβα ∧ ω1 − β hβ 11ωβα ∧ ω2 = −dhα 12 ∧ ω1 + dhα 11 ∧ ω2 − 2hα 12dω1 + 2hα 11dω2. Epeid H α 1 = hα 11 − ihα 12, apì ta parapˆnw kai apì tic sqèseic (2.3), (2.5) eÔkola sumperaÐnoume ìti (dH α 1 − 2iH α 1 ω12 + β H β 1 ωβα) ∧ φ = 0. (ii) UpologÐzoume: B(2,0) = B( ∂ ∂z , ∂ ∂z ) = E 2 B(e1, e1) − iB(e1, e2) . To sÔnolo {eα} apoteleÐ orjomonadiaÐo plaÐsio thc kˆjethc dèsmhc, ˆra B(2,0) = α B(2,0) , eα eα = E 2 α B(e1, e1), eα eα − i E 2 α B(e1, e2), eα eα = E 2 α hα 11eα − i E 2 α hα 12eα = E 2 α H α 1 eα. Telikˆ eÐnai Φ1 = B(2,0) , B(2,0) dz4 = E2 4 α (H α 1 )2 dz4 . Jètoume f1 := E2 4 α(H α 1 )2. Ja apodeÐxoume ìti h f1 eÐnai olìmorfh. Pollaplasiˆzontac th sqèsh pou apodeÐxame sto (i) me H α 1 , ajroÐzontac wc proc α kai kˆnontac qr sh twn sqèsewn (2.3), (2.4), (2.5) kai thc sqèshc α,β H β 1 H α 1 ωβα ∧ φ = 0, paÐrnoume df1 ∧ dz = 0 pou shmaÐnei ìti ∂f1 ∂z = 0. 'Ara to Φ1 eÐnai olìmorfo 4-diaforikì sth M2. Epiplèon, an h epifˆneia Riemann M2 eÐnai omoiomorfik me thn S2, tìte sÔmfwna me to Je¸rhma Riemann-Roch, lam- bˆnoume Φ1 = 0. Apì ed¸ kai sto ex c upojètoume ìti h f : M2 −→ Sn, n ≥ 3, eÐnai sumpag c, koresmènh, elaqistik epifˆneia gènouc mhdèn, dhlad to M2 eÐnai omoiomorfikì me thn S2.
- 31. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 23 Epeid Φ1|p = 0 gia kˆje p ∈ M2, èqoume B(2,0), B(2,0) (p) = 0, isodÔnama |B(e1, e1)|2 (p) − |B(e1, e2)|2 (p) − 2i B(e1, e1), B(e1, e2) (p) = 0. Apì ed¸ sumperaÐnoume ìti se kˆje shmeÐo tou M2 ta dianÔsmata B(e1, e1)|p kai B(e1, e2)|p eÐnai tou idÐou m kouc kai kˆjeta metaxÔ touc. SÔmfwna me thn Prìtash 1.3.1 se kˆje shmeÐo p tou M2 h èlleiyh E1(p) eÐnai kÔkloc me aktÐna κ1(p) = |B(e1, e1)|(p) = |B(e1, e2)|(p). Autì shmaÐnei ìti se kˆje shmeÐo p ∈ M2 èqoume dimN1 p f ∈ {0, 2}. An dimN1 p f = 0 gia kˆje shmeÐo p ∈ M2, tìte h deÔterh jemeli¸dhc morf B eÐnai tautotikˆ 0. Se aut thn perÐptwsh h embˆptish f eÐnai olikˆ gewdaisiak kai sÔmfwna me thn Parat rhsh 1.3.1, to f(M2) perièqetai se èna olikˆ gewdaisiakì didiˆstato upopolÔptugma thc Sn kai epomènwc to f(M2) eÐnai mia mègisth 2-sfaÐra thc Sn. 'Atopo, afoÔ h embˆptish eÐnai koresmènh. Epomènwc maxq∈M2 dimN1 q f = 2, to opoÐo shmaÐnei ìti n ≥ 4. An n = 4, tìte h diadikasÐa stamatˆ ed¸. An n ≥ 5, tìte suneqÐzoume th diadika- sÐa. 'Eqoume dh orÐsei to m koc thc deÔterhc jemeli¸douc morf c wc th sunˆrthsh B : M2 −→ R, me tÔpo B = |B(e1, e1)|2 + 2|B(e1, e2)|2 + |B(e2, e2)|2. Profan¸c h sunˆrthsh B eÐnai suneq c kai B = 2κ1. OrÐzoume to sÔnolo M1 := {p ∈ M2 : dimN1 p f = maxq∈M2 dimN1 q f}. Profan¸c isqÔei M1 = {p ∈ M2 : dimN1 p f = 2} = {p ∈ M2 : B (p) > 0} = {p ∈ M2 : κ1(p) > 0}. To M1 eÐnai mh kenì, afoÔ h f eÐnai koresmènh. Epiplèon, eÐnai anoiktì upo- sÔnolo tou M2, afoÔ eÐnai h antÐstrofh eikìna tou anoiktoÔ sunìlou (0, +∞) mèsw thc suneqoÔc sunˆrthshc B . Epomènwc, to M1 eÐnai prosanatolismèno didiˆstato polÔptugma Riemann kai sÔmfwna me thn Prìtash 2.1.1, eÐnai epifˆneia Riemann. Parat rhsh 2.2.1. To sÔnolo M1 eÐnai puknì uposÔnolo tou M2. Prˆgmati, ac upojèsoume ìti to eswterikì tou sunìlou Mc 1 = M2 − M1 den eÐnai kenì sÔnolo kai èstw U mia sunektik sunist¸sa tou int(Mc 1). Sto U isqÔei B|U = 0 kai sÔmfwna me thn Parat rhsh 1.3.1 to f(M2) eÐnai mia mègisth 2-sfaÐra thc Sn ìpwc parapˆnw. 'Atopo, afoÔ h f eÐnai koresmènh. H N1f|M1 eÐnai dianusmatik dèsmh me bajmÐda 2. Epomènwc h (T1f|M1 ⊕N1f|M1 )⊥ eÐnai dianusmatik dèsmh me bajmÐda n − 4 kai h jemeli¸dhc morf trÐthc tˆxhc thc f periorismènh sto M1 eÐnai (3,1)-tanustikì pedÐo B2|M1 : ∆(M1) × ∆(M1) × ∆(M1) −→ Γ (T1 f|M1 ⊕ N1 f|M1 )⊥ ,
- 32. 24 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 (X1, X2, X3) −→ B2|M1 (X1, X2, X3) = f X1 f X2 df(X3) (T1f|M1 ⊕N1f|M1 )⊥ . MigadikopoioÔme tic dianusmatikèc dèsmec TM1, T1f|M1 , N1f|M1 , epekteÐnoume C- grammikˆ th B2|M1 kai apoktoÔme to migadikì (3,1)-tanustikì pedÐo B2|M1 : Γ(TM1 ⊗C)×Γ(TM1 ⊗C)×Γ(TM1 ⊗C) −→ Γ (T1 f|M1 ⊕N1 f|M1 )⊥ ⊗C . Gia kˆje migadikì qˆrth (U, z) tou M1 me z = x + iy kai , = E|dz|2 to migadikì (3,1)-tanustikì pedÐo B2|M1 analÔetai sto U wc ex c: B2|M1 = B2| (3,0) M1 dz3 + B2| (2,1) M1 dz2 dz + B2| (1,2) M1 dzdz2 + B2| (0,3) M1 dz3 . GnwrÐzoume apì to L mma 1.3.2 ìti B2|M1 (X, e1, e1) + B2|M1 (X, e2, e2) = 0. Gia e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y h teleutaÐa sqèsh gÐnetai B2|M1 (X, ∂ ∂z , ∂ ∂z ) = 0. Epomènwc, eÐnai B2|M1 = B2| (3,0) M1 dz3 + B2| (0,3) M1 dz3 , ìpou B2| (3,0) M1 = B2|M1 ∂ ∂z , ∂ ∂z , ∂ ∂z , B2| (0,3) M1 = B2|M1 ∂ ∂z , ∂ ∂z , ∂ ∂z kai isqÔei B2| (3,0) M1 = B2| (0,3) M1 . Sto M1 orÐzoume to 6-diaforikì Φ2 := B2| (3,0) M1 , B2| (3,0) M1 dz6 . Ja deÐxoume ìti to Φ2 eÐnai kalˆ orismèno, dhlad anexˆrthto thc epilog c qˆrth. Gia to lìgo autì jewroÔme migadikoÔc qˆrtec tou M1, (U, ϕ) me migadik suntetagmènh z kai (V, ψ) me migadik suntetagmènh ζ ¸ste U ∩ V = ∅. To M1 eÐnai epifˆneia Riemann sunep¸c oi apeikonÐseic ψ ◦ϕ−1, ϕ◦ψ−1 eÐnai olìmorfec, dhlad ∂ζ ∂z = 0 kai ∂z ∂ζ = 0. Sto U ∩ V èqoume tic sqèseic ∂ ∂z = ∂ζ ∂z ∂ ∂ζ , dζ = ∂ζ ∂z dz kai isqÔei B2|M1 ∂ ∂ζ , ∂ ∂ζ , ∂ ∂ζ , B2|M1 ∂ ∂ζ , ∂ ∂ζ , ∂ ∂ζ dζ6 = = B2|M1 ∂ ∂ζ , ∂ ∂ζ , ∂ ∂ζ , B2|M1 ∂ ∂ζ , ∂ ∂ζ , ∂ ∂ζ ( ∂ζ ∂z )6 dz6 = ( ∂ζ ∂z )3 B2|M1 ∂ ∂ζ , ∂ ∂ζ , ∂ ∂ζ , ( ∂ζ ∂z )3 B2|M1 ∂ ∂ζ , ∂ ∂ζ , ∂ ∂ζ dz6 = B2|M1 ∂ ∂z , ∂ ∂z , ∂ ∂z , B2|M1 ∂ ∂z , ∂ ∂z , ∂ ∂z dz6 . ToÔto dhl¸nei ìti to Φ2 eÐnai kalˆ orismèno 6-diaforikì sto M1.
- 33. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 25 L mma 2.2.2. 'Estw f : M2 −→ Sn, n ≥ 5, sumpag c, koresmènh, elaqistik epifˆneia gènouc mhdèn. Gia kˆje migadikì qˆrth (U, z) tou M2 me z = x + iy kai , = E|dz|2 upˆrqei orjomonadiaÐo plaÐsio katˆ m koc thc f|M1∩U me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y kai me kˆjeto mèroc {eα}, ìpou e3 = 1 κ1 B(e1, e1), e4 = 1 κ1 B(e1, e2). Epiplèon, sto M1 ∩ U isqÔoun 1: (i) (d log κ1 + iω34) ∧ φ = 2iω12 ∧ φ, (ii) ω3α(e1) = −ω4α(e2), ω3α(e2) = ω4α(e1), α = 5, ..., n, (iii) (dH α 2 − 3iH α 2 ω12 + β≥5 H β 2 ωβα) ∧ φ = 0, α = 5, ..., n, (iv) To 6-diaforikì Φ2 eÐnai olìmorfo. Apìdeixh. Lìgw tou L mmatoc 2.2.1 èqoume Φ1 = 0. Sunep¸c ta B(e1, e1), B(e1, e2) eÐnai isom kh kai kˆjeta metaxÔ touc. Epomènwc, mporoÔme na epilèxoume orjo- monadiaÐo plaÐsio {e3, e4} thc dianusmatik c dèsmhc N1f|M1∩U tètoio ¸ste e3 = 1 κ1 B(e1, e1), e4 = 1 κ1 B(e1, e2). (i) Lìgw thc epilog c tou plaisÐou {eα}, eÐnai H3 1 |M1∩U = κ1, H4 1 |M1∩U = iκ1 kai Hα 1 |M1∩U = 0 gia α ≥ 5. Apì to L mma 2.2.1(i) gia α = 3 lambˆnoume sto M1 ∩ U th sqèsh (d log κ1 + iω34) ∧ φ = 2iω12 ∧ φ. (ii) Sto M1 ∩U lambˆnontac upìyh ton orismì thc B2 kai touc tÔpouc twn Gauss kai Weingarten èqoume B2|M1 (e1, e1, e1) = f e1 f e1 df(e1) (T1f|M1 ⊕N1f|M1 )⊥ = f e1 B(e1, e1) (T1f|M1 ⊕N1f|M1 )⊥ = f e1 (κ1e3) (T1f|M1 ⊕N1f|M1 )⊥ = e1(κ1)e3 ∈N1f|M1 +κ1 f e1 e3 (T1f|M1 ⊕N1f|M1 )⊥ = κ1( f e1 e3)(T1f|M1 ⊕N1f|M1 )⊥ = κ1( ⊥ e1 e3)(T1f|M1 ⊕N1f|M1 )⊥ . Epeid o deÔteroc kˆjetoc q¸roc thc f eÐnai upìqwroc tou dianusmatikoÔ q¸rou pou parˆgetai apì ta {e5, ..., en} èqoume B2|M1 (e1, e1, e1) = κ1 n α=5 ⊥ e1 e3, eα eα = κ1 n α=5 ω3α(e1)eα. 'Omwc, lìgw tou L mmatoc 1.3.2 eÐnai B2|M1 (e1, e1, e1) = −B2|M1 (e2, e1, e2) = −κ1( ⊥ e2 e4)(T1f|M1 ⊕N1f|M1 )⊥ = −κ1 n α=5 ω4α(e2)eα. 1 Epeid to M1 eÐnai puknì uposÔnolo tou M2 (Parat rhsh 2.2.1) eÐnai M1 ∩ U = ∅.
- 34. 26 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 Sunep¸c isqÔei ω3α(e1) = −ω4α(e2), α = 5, ..., n. 'Omoia, B2|M1 (e1, e1, e2) = κ1 n α=5 ω4α(e1)eα = κ1 n α=5 ω3α(e2)eα kai ω3α(e2) = ω4α(e1), α = 5, ..., n. (iii) Sto M1 ∩ U gia α ≥ 5 eÐnai hα (2),1 = κ1ω3α(e1) kai hα (2),2 = κ1ω3α(e2). EÐnai fanerì ìti oi sunart seic hα (2),1, hα (2),2 : M1 ∩ U −→ R eÐnai diaforÐsimec kai tètoiec ¸ste B2|M1 (e1, e1, e1) = n α=5 hα (2),1eα, B2|M1 (e1, e1, e2) = n α=5 hα (2),2eα. Epiplèon, sto M1 ∩ U oi migadikèc sunart seic Hα 2 eÐnai diaforÐsimec. Lìgw tou (ii) isqÔei h sqèsh Hα 2 φ = κ1ω3α + iκ1ω4α. EpÐshc, sto M1 ∩ U gia α ≥ 5 kai gia j ∈ {1, 2} èqoume hα 1j = B(e1, ej), eα = 0, hα 2j = B(e2, ej), eα = 0. Epomènwc sto M1 ∩ U isqÔei ω1α = j hα 1jωj = 0 kai ω2α = j hα 2jωj = 0 kai oi exis¸seic Ricci dÐnoun dω3α = ω34 ∧ ω4α + β≥5 ω3β ∧ ωβα, dω4α = −ω34 ∧ ω3α + β≥5 ω4β ∧ ωβα. ParagwgÐzontac exwterikˆ th sqèsh Hα 2 φ = κ1ω3α +iκ1ω4α, qrhsimopoi¸ntac tic parapˆnw exis¸seic Ricci kai th sqèsh (2.5) paÐrnoume dHα 2 ∧ φ + iHα 2 ω12 ∧ φ = Hα 2 d(log κ1) ∧ φ − iHα 2 ω34 ∧ φ + β≥5 Hβ 2 φ ∧ ωβα. PaÐrnoume th suzug sqèsh aut c kai lambˆnontac upìyh th sqèsh pou apodeÐxame sto (i) èqoume to zhtoÔmeno. (iv) Epeid B2| (3,0) M1 = B2|M1 ( ∂ ∂z , ∂ ∂z , ∂ ∂z ) = 1 2 E 3 2 α≥5 H α 2 eα, èqoume Φ2 = B2| (3,0) M1 , B2| (3,0) M1 dz6 = 1 4 E3 α,β≥5 H α 2 H β 2 eα, eβ dz6 = 1 4 E3 α≥5 (H α 2 )2 dz6 . Jètoume f2 := 1 4E3 α≥5(H α 2 )2. Ja apodeÐxoume ìti h f2 eÐnai olìmorfh. Pollaplasi- ˆzoume th sqèsh pou apodeÐxame sto (iii) me H α 2 , ajroÐzoume gia α ≥ 5, kˆnoume qr sh twn sqèsewn (2.4), (2.5) kai α,β≥5 H β 2 H α 2 ωβα ∧ φ = 0 kai brÐskoume df2 ∧ dz = 0. Autì shmaÐnei ìti ∂f2 ∂z = 0 kai epomènwc to Φ2 eÐnai olìmorfo sto M1 ∩ U.
- 35. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 27 Stìqoc mac eÐnai na deÐxoume ìti to Φ2 eÐnai tautotikˆ mhdèn. Gia autì ja deÐxoume ìti to Mc 1 eÐnai peperasmèno sÔnolo kai ìti upˆrqei upodèsmh thc kˆjethc dèsmhc me bajmÐda 2 orismènh sto M2, h opoÐa sto M1 tautÐzetai me th dèsmh N1f|M1 . Gia to skopì autì qreiazìmaste to epìmeno l mma to opoÐo ofeÐletai ston Chern [8] (blèpe epÐshc [5] [11]). L mma 2.2.3. An f1, ..., fm : U ⊂ C −→ C eÐnai migadikèc sunart seic oi opoÐec se mia perioq U tou mhdenìc ikanopoioÔn to diaforikì sÔsthma ∂fi ∂z = j aijfj, ìpou aij : U −→ C eÐnai diaforÐsimec sunart seic kai i, j = 1, ..., m, tìte eÐte f1 = ... = fm = 0, eÐte oi koinèc rÐzec twn f1, ..., fm eÐnai memonwmènec kai mˆlista upˆrqei jetikìc akèraioc l kai diaforÐsimec sunart seic f∗ 1 , ..., f∗ m : U ⊂ C −→ C ¸ste na isqÔei fi(z) = zlf∗ i (z) me f∗ 1 (0), ..., f∗ m(0) = (0, ..., 0). L mma 2.2.4. An f : M2 −→ Sn, n ≥ 5, sumpag c, koresmènh, elaqistik epi- fˆneia gènouc mhdèn, tìte to sÔnolo Mc 1 eÐnai peperasmèno. Epiplèon, upˆrqei dia- nusmatik upodèsmh N∗1f thc kˆjethc dèsmhc Nf me bajmÐda 2 ¸ste N∗1f|M1 = N1f|M1 . Apìdeixh. 'Estw p ∈ Mc 1. JewroÔme gÔrw apì to p migadikì qˆrth (U, z) me z(p) = 0, z = x + iy, , = E|dz|2 kai orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y , kai kˆjeto mèroc {eα}. Sto U èqoume to migadikì (1,0)-tanustikì pedÐo φ = ω1 + iω2 kai tic migadikèc sunart seic Hα 1 = hα 11 + ihα 12 gia α = 3, ..., n. Me th bo jeia twn sqèsewn dH α 1 = ∂H α 1 ∂z dz + ∂H α 1 ∂z dz, ωαβ = ωαβ( ∂ ∂z )dz + ωαβ( ∂ ∂z )dz kai twn sqèsewn (2.4), (2.5), apì to L mma 2.2.1(i) lambˆnoume ∂H α 1 ∂z = β gαβH β 1 , ìpou gαβ := −ωβα( ∂ ∂z ) − δαβd log E( ∂ ∂z ) eÐnai diaforÐsimec sunart seic kai δαβ eÐnai to dèlta tou Kronecker. SÔmfwna me to L mma 2.2.3 eÐte H α 1 = 0 gia kˆje α = 3, ..., n, eÐte oi koinèc rÐzec twn H α 1 eÐnai memonwmènec kai upˆrqei jetikìc akèraioc l1 kai diaforÐsimec sunart seic H ∗α 1 : U −→ C ¸ste na isqÔei H α 1 = zl1 H ∗α 1 (2.6) gia kˆje α kai H ∗3 1 (0), ..., H ∗n 1 (0) = (0, ..., 0). An tan H α 1 = 0 gia kˆje α = 3, ..., n, tìte B|U = 0. 'Atopo, afoÔ sÔmfwna me thn Parat rhsh 1.3.1 h f den ja tan koresmènh. 'Ara oi koinèc rÐzec twn H α 1 eÐnai memonwmènec. Oi koinèc rÐzec twn H α 1 , α = 3, ..., n, eÐnai akrib¸c ta shmeÐa ìpou o pr¸toc kˆjetoc q¸roc gÐnetai mhdenikìc. Autì shmaÐnei ìti to sÔnolo Mc 1 apoteleÐtai apì memonwmèna shmeÐa kai afoÔ to M2 eÐnai sumpagèc, eÐnai peperasmèno sÔnolo.
- 36. 28 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 Epeid to Mc 1 eÐnai peperasmèno sÔnolo, mporoÔme na jewr soume gÔrw apì to tuqìn shmeÐo p ∈ Mc 1 migadikì qˆrth (U, z) me z(p) = 0 gia ton opoÐo ìmwc isqÔei U ∩ Mc 1 = {p}. Epilègoume me ton sun jh trìpo topikì orjomonadiaÐo plaÐsio katˆ m koc thc f. 'Eqoume dh dei ìti se kˆje shmeÐo q tou M2 h èlleiyh E1(q) eÐnai kÔkloc, opìte ta B(e1, e1)|q, B(e1, e2)|q eÐnai tou idÐou m kouc kai kˆjeta metaxÔ touc gia kˆje q ∈ U. Autì shmaÐnei ìti to diˆnusma B(e1, e1)|q −iB(e1, e2)|q eÐnai isotropikì, dhlad B(e1, e1)|q − iB(e1, e2)|q, B(e1, e1)|q − iB(e1, e2)|q = 0 gia kˆje q ∈ U. 'Omwc lìgw thc (2.6) èqoume B(e1, e1) − iB(e1, e2) = α H α 1 eα = zl1 α H ∗α 1 eα, ˆra sto U èqoume z2l1 α H ∗α 1 eα, α H ∗α 1 eα = 0. Sto U −{p} eÐnai z = 0, epomènwc sto U − {p} èqoume th sqèsh α H ∗α 1 eα, α H ∗α 1 eα = 0. Lìgw sunèqeiac, h sqèsh aut isqÔei kai sto p. Epomènwc gia kˆje q ∈ U isqÔei Re α H ∗α 1 eα (q) = Im α H ∗α 1 eα (q) = 0 kai Re α H ∗α 1 eα |q, Im α H ∗α 1 eα |q = 0. Gia kˆje q ∈ U orÐzoume ton didiˆstato upìqwro N∗1 q f tou kˆjetou q¸rou thc f sto q N∗1 q f := span Re n α=3 H ∗α 1 eα |q, Im n α=3 H ∗α 1 eα |q . Sto U èqoume th dèsmh bajmÐdac 2 N∗1 f|U = q∈U N∗1 q f. Epiplèon, epeid gia q ∈ U − {p} èqoume N1 q f = span{B(e1, e1)|q, B(e1, e2)|q} = span Re B(e1, e1)|q − iB(e1, e2)|q , Im B(e1, e1)|q − iB(e1, e2)|q = span Re zl1 α H ∗α 1 eα |q, Im zl1 α H ∗α 1 eα |q = span Re α H ∗α 1 eα |q, Im α H ∗α 1 eα |q , isqÔei N1 q f = span{Re( α H ∗α 1 eα)|q, Im( α H ∗α 1 eα)|q}, q ∈ U − {p}, {0}, q = p, ˆra N∗1f|U−{p} = N1f|U−{p}. An epanalˆboume thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou Mc 1, pou ìpwc eÐdame eÐnai peperasmèna to pl joc, apoktoÔme th dianusmatik dèsmh N∗1f me bajmÐda 2 sto M2 gia thn opoÐa isqÔei N∗1f|M1 = N1f|M1 .
- 37. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 29 Sto M1 èqoume to migadikì (3,1)-tanustikì pedÐo B2|M1 kai to olìmorfo 6- diaforikì Φ2. Ja deÐxoume ìti Φ2 = 0. Gia to skopì autì orÐzoume thn apeikìnish B∗ 2 : ∆(M2 ) × ∆(M2 ) × ∆(M2 ) −→ Γ (T1 f ⊕ N∗1 f)⊥ , (X1, X2, X3) −→ B∗ 2(X1, X2, X3) = f X1 f X2 df(X3) (T1f⊕N∗1f)⊥ . H B∗ 2 eÐnai D(M2)-grammik wc proc thn pr¸th metablht thc. Ja deÐxoume ìti eÐnai summetrik kai epomènwc ja eÐnai D(M2)-grammik wc proc ìlec tic metablhtèc thc. L mma 2.2.5. IsqÔoun ta akìlouja: (i) To B∗ 2 eÐnai summetrikì (3, 1)-tanustikì pedÐo. Epiplèon, gia kˆje X ∈ ∆(M2) kai {e1, e2} tuqìn topikì orjomonadiaÐo plaÐsio tou M2 isqÔei B∗ 2(X, e1, e1) + B∗ 2(X, e2, e2) = 0. (ii) Gia X1, X2, X3 ∈ ∆(M2) isqÔei B∗ 2(X1, X2, X3) = f X1 B(X2, X3) (T1f⊕N∗1f)⊥ . (iii) Gia kˆje p ∈ M2 èqoume B∗ 2|p = B2|p, p ∈ M1, 0, p ∈ Mc 1. Apìdeixh. (i) Lìgw tou L mmatoc 2.2.4, isqÔei N∗1f|M1 = N1f|M1 . Epomènwc apì touc orismoÔc twn B2 kai B∗ 2 eÐnai B∗ 2|M1 = B2|M1 . 'Ara h B∗ 2|M1 eÐnai summetrik kai plhroÐ th sqèsh B∗ 2|M1 (X, e1, e1) + B∗ 2|M1 (X, e2, e2) = 0. GnwrÐzoume apì to L mma 2.2.4 ìti to Mc 1 eÐnai peperasmèno sÔnolo kai epeid h B∗ 2|M1 eÐnai summetrikì (3,1)- tanustikì pedÐo, lìgw sunèqeiac, h B∗ 2 eÐnai summetrik kai plhroÐ thn en lìgw sqèsh kai sta shmeÐa tou Mc 1. (ii) 'Estw X1, X2, X3 ∈ ∆(M2). Qrhsimopoi¸ntac ton tÔpo tou Gauss èqoume B∗ 2(X1, X2, X3) = f X1 f X2 df(X3) (T1f⊕N∗1f)⊥ = f X1 df( f X2 X3) ∈T2f + f X1 B(X2, X3) (T1f⊕N∗1f)⊥ = f X1 B(X2, X3) (T1f⊕N∗1f)⊥ . (iii) 'Eqoume dh dei ìti B∗ 2|M1 = B2|M1 . Ja deÐxoume ìti B∗ 2|Mc 1 = 0. Gia to skopì autì jewroÔme tuqìn topikì orjomonadiaÐo plaÐsio {e∗ 3, e∗ 4} thc dianusmatik c dèsmhc N∗1f. Tìte lìgw twn Lhmmˆtwn 1.3.1 kai 2.2.4 èqoume B(e1, e1) = h∗3 11e∗ 3 + h∗4 11e∗ 4, B(e1, e2) = h∗3 12e∗ 3 + h∗4 12e∗ 4,
- 38. 30 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 ìpou h∗3 11, h∗4 11, h∗3 12, h∗4 12 eÐnai diaforÐsimec sunart seic. Gia kˆje p ∈ Mc 1 isqÔei B|p = 0, epomènwc h∗3 11(p) = h∗4 11(p) = h∗3 12(p) = h∗4 12(p) = 0. Kˆnontac qr sh tou tÔpou tou Weingarten, upologÐzoume B∗ 2(e1, e1, e1) = f e1 B(e1, e1) (T1f⊕N∗1f)⊥ = ⊥ e1 (h∗3 11e∗ 3 + h∗4 11e∗ 4) (T1f⊕N∗1f)⊥ = e1(h∗3 11)e∗ 3 + e1(h∗4 11)e∗ 4 (T1f⊕N∗1f)⊥ + h∗3 11( ⊥ e1 e∗ 3)(T1f⊕N∗1f)⊥ + h∗4 11( ⊥ e1 e∗ 4)(T1f⊕N∗1f)⊥ = h∗3 11( ⊥ e1 e∗ 3)(T1f⊕N∗1f)⊥ + h∗4 11( ⊥ e1 e∗ 4)(T1f⊕N∗1f)⊥ . 'Omoia B∗ 2(e1, e1, e2) = h∗3 12( ⊥ e1 e∗ 3)(T1f⊕N∗1f)⊥ + h∗4 12( ⊥ e1 e∗ 4)(T1f⊕N∗1f)⊥ . Sunep¸c gia kˆje shmeÐo p tou Mc 1 isqÔei B∗ 2(e1, e1, e1)|p = B∗ 2(e1, e1, e2)|p = 0 kai lìgw tou (i) èqoume ìti B∗ 2|p = 0. MigadikopoioÔme tic dianusmatikèc dèsmec TM2, T1f, N∗1f kai epekteÐnoume C- grammikˆ to (3,1)-tanustikì pedÐo B∗ 2, opìte apoktoÔme to migadikì (3,1)-tanustikì pedÐo B∗ 2 : Γ(TM2 ⊗ C) × Γ(TM2 ⊗ C) × Γ(TM2 ⊗ C) −→ Γ (T1 f ⊕ N∗1 f)⊥ ⊗ C . To migadikì (3,1)-tanustikì pedÐo B∗ 2 dèqetai thn anˆlush B∗ 2 = B ∗ (3,0) 2 dz3 + B ∗ (2,1) 2 dz2 dz + B ∗ (1,2) 2 dzdz2 + B ∗ (0,3) 2 dz3 . Apì to L mma 2.2.5(i) èqoume B∗ 2(X, e1, e1) + B∗ 2(X, e2, e2) = 0 gia X ∈ ∆(U) kai {e1, e2} topikì orjomonadiaÐo plaÐsio tou U. Gia e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y , epeid ∂ ∂x = ∂ ∂z + ∂ ∂z kai ∂ ∂y = i( ∂ ∂z − ∂ ∂z ), h teleutaÐa sqèsh gÐnetai B∗ 2(X, ∂ ∂z , ∂ ∂z ) = 0. Epomènwc B∗ 2 = B ∗ (3,0) 2 dz3 + B ∗ (0,3) 2 dz3 , ìpou B ∗ (3,0) 2 = B∗ 2 ∂ ∂z , ∂ ∂z , ∂ ∂z , B ∗ (0,3) 2 = B∗ 2 ∂ ∂z , ∂ ∂z , ∂ ∂z kai isqÔei B ∗ (3,0) 2 = B ∗ (0,3) 2 . Sto U orÐzoume to 6-diaforikì Φ∗ 2 := B ∗ (3,0) 2 , B ∗ (3,0) 2 dz6 .
- 39. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 31 To Φ∗ 2 eÐnai kalˆ orismèno se ìlo to M2 kai lìgw tou L mmatoc 2.2.5(iii), isqÔei Φ∗ 2|p = Φ2|p, p ∈ M1, 0, p ∈ Mc 1. JewroÔme èna shmeÐo p ∈ Mc 1 kai èstw (U, z) migadikìc qˆrthc gÔrw apì to p me z(p) = 0 kai U ∩ Mc 1 = {p}. Apì to L mma 2.2.2(iv) to Φ∗ 2 eÐnai olìmorfo sto U −{p}. Epeid eÐnai kai suneqèc sto shmeÐo p, sumperaÐnoume ìti to Φ∗ 2 eÐnai olìmorfo sto U [1]. Epanalambˆnontac thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou Mc 1, èqoume ìti to Φ∗ 2 eÐnai olìmorfo 6-diaforikì se ìlo to M2. Apì to Je¸rhma Riemann-Roch èqoume Φ∗ 2 = 0. Sunep¸c deÐxame ìti Φ2 = 0. Katˆ sunèpeia gia kˆje p ∈ M1 isqÔei Φ2|p = 0 isodÔnama B2(e1, e1, e1) 2 (p) − B2(e1, e1, e2) 2 (p) − 2i B2(e1, e1, e1), B2(e1, e1, e2) (p) = 0. Apì thn teleutaÐa sqèsh, lìgw thc Prìtashc 1.3.1 prokÔptei ìti se kˆje shmeÐo p ∈ M1 h èlleiyh E2(p) eÐnai kÔkloc me aktÐna κ2(p) = B2(e1, e1, e1) (p) = B2(e1, e1, e2) (p). Epomènwc, h diˆstash tou kˆjetou q¸rou deÔterhc tˆxhc sto tuqìn p ∈ M1 eÐnai 0 2. An dimN2 p f = 0 gia kˆje p ∈ M1, tìte sÔmfwna me thn Parat rhsh 1.3.1, f(M2) ⊂ S4, ìpou S4 eÐnai mia mègisth 4-sfaÐra thc Sn. 'Atopo, afoÔ èqoume dh anafèrei pwc exetˆzoume koresmènh elaqistik epifˆneia f : M2 −→ Sn, n ≥ 5. Sunep¸c, maxp∈M2 dimN2 p f = 2 to opoÐo shmaÐnei ìti n ≥ 6. An n = 6, tìte h diadikasÐa stamatˆ ed¸ kai an n ≥ 7, tìte suneqÐzei. EÐnai t¸ra eÔlogo h diadikasÐa aut na genikeÔetai epagwgikˆ. Gia lìgouc plh- rìthtac thc apìdeixhc twn apotelesmˆtwn, ja perigrˆyoume leptomer¸c to epagwgikì b ma. 'Estw f : M2 −→ Sn, n ≥ 7, sumpag c, koresmènh, elaqistik epifˆneia gènouc mhdèn kai jetikìc akèraioc r me 2 ≤ r ≤ [n−1 2 ] − 1. Upojètoume ìti gia kˆje s ∈ {2, ..., r} isqÔoun ta akìlouja: (I) Upˆrqoun dianusmatikèc upodèsmec N∗1f, ..., N∗r−1f thc kˆjethc dèsmhc Nf me bajmÐda 2 uperˆnw tou M2 ètsi ¸ste N∗s−1f|Ms−1 = Ns−1f|Ms−1 , ìpou Ms−1 := {p ∈ M2 : B∗ s−1 (p) > 0} kai B∗ s := 2 j1,...,js+1=1 |B∗ s (ej1 , ..., ejs+1 )|2 eÐnai to m koc tou summetrikoÔ (s + 1, 1)-tanustikoÔ pedÐou B∗ s : ∆(M2 ) × ... × ∆(M2 ) s+1 −→ Γ (T1 f ⊕ N∗1 f ⊕ ... ⊕ N∗s−1 f)⊥ , (X1, ..., Xs+1) −→ B∗ s (X1, ..., Xs+1) = f X1 ... f Xs df(Xs+1) (T1f⊕N∗1f⊕...⊕N∗s−1f)⊥ .
- 40. 32 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 Jètoume B∗ 1 = B. Gia to (s + 1, 1)-tanustikì pedÐo B∗ s isqÔoun: (i) An X1, ..., Xs−1 ∈ ∆(M2) kai {e1, e2} eÐnai opoiod pote topikì orjomonadiaÐo plaÐsio tou M2, tìte B∗ s (X1, ..., Xs−1, e1, e1) + B∗ s (X1, ..., Xs−1, e2, e2) = 0. (ii) An X1, ..., Xs+1 ∈ ∆(M2), tìte B∗ s (X1, ..., Xs+1) = f X1 B∗ s−1(X2, ..., Xs+1) (T1f⊕N∗1f⊕...⊕N∗s−1f)⊥ . (iii) B∗ s |Ms−1 = Bs|Ms−1 kai B∗ s |Mc s−1 = 0. Epiplèon, ta sÔnola Ms−1 eÐnai mh kenˆ, anoiktˆ kai puknˆ uposÔnola tou M2, me M1 ⊃ M2 ⊃ ... ⊃ Mr−1 kai to sumpl rwma kajenìc apì autˆ sto M2 eÐnai peperasmèno sÔnolo. (II) MigadikopoioÔme tic dianusmatikèc dèsmec T1f, N∗1f, ..., N∗r−1f, epekteÐnoume C-grammikˆ ta B∗ s , opìte apoktoÔme ta migadikˆ (s + 1, 1)-tanustikˆ pedÐa B∗ s : Γ(TM2 ⊗ C) × ... × Γ(TM2 ⊗ C) s+1 −→ Γ (T1 f ⊕ N∗1 f ⊕ ... ⊕ N∗s−1 f)⊥ ⊗ C . To B∗ s wc migadikì (s + 1, 1)-tanustikì pedÐo dèqetai thn anˆlush B∗ s = p+q=s+1 B∗ (p,q) s dzp dzq . Se èna migadikì qˆrth (U, z) tou M2 me z = x + iy kai , = E|dz|2 jewroÔme topikì orjomonadiaÐo plaÐsio {e1, e2} me e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y . IsqÔei h sqèsh B∗ s (X1, ..., Xs−1, e1, e1) + B∗ s (X1, ..., Xs−1, e2, e2) = 0, gia kˆje X1, ..., Xs−1 ∈ ∆(U). Epeid ∂ ∂x = ∂ ∂z + ∂ ∂z , ∂ ∂y = i( ∂ ∂z − ∂ ∂z ), h teleutaÐa sqèsh gÐnetai B∗ s (X1, ..., Xs−1, ∂ ∂z , ∂ ∂z ) = 0. Epomènwc, to B∗ s èqei thn anˆlush B∗ s = B∗ (s+1,0) s dzs+1 + B∗ (0,s+1) s dzs+1 , ìpou B∗ (s+1,0) s = B∗ s ∂ ∂z , ..., ∂ ∂z , B∗ (0,s+1) s = B∗ s ∂ ∂z , ..., ∂ ∂z kai isqÔei B ∗ (s+1,0) s = B ∗ (0,s+1) s . Upojètoume ìti ta kalˆ orismèna migadikˆ (2s + 2)-diaforikˆ Φ∗ s := B∗ (s+1,0) s , B∗ (s+1,0) s dz2s+2 eÐnai ek tautìthtoc mhdèn.
- 41. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 33 (III) Gia kˆje migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2 upˆrqei orjomonadiaÐo plaÐsio katˆ m koc thc f|Mr−1 me efaptìmeno mèroc {e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y } kai kˆjeto mèroc {eα} ètsi ¸ste e3 = 1 κ1 B(e1, e1), e4 = 1 κ1 B(e1, e2), e5 = 1 κ2 B2(e1, e1, e1), e6 = 1 κ2 B2(e1, e1, e2), . . . e2r−1 = 1 κr−1 Br−1(e1, ..., e1), e2r = 1 κr−1 Br−1(e1, ..., e1, e2), ìpou κs−1 > 0, gia to opoÐo upojètoume ìti sto Ms−1 ∩ U isqÔoun oi sqèseic (d log κs−1 + iω2s−1,2s) ∧ φ = isω12 ∧ φ kai dH α s − i(s + 1)H α s ω12 + β≥2s+1 H β s ωβα ∧ φ = 0, α ≥ 2s + 1. Ja deÐxoume ìti ta (I), (II), (III) isqÔoun kai gia s = r + 1. Apìdeixh. Lìgw thc epagwgik c upìjeshc, isqÔei Φ∗ r = 0 sto M2 kai sunep¸c Φ∗ r|Mr−1 = 0. Ja deÐxoume ìti h èlleiyh Er(p) eÐnai kÔkloc se kˆje shmeÐo p ∈ Mr−1. To Mr−1, wc mh kenì kai anoiktì uposÔnolo tou prosanatolismènou poluptÔg- matoc Riemann M2, eÐnai kai autì prosanatolismèno polÔptugma Riemann. 'Estw p èna tuqìn shmeÐo tou Mr−1. GÔrw apì to p ∈ Mr−1 jewroÔme migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2. 'Estw {e1, e2} topikì orjomonadiaÐo plaÐsio sto U me e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y . Sto U èqoume ∂ ∂z = 1 2( ∂ ∂x − i ∂ ∂y ) = 1 2 √ E(e1 − ie2). Gia to (2r + 2)-diaforikì Φ∗ r isqÔei Φ∗ r|p = 0 an kai mìno an B∗ r ( ∂ ∂z , ..., ∂ ∂z )|p, B∗ r ( ∂ ∂z , ..., ∂ ∂z )|p = 0, isodÔnama lìgw thc sqèshc B∗ r |Mr−1 = Br|Mr−1 , Br( ∂ ∂z , ..., ∂ ∂z )|p, Br( ∂ ∂z , ..., ∂ ∂z )|p = 0. 'Omwc èqoume Br( ∂ ∂z , ..., ∂ ∂z ) = Br √ E 2 (e1 − ie2), ..., √ E 2 (e1 − ie2)
- 42. 34 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 = 1 2r+1 E r+1 2 r+1 m=0 r + 1 m Br(e1, ..., e1, −ie2, ..., −ie2 m ). SumbolÐzoume me I to sÔnolo twn ˆrtiwn arijm¸n tou sunìlou {1, ..., r + 1} kai me J to sÔnolo twn peritt¸n arijm¸n tou, epomènwc Br( ∂ ∂z , ..., ∂ ∂z ) = 1 2r+1 E r+1 2 m∈I,m=2l (−1)l r + 1 m Br(e1, ..., e1, e2, ..., e2 m ) − i 1 2r+1 E r+1 2 m∈J,m=2l+1 (−1)l r + 1 m Br(e1, ..., e1, e2, ..., e2 m ). Epeid apì to L mma 1.3.2 isqÔei Br(e1, ..., e1) + Br(e1, ..., e1, e2, e2) = 0, eÐnai Br(e1, ...e1, e2, ..., e2 m ) = (−1)lBr(e1, ..., e1), m = 2l, (−1)lBr(e1, ..., e1, e2), m = 2l + 1, kai epomènwc, Br( ∂ ∂z , ..., ∂ ∂z ) = 1 2r+1 E r+1 2 Br(e1, ..., e1) m∈I r + 1 m − i 1 2r+1 E r+1 2 Br(e1, ..., e1, e2) m∈J r + 1 m . 'Omwc isqÔei m∈I r + 1 m = m∈J r + 1 m = 2r kai telikˆ èqoume Br( ∂ ∂z , ..., ∂ ∂z ) = 1 2 E r+1 2 Br(e1, ..., e1) − i 1 2 E r+1 2 Br(e1, ..., e1, e2). Opìte, apì th sqèsh Φ∗ r|p = 0 sunˆgetai ìti sto Mr−1 isqÔoun |Br(e1, ..., e1)| = |Br(e1, ..., e1, e2)|, Br(e1, ..., e1), Br(e1, ..., e1, e2) = 0. Lìgw thc Prìtashc 1.3.1 èqoume ìti h èlleiyh Er(p) eÐnai kÔkloc se kˆje shmeÐo tou Mr−1. Autì shmaÐnei ìti dimNr p f ∈ {0, 2} gia kˆje p ∈ Mr−1. An upojèsoume ìti dimNr p f = 0 gia kˆje p ∈ Mr−1, tìte apì thn Parat rhsh 1.3.1 upˆrqei mègisth 2r-sfaÐra thc Sn ¸ste f(M2) ⊂ S2r. 'Atopo afoÔ h embˆptish eÐnai koresmènh sthn Sn. 'Ara, upˆrqei shmeÐo p ∈ Mr−1 ¸ste dimNr p f = 2, iso- dÔnama B∗ r (p) > 0, to opoÐo shmaÐnei ìti to sÔnolo Mr := {p ∈ M2 : B∗ r (p) > 0} eÐnai mh kenì.
- 43. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 35 Epeid h B∗ r eÐnai suneq c sunˆrthsh, to sÔnolo B∗ r −1 (0, +∞) = Mr eÐnai anoiktì sto M2. Epiplèon, afoÔ B∗ r |p = Br|p, p ∈ Mr−1, 0, p ∈ Mc r−1, èqoume Mr ⊂ Mr−1. Sth sunèqeia, ja deÐxoume ìti to Mr eÐnai puknì sÔnolo tou M2. 'Estw int(Mc r ) = ∅. JewroÔme èna mh kenì, anoiktì kai sunektikì uposÔnolo V tou Mc r . 'Eqoume B∗ r |V = 0. An V ∩ Mr−1 = ∅, tìte ja isqÔei V ⊂ Mc r−1. AdÔnato, afoÔ to Mc r−1 apoteleÐtai apì peperasmèna shmeÐa. To V ∩ Mr−1 eÐnai mh kenì kai anoiktì sÔnolo sto opoÐo èqoume Br|V ∩Mr−1 = 0. Tìte, sÔmfwna me thn Parat rhsh 1.3.1, h f de ja tan koresmènh, ˆtopo. Epomènwc to Mr eÐnai puknì uposÔnolo tou M2. 'Estw p0 ∈ Mc r . JewroÔme gÔrw apì to p0 migadikì qˆrth (U, z) me z = x + iy, , = E|dz|2 kai z(p0) = 0. Epilègoume orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y kai kˆjeto mèroc {eα}. Jètoume h∗α (r),1 := B∗ r (e1, ..., e1), eα , h∗α (r),2 := B∗ r (e1, ..., e1, e2), eα kai H∗α r := h∗α (r),1 +ih∗α (r),2, gia α ≥ 2r +1. Profan¸c, h H∗α r eÐnai diaforÐsimh gia kˆje α ≥ 2r +1 kai B∗ r (e1, ..., e1) = α≥2r+1 h∗α (r),1eα, B∗ r (e1, ..., e1, e2) = α≥2r+1 h∗α (r),2eα. Sto Mr−1 ∩ U èqoume B∗ r |Mr−1∩U = Br|Mr−1∩U , sunep¸c h sqèsh dH α r − i(r + 1)H α r ω12 + β≥2r+1 H β r ωβα ∧ φ = 0 mac dÐnei th sqèsh dH ∗α r − i(r + 1)H ∗α r ω12 + β≥2r+1 H ∗β r ωβα ∧ φ = 0 (2.7) gia kˆje α ≥ 2r + 1. Epeid to Mc r−1 eÐnai peperasmèno sÔnolo, lìgw sunèqeiac h teleutaÐa sqèsh isqÔei gÔrw apì to tuqìn shmeÐo tou U. Me th bo jeia twn sqèsewn (2.3), (2.4) kai twn sqèsewn dH ∗α r = ∂H ∗α r ∂z dz + ∂H ∗α 1 ∂z dz, ωαβ = ωαβ( ∂ ∂z )dz + ωαβ( ∂ ∂z )dz, h (2.7) gÐnetai ∂H ∗α r ∂z = β gαβH ∗β r , sto U gÔrw apì to 0, ìpou gαβ = −ωβα( ∂ ∂z ) − r+1 2 δαβd log E( ∂ ∂z ) eÐnai diaforÐsimec sunart seic kai δαβ eÐnai to dèlta tou Kronecker.
- 44. 36 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 SÔmfwna me to L mma 2.2.3 eÐte H ∗α r = 0 gia kˆje α = 2r + 1, ..., n oi koinèc rÐzec twn H ∗α r eÐnai memonwmènec kai upˆrqei jetikìc akèraioc lr kai diaforÐsimec sunart seic G ∗α r : U −→ C ¸ste na isqÔei H ∗α r = zlr G ∗α r (2.8) gia kˆje α ≥ 2r + 1 me G ∗2r+1 1 (0), ..., G ∗n 1 (0) = (0, ..., 0). An tan H ∗α r = 0 gia kˆje α = 2r + 1, ..., n, tìte Br|U∩Mr−1 = 0, ˆtopo afoÔ lìgw thc Parat rhshc 1.3.1, h f den ja tan koresmènh. Epomènwc, oi koinèc rÐzec twn H ∗α r eÐnai memonwmènec sto U kai to sÔnolo {p ∈ U : B∗ r |p = 0} apoteleÐtai apì memonwmèna shmeÐa. 'Ara kai to sÔnolo Mc r = {p ∈ M2 : B∗ r |p = 0} apoteleÐtai apì memonwmèna shmeÐa kai afoÔ to M2 eÐnai sumpagèc, eÐnai peperasmèno. MporoÔme na jewr soume loipìn ìti isqÔei U ∩ Mc r = {p0}. Apì thn upìjesh, gnwrÐzoume ìti se kˆje shmeÐo p tou U isqÔei Φ∗ r|p = 0, isodÔnama B∗ r (e1, ..., e1) (p) = B∗ r (e1, ..., e1, e2) (p) kai B∗ r (e1, ..., e1), B∗ r (e1, ..., e1, e2) (p) = 0. Sunep¸c sto U isqÔei B∗ r (e1, ..., e1) − iB∗ r (e1, ..., e1, e2), B∗ r (e1, ..., e1) − iB∗ r (e1, ..., e1, e2) = 0. 'Omwc lìgw thc (2.8) èqoume B∗ r (e1, ..., e1) − iB∗ r (e1, ..., e1, e2) = α≥2r+1 H ∗α r eα = zlr α≥2r+1 G ∗α r eα. Sunep¸c sto U − {p0} epeid z = 0, isqÔei h isìthta α≥2r+1 G ∗α r eα, α≥2r+1 G ∗α r eα = 0, h opoÐa lìgw sunèqeiac isqÔei kai sto p0. Epomènwc gia kˆje q ∈ U isqÔei Re n α=2r+1 G ∗α r eα (q) = Im n α=2r+1 G ∗α r eα (q) = 0, Re n α=2r+1 G ∗α r eα , Im n α=2r+1 G ∗α r eα (q) = 0. Gia kˆje q ∈ U orÐzoume ton didiˆstato upìqwro N∗r q f tou kajètou q¸rou thc f sto q N∗r q f := span Re n α=2r+1 G ∗α r eα |q, Im n α=2r+1 G ∗α r eα |q . Epomènwc sto U èqoume th dianusmatik dèsmh bajmÐdac 2 N∗r f|U = q∈U N∗r q f.
- 45. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 37 Epeid Br|Mr = B∗ r |Mr , èqoume gia q ∈ U − {p0} Nr q f = span{Br(e1, ..., e1)|q, Br(e1, ..., e1, e2)|q} = span{B∗ r (e1, ..., e1)|q, B∗ r (e1, ..., e1, e2)|q} = span Re B∗ r (e1, ..., e1)|q − iB∗ r (e1, ..., e1, e2)|q , Im B∗ r (e1, ..., e1)|q − iB∗ r (e1, ..., e1, e2)|q = span Re zlr α≥2r+1 G ∗α r eα |q, Im zlr α≥2r+1 G ∗α r eα |q = span Re α≥2r+1 G ∗α r eα |q, Im α≥2r+1 G ∗α r eα |q , isqÔei N∗rf|U−{p0} = Nrf|U−{p0}. An epanalˆboume thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou Mc r , pou ìpwc eÐdame eÐnai peperasmèna to pl joc, apoktoÔme th dianusmatik dèsmh N∗rf me bajmÐda 2 sto M2 gia thn opoÐa isqÔei N∗rf|Mr = Nrf|Mr . OrÐzoume thn apeikìnish B∗ r+1 : ∆(M2 ) × ... × ∆(M2 ) r+2 −→ Γ (T1 f ⊕ N∗1 f ⊕ ... ⊕ N∗r f)⊥ , (X1, ..., Xr+2) −→ B∗ r+1(X1, ..., Xr+2) := ( f X1 ... f Xr+1 df(Xr+2))(T1f⊕N∗1f⊕...⊕N∗rf)⊥ . H B∗ r+1 eÐnai D(M2)-grammik wc proc thn pr¸th metablht thc. Ja deÐxoume ìti h B∗ r+1 eÐnai summetrik kai epomènwc ja eÐnai D(M2)-grammik wc proc ìlec tic metablhtèc thc. Katarq n, epeid M1 ⊃ M2 ⊃ ... ⊃ Mr, isqÔei N∗1f|Mr = N1f|Mr , ..., N∗rf|Mr = Nrf|Mr kai epomènwc èqoume B∗ r+1|Mr = Br+1|Mr . Sunep¸c, h B∗ r+1 eÐnai summetrik sto Mr. To Mc r apoteleÐtai apì peperasmèna shmeÐa kai epomènwc, lìgw sunèqeiac, h B∗ r+1 eÐnai summetrik kai sto Mc r . Gia touc Ðdiouc lìgouc isqÔei B∗ r+1(X1, ..., Xr, e1, e1) + B∗ r+1(X1, ..., Xr, e2, e2) = 0, gia X1, ..., Xr ∈ ∆(M2) kai {e1, e2} opoiod pote topikì orjomonadiaÐo plaÐsio tou M2. Gia X1, ..., Xr+2 ∈ ∆(M2) èqoume B∗ r+1|Mr (X1, ..., Xr+2) = = Br+1|Mr (X1, ..., Xr+2) = f X1 ... f Xr+1 df(Xr+2) (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = f X1 f X2 ... f Xr+1 df(Xr+2) Trf|Mr ∈Tr+1f|Mr (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ + f X1 f X2 ... f Xr+1 df(Xr+2) Nrf|Mr (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥
- 46. 38 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 = f X1 Br|Mr (X2, ..., Xr+2) (T1f⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = f X1 B∗ r |Mr (X2, ..., Xr+2) (T1f|Mr ⊕N∗1f|Mr ⊕...⊕N∗rf|Mr )⊥ kai lìgw sunèqeiac, isqÔei kai sto Mc r . 'Ara sto M2 èqoume th sqèsh B∗ r+1(X1, ..., Xr+2) = f X1 B∗ r (X2, ..., Xr+2) (T1f⊕N∗1f⊕...⊕N∗rf)⊥ . En suneqeÐa, ja deÐxoume ìti an B∗ r |p = 0, tìte B∗ r+1|p = 0. Gia to skopì autì, jewroÔme migadikì qˆrth (U, z) tou M2 kai topikì orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno mèroc {e1, e2} kai kˆjeto mèroc {eα} ètsi ¸ste eα = e∗ α gia α = 3, ..., 2r+2, ìpou ta e∗ 3, ..., e∗ 2r+2 eÐnai tètoia ¸ste N∗1f = span{e∗ 3, e∗ 4},...,N∗rf = span{e∗ 2r+1, e∗ 2r+2}. OrÐzoume tic sunart seic h∗2r+1 (r),1 , h∗2r+2 (r),1 , h∗2r+1 (r),2 , h∗2r+2 (r),2 : U −→ R, oi opoÐec eÐnai tètoiec ¸ste B∗ r (e1, ..., e1) = h∗2r+1 (r),1 e∗ 2r+1 + h∗2r+2 (r),1 e∗ 2r+2 kai B∗ r (e1, ..., e1, e2) = h∗2r+1 (r),2 e∗ 2r+1 + h∗2r+2 (r),2 e∗ 2r+2. Prˆgmati, autì gÐnetai diìti gia kˆje p ∈ M2 o dianusmatikìc q¸roc spanImB∗ r |p eÐnai upìqwroc tou N∗r p f. ToÔto isqÔei apì to L mma 1.3.1 gia kˆje p ∈ Mr−1, afoÔ B∗ r |Mr−1 = Br|Mr−1 . An p ∈ Mc r−1, tìte isqÔei tetrimmèna epeid B∗ r |p = 0. Lìgw tou tÔpou tou Weingarten èqoume B∗ r+1(e1, ..., e1) = = f e1 B∗ r (e1, ..., e1) (T1f⊕N∗1f⊕...⊕N∗rf)⊥ = − df(AB∗ r (e1,...,e1)e1) (T1f⊕N∗1f⊕...⊕N∗rf)⊥ + ⊥ e1 B∗ r (e1, ..., e1) (T1f⊕N∗1f⊕...⊕N∗rf)⊥ = ⊥ e1 (h∗2r+1 (r),1 e∗ 2r+1 + h∗2r+2 (r),1 e∗ 2r+2) (T1f⊕N∗1f⊕...⊕N∗rf)⊥ = h∗2r+1 (r),1 ( ⊥ e1 e∗ 2r+1)(T1f⊕N∗1f⊕...⊕N∗rf)⊥ + h∗2r+2 (r),1 ( ⊥ e1 e∗ 2r+2)(T1f⊕N∗1f⊕...⊕N∗rf)⊥ . 'Omoia, B∗ r+1(e1, ..., e1, e2) = h∗2r+1 (r),2 ( ⊥ e1 e∗ 2r+1)(T1f⊕N∗1f⊕...⊕N∗rf)⊥ + h∗2r+2 (r),2 ( ⊥ e1 e∗ 2r+2)(T1f⊕N∗1f⊕...⊕N∗rf)⊥ . Upojèsame pwc B∗ r |p = 0, ˆra h∗2r+1 (r),1 (p) = h∗2r+2 (r),1 (p) = h∗2r+1 (r),2 (p) = h∗2r+2 (r),2 (p) = 0
- 47. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 39 kai sÔmfwna me tic parapˆnw sqèseic, B∗ r+1(e1, ..., e1)|p = B∗ r+1(e1, ..., e1, e2)|p = 0. Apì aut th sqèsh kai epeid h B∗ r+1 eÐnai summetrik kai tètoia ¸ste na isqÔei B∗ r+1(X1, ..., Xr, e1, e1) + B∗ r+1(X1, ..., Xr, e2, e2) = 0, èqoume B∗ r+1|p = 0. Sunep¸c apodeÐxame ìti an p eÐnai èna shmeÐo tou Mc r , tìte B∗ r+1|p = 0. Telikˆ, gia thn B∗ r+1 isqÔei B∗ r+1|p = Br+1|p, p ∈ Mr, 0, p ∈ Mc r . MigadikopoioÔme tic dianusmatikèc dèsmec T1f, N∗1f, ..., N∗rf, epekteÐnoume C- grammikˆ thn B∗ r+1 kai apoktoÔme to migadikì (r + 2, 1)-tanustikì pedÐo B∗ r+1 : Γ(TM2 ⊗ C) × ... × Γ(TM2 ⊗ C) r+2 −→ Γ (T1 f ⊕ N∗1 f ⊕ ... ⊕ N∗r f)⊥ ⊗ C) . H B∗ r+1 wc migadikì (r + 2, 1)-tanustikì pedÐo dèqetai thn anˆlush B∗ r+1 = p+q=r+2 B ∗ (p,q) r+1 dzp dzq . Se migadikì qˆrth (U, z) tou M2 me z = x+iy kai , = E|dz|2 gnwrÐzoume ìti isqÔei B∗ r+1(X1, ..., Xr, e1, e1) + B∗ r+1(X1, ..., Xr, e2, e2) = 0, gia X1, ..., Xr ∈ ∆(U) kai {e1, e2} opoiod pote topikì orjomonadiaÐo plaÐsio tou U. An jewr soume e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y , tìte epeid ∂ ∂x = ∂ ∂z + ∂ ∂z , ∂ ∂y = i( ∂ ∂z − ∂ ∂z ), h teleutaÐa sqèsh gÐnetai B∗ r+1 X1, ..., Xr, ∂ ∂z , ∂ ∂z = 0. Epomènwc, B∗ r+1 = B ∗ (r+2,0) r+1 dzr+2 + B ∗ (0,r+2) r+1 dzr+2 , ìpou B ∗ (r+2,0) r+1 = B∗ r+1 ∂ ∂z , ..., ∂ ∂z , B ∗ (0,r+2) r+1 = B∗ r+1 ∂ ∂z , ..., ∂ ∂z kai isqÔei B ∗ (r+2,0) r+1 = B ∗ (0,r+2) r+1 . OrÐzoume topikˆ to (2r + 4)-diaforikì Φ∗ r+1 := B ∗ (r+2,0) r+1 , B ∗ (r+2,0) r+1 dz2r+4 . Ja deÐxoume ìti eÐnai kalˆ orismèno se ìlo to M2. Gia to lìgo autì jewroÔme qˆrtec tou M2 (U, ϕ) me migadik suntetagmènh z kai (V, ψ) me migadik suntetagmènh ζ me U ∩ V = ∅. Epeid to M2 eÐnai epifˆneia Riemann, oi apeikonÐseic ψ ◦ ϕ−1, ϕ ◦ ψ−1
- 48. 40 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 eÐnai olìmorfec, dhlad ∂ζ ∂z = 0 kai ∂z ∂ζ = 0. Sto U ∩ V èqoume tic sqèseic ∂ ∂z = ∂ζ ∂z ∂ ∂ζ , dζ = ∂ζ ∂z dz kai epomènwc B∗ r+1( ∂ ∂ζ , ..., ∂ ∂ζ ), B∗ r+1( ∂ ∂ζ , ..., ∂ ∂ζ ) dζ2r+4 = = B∗ r+1( ∂ ∂ζ , ..., ∂ ∂ζ ), B∗ r+1( ∂ ∂ζ , ..., ∂ ∂ζ ) ( ∂ζ ∂z )2r+4 dz2r+4 = ( ∂ζ ∂z )r+2 B∗ r+1( ∂ ∂ζ , ..., ∂ ∂ζ ), ( ∂ζ ∂z )r+2 B∗ r+1( ∂ ∂ζ , ..., ∂ ∂ζ ) dz2r+4 = B∗ r+1( ∂ ∂z , ..., ∂ ∂z ), B∗ r+1( ∂ ∂z , ..., ∂ ∂z ) dz2r+4 . 'Ara to Φ∗ r+1 eÐnai kalˆ orismèno (2r + 4)-diaforikì sto M2. 'Estw p ∈ Mr. JewroÔme gÔrw apì to p migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2. Epilègoume orjomonadiaÐo plaÐsio katˆ m koc thc f|Mr∩U me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y kai kˆjeto mèroc {eα}, ìpou e3 = 1 κ1 B(e1, e1), e4 = 1 κ1 B(e1, e2), e5 = 1 κ2 B2(e1, e1, e1), e6 = 1 κ2 B2(e1, e1, e2), . . . e2r+1 = 1 κr Br(e1, ..., e1, e1), e2r+2 = 1 κr Br(e1, ..., e1, e2) kai eα gia α ≥ 2r + 3 eÐnai tuqìnta. SumbolÐzoume me {ω1, ω2} to sumplaÐsio tou {e1, e2}. Sto Mr ∩ U èqoume to migadikì (1,0)-tanustikì pedÐo φ = ω1 + iω2 = √ Edz gia to opoÐo isqÔoun oi sqèseic (2.4) kai (2.5). 'Eqoume Br(e1, ..., e1) = α≥2r+1 hα (r),1eα, Br(e1, ..., e1, e2) = α≥2r+1 hα (r),2eα kai Hα r = hα (r),1 + ihα (r),2 gia α = 2r + 1, ..., n. Epeid epilèxame wc orjomonadiaÐo plaÐsio thc Nrf|Mr∩U to {e2r+1 = 1 κr Br(e1, ..., e1, e1), e2r+2 = 1 κr Br(e1, ..., e1, e2)}, èqoume h2r+1 (r),1 = κr, hα (r),1 = 0, α ≥ 2r + 2, h2r+2 (r),2 = κr, hα (r),2 = 0, α = 2r + 2
- 49. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 41 kai sunep¸c H2r+1 r = κr, H2r+2 r = iκr. Sto Mr ∩ U isqÔei h sqèsh (dH α r − i(r + 1)H α r ω12 + β≥2r+1 H β r ωβα) ∧ φ = 0 gia kˆje α ≥ 2r + 1. Gia α = 2r + 1 lambˆnoume dκr − i(r + 1)κrω12 + β≥2r+1 H β r ωβ,2r+1 ∧ φ = 0 kai an qrhsimopoi soume to gegonìc ìti gia β ≥ 2r + 3 isqÔei Hβ r = hβ (r),1 + ihβ (r),2 = Br(e1, ..., e1), eβ + i Br(e1, ..., e1, e2), eβ = κre2r+1, eβ + i κre2r+2, eβ = 0, paÐrnoume th sqèsh (d log κr + iω2r+1,2r+2) ∧ φ = i(r + 1)ω12 ∧ φ. (2.9) An periorioristoÔme sto Mr, h (T1f|Mr ⊕ N1f|Mr ⊕ ... ⊕ Nrf|Mr )⊥ eÐnai dianu- smatik dèsmh me bajmÐda n − 2r − 2 kai h Br+1|Mr eÐnai (r + 2, 1)-tanustikì pedÐo. Sto Mr ∩ U, lambˆnontac upìyh ton orismì thc Br+1 kai touc tÔpouc twn Gauss kai Weingarten, èqoume Br+1|Mr (e1, ..., e1) = f e1 f e1 ... f e1 df(e1) (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = f e1 Br(e1, ..., e1) (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = f e1 (κre2r+1) (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = e1(κr)e2r+1 (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ + (κr f e1 e2r+1)(T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = κr( f e1 e2r+1)(T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = κr − df(Ae2r+1 e1) (T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ + ( ⊥ e1 e2r+1)(T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥ = κr α≥2r+3 ⊥ e1 e2r+1, eα eα = κr α≥2r+3 ω2r+1,α(e1)eα. Epiplèon, Br+1|Mr (e1, ..., e1) = −Br+1|Mr (e2, e1, ..., e1, e2) = −κr α≥2r+3 ω2r+2,α(e2)eα.
- 50. 42 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 'Omoia apodeiknÔetai ìti Br+1|Mr (e1, ..., e1, e2) = κr α≥2r+3 ω2r+2,α(e1)eα = κr α≥2r+3 ω2r+1,α(e2)eα. Epomènwc sto Mr ∩ U èqoume gia α ≥ 2r + 3 ω2r+1,α(e1) = −ω2r+2,α(e2), ω2r+2,α(e1) = ω2r+1,α(e2) kai hα (r+1),1 = κrω2r+1,α(e1), hα (r+1),2 = κrω2r+1,α(e2). Apì tic parapˆnw sqèseic paÐrnoume sto Mr ∩ U Hα r+1 = hα (r+1),1 + ihα (r+1),2 = κrω2r+1,α(e1) + iκrω2r+1,α(e2). EÔkola diapist¸noume ìti Hα r+1φ = κrω2r+1,α + iκrω2r+2,α. (2.10) Oi exis¸seic Ricci gia α ≥ 2r + 3 dÐnoun dω2r+1,α = 2 j=1 ω2r+1,j ∧ ωjα + n β=3 ω2r+1,β ∧ ωβα, dω2r+2,α = 2 j=1 ω2r+2,j ∧ ωjα + n β=3 ω2r+2,β ∧ ωβα. Gia s = 2, ..., r èqoume topikˆ sto Ms−1 Bs(e1, ..., e1) = κs−1 α≥2s+1 ω2s−1,α(e1)eα = −κs−1 α≥2s+1 ω2s,α(e2)eα kai Bs(e1, ..., e1, e2) = κs−1 α≥2s+1 ω2s−1,α(e2)eα = κs−1 α≥2s+1 ω2s,α(e1)eα, ˆra ω2s−1,α(e1) = −ω2s,α(e2) kai ω2s−1,α(e2) = ω2s,α(e1) gia α = 2s + 1, ..., n. 'Omwc, Bs(e1, ..., e1) = κse2s+1 kai Bs(e1, ..., e1, e2) = κse2s+2, epomènwc èqoume ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) = κs κs−1 , (2.11) ω2s,2s+2(e1) = ω2s−1,2s+2(e2) = κs κs−1 (2.12) kai ω2s−1,α(e1) = ω2s,α(e2) = 0, (2.13)
- 51. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 43 gia α = 2s + 1, ω2s−1,α(e2) = ω2s,α(e1) = 0, (2.14) gia α = 2s + 2. 'Ara gia kˆje α ≥ 2r + 3 isqÔei ω2s−1,α = ω2s,α = 0. EpÐshc, gia α ≥ 2r + 3 eÐnai ω1α = 2 j=1 hα 1jωj me hα 1j = B(e1, ej), eα = 0 kai ω2α = 2 j=1 hα 2jωj me hα 2j = B(e2, ej), eα = 0, sunep¸c ω1α = ω2α = 0. Telikˆ paÐrnoume ωsα = 0, (2.15) gia s = 1, ..., 2r, α ≥ 2r + 3. Oi exis¸seic Ricci apì tic sqèseic (2.15) gÐnontai dω2r+1,α = ω2r+1,2r+2 ∧ ω2r+2,α + β≥2r+3 ω2r+1,β ∧ ωβα, (2.16) dω2r+2,α = ω2r+2,2r+1 ∧ ω2r+1,α + β≥2r+3 ω2r+2,β ∧ ωβα. (2.17) ParagwgÐzontac exwterikˆ th sqèsh (2.10) kai kˆnontac qr sh twn sqèsewn (2.5), (2.9), (2.16), (2.17) paÐrnoume th zhtoÔmenh isìthta dH α r+1 − i(r + 2)H α r+1ω12 + β≥2r+3 H β r+1ωβα ∧ φ = 0 gia kˆje α ≥ 2r + 3 sto Mr ∩ U. MigadikopoioÔme tic dianusmatikèc dèsmec T1 f|Mr , N1 f|Mr , ..., Nr f|Mr , epekteÐnoume C-grammikˆ thn Br+1|Mr kai apoktoÔme to migadikì (r + 2, 1)-tanustikì pedÐo Br+1|Mr : Γ(TMr ⊗ C) × ... × Γ(TMr ⊗ C) r+2 −→ Γ (T1 f|Mr ⊕ N1 f|Mr ⊕ ... ⊕ Nr f|Mr )⊥ ⊗ C . H Br+1|Mr wc migadikì (r + 2, 1)-tanustikì pedÐo dèqetai thn anˆlush Br+1|Mr = p+q=r+2 Br+1| (p,q) Mr dzp dzq . Se èna migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2 tou M2, sÔmfwna me to L mma 1.3.2, isqÔei Br+1|Mr (X1, ..., Xr, e1, e1) + Br+1|Mr (X1, ..., Xr, e2, e2) = 0, gia X1, ..., Xr ∈ ∆(U) kai {e1, e2} opoiod pote topikì orjomonadiaÐo plaÐsio tou U. An jewr soume e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y , tìte epeid ∂ ∂x = ∂ ∂z + ∂ ∂z , ∂ ∂y = i( ∂ ∂z − ∂ ∂z ), h teleutaÐa sqèsh gÐnetai Br+1|Mr (X1, ..., Xr, ∂ ∂z , ∂ ∂z ) = 0.
- 52. 44 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 Epomènwc, h Br+1|Mr èqei thn anˆlush Br+1|Mr = Br+1| (r+2,0) Mr dzr+2 + Br+1| (0,r+2) Mr dzr+2 , ìpou Br+1| (r+2,0) Mr = Br+1|Mr ∂ ∂z , ..., ∂ ∂z , Br+1| (0,r+2) Mr = Br+1|Mr ∂ ∂z , ..., ∂ ∂z kai isqÔei Br+1| (r+2,0) Mr = Br+1| (0,r+2) Mr . OrÐzoume to (2r + 4)-diaforikì Φr+1 := Br+1| (r+2,0) Mr , Br+1| (r+2,0) Mr dz2r+4 . Epeid B∗ r+1|p = Br+1|p, p ∈ Mr, 0, p ∈ Mc r , isqÔei Φ∗ r+1|p = Φr+1|p, p ∈ Mr, 0, p ∈ Mc r kai epomènwc to Φr+1 eÐnai kalˆ orismèno se ìlo to Mr. SumbolÐzoume me {ω1, ω2} to sumplaÐsio tou {e1, e2}. Sto Mr ∩ U èqoume to migadikì (1,0)-tanustikì pedÐo φ = ω1 + iω2 = √ Edz kai tic migadikèc sunart seic Hα r+1 gia α = 2s + 3, ..., n. Ja apodeÐxoume ìti Φr+1 = 1 4Er+2 α≥2r+3(H α r+1)2dz2r+4 sto Mr ∩ U kai ìti eÐnai olìmorfo. JewroÔme migadikì qˆrth (U, z) me z = x+iy kai , = E|dz|2 tou M2. Epilègoume orjomonadiaÐo plaÐsio katˆ m koc thc f|Mr∩U me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y kai kˆjeto mèroc {eα}, ìpou e3 = 1 κ1 B(e1, e1), e4 = 1 κ1 B(e1, e2), e5 = 1 κ2 B2(e1, e1, e1), e6 = 1 κ2 B2(e1, e1, e2), . . . e2r+1 = 1 κr Br(e1, ..., e1, e1), e2r+2 = 1 κr Br(e1, ..., e1, e2) kai eα gia α ≥ 2r + 3 eÐnai tuqìnta. Sto Mr ∩ U, epiqeirhmatolog¸ntac ìpwc sthn arq thc apìdeixhc (sel. 33), apodeiknÔetai ìti Br+1( ∂ ∂z , ..., ∂ ∂z ) = 1 2 E r+2 2 Br(e1, ..., e1) − i 1 2 E r+2 2 Br+1(e1, ..., e1, e2).
- 53. SumpageÐc elaqistikèc epifˆneiecgènouc mhdèn 45 Epomènwc, èqoume B (r+2,0) r+1 = Br+1 ∂ ∂z , ..., ∂ ∂z = 1 2 E r+2 2 Br(e1, ..., e1) − iBr+1(e1, ..., e1, e2) = 1 2 E r+2 2 α≥2r+3 hα (r+1),1eα − i α≥2r+3 hα (r+1),2eα = 1 2 E r+2 2 α≥2r+3 H α r+1eα kai sunep¸c Φr+1 = Br+1| (r+2,0) Mr , Br+1| (r+2,0) Mr dz2r+4 = 1 4 Er+2 α≥2r+3 H α r+1eα, α≥2r+3 H α r+1eα dz2r+4 = 1 4 Er+2 α≥2r+3 (H α r+1)2 dz2r+4 . Jètoume fr+1 := 1 4 Er+2 α≥2r+3 (H α r+1)2 . ApodeÐxame sto Mr ∩ U th sqèsh dH α r+1 − i(r + 2)H α r+1ω12 + β≥2r+3 H β r+1ωβα ∧ φ = 0 gia kˆje α ≥ 2r + 3. Pollaplasiˆzontac aut th sqèsh me H α r+1, ajroÐzontac wc proc α ≥ 2r + 3, lambˆnontac upìyh thn (2.5) kai epeid α,β≥2r+3 H α r+1H β r+1ωβα ∧ φ = 0, ftˆnoume sth sqèsh d 1 4 α≥2r+3 (H α r+1)2 ∧ φ + r + 2 2 α≥2r+3 (H α r+1)2 dφ = 0, isodÔnama d fr+1 Er+2 ∧ φ + 2r + 4 Er+2 fr+1dφ = 0. Lìgw twn (2.3), (2.4) metˆ apì prˆxeic brÐskoume ∂fr+1 ∂z = 0. Sunep¸c to Φr+1 eÐnai olìmorfo diaforikì sto U.
- 54. 46 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 JewroÔme èna shmeÐo p ∈ Mc r kai èstw (U, z) migadikìc qˆrthc gÔrw apì to p me z(p) = 0 kai U ∩ Mc r = {p}. To Φ∗ r+1 eÐnai olìmorfo sto U − {p} kai suneqèc sto shmeÐo p, ˆra Φ∗ r+1 eÐnai olìmorfo sto U [1]. Epanalambˆnontac thn parapˆnw diadikasÐa gÔrw apì ìla ta peperasmèna shmeÐa tou Mc r , èqoume to Φ∗ r+1 olìmorfo (2r + 4)-diaforikì se ìlo to M2 kai apì to Je¸rhma Riemann-Roch Φ∗ r+1 = 0. 2.3 Bohjhtikˆ apotelèsmata Ja sunoyÐsoume ìsa mèqri stigm c èqoume apodeÐxei kai ja qreiastoÔme sth su- nèqeia gia tic sumpageÐc, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra. Gia to skopì autì qreiazìmaste ton orismì kai to l mma pou akoloujoÔn. Orismìc 2.3.1. Se prosanatolismèno polÔptugma Riemann M2 jewroÔme topikì orjomonadiaÐo plaÐsio {e1, e2} tou prosanatolismoÔ, to sumplaÐsiì tou {ω1, ω2} kai to sÔnolo twn 1-morf¸n Λ1(M2). O telest c tou Hodge orÐzetai na eÐnai h apeikìnish ∗ : Λ1 (M2 ) −→ Λ1 (M2 ), ω = ω(e1)ω1 + ω(e2)ω2 −→ ∗ω := −ω(e2)ω1 + ω(e1)ω2. ApodeiknÔetai ìti o anwtèrw orismìc eÐnai kalìc, dhlad anexˆrthtoc tou plaisÐou {e1, e2}. Profan¸c isqÔei ∗ω1 = ω2 kai ∗ω2 = −ω1. JumÐzoume ìti gia mia sunˆrthsh g ∈ D(M2) h klÐsh thc gradg kai h Laplasian thc ∆g, ìpou ∆ eÐnai o telest c Laplace tou M2, dÐnontai wc gradg = e1(g)e1+e2(g)e2 kai ∆g = e1 e1(g) + e2 e2(g) − e1 e1 (g) − e2 e2 (g). L mma 2.3.1. 'Estw {e1, e2} topikì orjomonadiaÐo plaÐsio tou M2 kai {ω1, ω2} to sumplaÐsiì tou. (i) Gia tuqoÔsa sunˆrthsh g ∈ D(M2) isqÔei h sqèsh d(∗dg) = ∆gω1 ∧ ω2. (ii) An η, ω ∈ Λ1(M2), tìte η = ∗ω ⇔ η ∧ φ = iω ∧ φ, ìpou φ = ω1 + iω2. Apìdeixh. (i) ArkeÐ na deÐxoume ìti d(∗dg)(e1, e2) = ∆g. Prˆgmati, epeid dg = dg(e1)ω1 + dg(e2)ω2 = e1(g)ω1 + e2(g)ω2, èqoume ∗dg = −e2(g)ω1 + e1(g)ω2. UpologÐzoume d(∗dg)(e1, e2) = = −d e2(g) ∧ ω1(e1, e2) + d e1(g) ∧ ω2(e1, e2)
- 55. Bohjhtikˆ apotelèsmata 47 −e2(g)ω12 ∧ ω2(e1, e2) + e1(g)ω21 ∧ ω1(e1, e2) = d e2(g) (e2) + d e1(g) (e1) − e2(g)ω12(e1) − e1(g)ω21(e2) = e2 e2(g) + e1 e1(g) − e2(g) e1 e1, e2 − e1(g) e2 e2, e1 = ∆g. (ii) H apìdeixh eÐnai aplìc upologismìc. Prìtash 2.3.1. 'Estw f : M2 −→ Sn, n ≥ 3, sumpag c, koresmènh, elaqistik epifˆneia gènouc mhdèn. Upˆrqoun anoiktˆ uposÔnola M1, ..., Mm−1 tou M2 me M1 ⊃ M2 ⊃ ... ⊃ Mm−1, ìpou m := 1 + [n−1 2 ], tètoia ¸ste ta sumplhr¸matˆ touc Mc s = M2 − Ms gia kˆje s ∈ {1, ..., m − 1} na eÐnai peperasmèna sÔnola kai isqÔoun: (i) H èlleiyh Es s-tˆxhc eÐnai kÔkloc aktÐnac κs > 0 se kˆje shmeÐo tou Mm−1 gia kˆje s ∈ {1, ..., m − 1}. Epiplèon, o arijmìc n eÐnai ˆrtioc kai n = 2m. (ii) Upˆrqoun diaforÐsimec sunart seic gs : M2 −→ [0, +∞) me gs|Mm−1 = κ2 s|Mm−1 , gs|Mc m−1 = 0 gia kˆje s ∈ {1, ..., m − 1}. Epiplèon, gia kˆje p ∈ Mc m−1, upˆrqei migadikìc qˆrthc (U, z) me z(p) = 0 kai gs = |z|2ls us sto U, ìpou ls jetikìc akèraioc kai us ∈ D(U) jetik sunˆrthsh. (iii) Gia kˆje migadikì qˆrth (U, z) me z = x + iy, , = E|dz|2 upˆrqei topikì orjomonadiaÐo plaÐsio {eA} katˆ m koc thc f|U∩Mm−1 me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y kai kˆjeto mèroc {eα} tètoio ¸ste e2s+1 = 1 κs Bs(e1, ..., e1), e2s+2 = 1 κs Bs(e1, ..., e1, e2), s ∈ {1, ..., m − 1}, gia to opoÐo isqÔoun ta akìlouja ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs, ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) = κs κs−1 , ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1, ω2s,2s+2(e1) = ω2s−1,2s+2(e2) = κs κs−1 , ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2, ωrα = 0, 1 ≤ r ≤ 2s, α ≥ 2s + 3. Apìdeixh. (i) 'Eqei apodeiqjeÐ sto epagwgikì b ma ìti se kˆje shmeÐo p ∈ Ms h èlleiyh mèqri kai s-tˆxhc sto p eÐnai kÔkloc jetik c aktÐnac, gia kˆje s ∈ {1, ..., m − 1}. Epeid M1 ⊃ M2 ⊃ ... ⊃ Mm−1, se kˆje shmeÐo tou Mm−1 h èlleiyh s-tˆxhc Es eÐnai kÔkloc aktÐnac κs > 0 gia ìla ta s ∈ {1, ..., m − 1}. Epeid h f eÐnai koresmènh, gia tuqìn shmeÐo p tou Mm−1 èqoume Tf(p)Sn = T1 p f ⊕ N1 p f ⊕ ... ⊕ Nm−1 p f. 'Olec oi elleÐyeic kˆje tˆxhc eÐnai kÔkloi me jetik aktÐna, ˆra ìloi oi kˆjetoi q¸roi thc f sto tuqìn p ∈ Mm−1 eÐnai didiˆstatoi. Epomènwc, h parapˆnw anˆlush mac dÐnei n = 2m, dhlad o arijmìc n eÐnai ˆrtioc.
- 56. 48 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 (ii) Gia kˆje s jètoume gs = 1 2s+1 B∗ s 2 . Epeid M1 ⊃ M2 ⊃ ... ⊃ Mm−1, isqÔei B∗ s |Mm−1 = Bs|Mm−1 kai B∗ s |Mc m−1 = 0. Epomènwc èqoume gs(p) = κ2 s(p), p ∈ Mm−1, 0, p ∈ Mc m−1. Sto epagwgikì b ma apodeÐxame ìti Φ∗ 1 = ... = Φ∗ m−1 = 0 apì ìpou sumperaÐnoume ìti |B∗ s (e1, ..., e1)| = |B∗ s (e1, ..., e1, e2)|. Lambˆnontac upìyh tic sqèseic (2.6) kai (2.8) upologÐzoume gs = 1 2 |B∗ s (e1, ..., e1)|2 + 1 2 |B∗ s (e1, ..., e1, e2)|2 = 1 2 n α=2s+1 (h∗α (s),1)2 + (h∗α (s),2)2 = 1 2 n α=2s+1 |H ∗α s |2 = |z|2ls us, ìpou us := 1 2 n α=2s+1 |G ∗α s |2. (iii) Efìson Φ1|Mm−1 = ... = Φm−1|Mm−1 = 0, ta Bs(e1, ..., e1), Bs(e1, ..., e1, e2) eÐnai isom kh kai kˆjeta metaxÔ touc gia kˆje s ∈ {1, ..., m−1}. Epomènwc, h epilog autoÔ tou plaisÐou eÐnai efikt . GnwrÐzoume ìti sto Mm−1 gia kˆje s ∈ {1, ..., m − 1} isqÔei h sqèsh ω2s+1,2s+2 − (s + 1)ω12 ∧ ϕ = id log κs ∧ ϕ, epomènwc apì to L mma 2.3.1(ii) èqoume ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs. Oi upìloipec sqèseic èqoun apodeiqjeÐ sto epagwgikì b ma (2.11), (2.12), (2.13), (2.14), (2.15) . Sto anoiktì kai puknì uposÔnolo Mm−1 tou M2 èqoume tic dianusmatikèc u- podèsmec N1f|Mm−1 , ..., Nm−1f|Mm−1 thc kˆjethc dèsmhc me bajmÐda 2. Gia kˆje s ∈ {1, ..., m − 1}, h dèsmh Nsf|Mm−1 èqei metrik , s th metrik thc Sn periori- smènh se kˆje èna apì autˆ kai sunoq s thn kˆjeth sunoq thc embˆptishc f, epÐshc periorismènh se kˆje èna apì autˆ. Epiplèon, h dèsmh Nsf|Mm−1 èqei tanust kampulìthtac Rs : ∆(Mm−1) × ∆(Mm−1) × Γ(Ns f|Mm−1 ) −→ Γ(Ns f|Mm−1 ), (X, Y, V ) −→ Rs (X, Y )V = s X s Y V − s Y s XV − s [X,Y ]V.
- 57. Bohjhtikˆ apotelèsmata 49 An{e1, e2} eÐnai topikì orjomonadiaÐo plaÐsio tou prosanatolismoÔ tou M2 kai {vs 1, vs 2} orjomonadiaÐo plaÐsio tou prosanatolismoÔ thc Nsf|Mm−1 pou epˆgei h Bs, dhlad o prosanatolismìc pou orÐzei sthn Nsf|Mm−1 to plaÐsiì thc {Bs(e1, ..., e1), Bs(e1, ..., e1, e2)}, tìte h kampulìthta Ks thc Nsf|Mm−1 orÐzetai wc Ks = − Rs (e1, e2)vs 1, vs 2 s. 'Eqoume epilèxei topikì orjomonadiaÐo plaÐsio {e2s+1, e2s+2} thc Nsf|Mm−1 ètsi ¸ste Bs(e1, ..., e1) = κse2s+1, Bs(e1, ..., e1, e2) = κse2s+2 gia kˆje s ∈ {1, ..., m − 1}. H morf sunoq c ω2s+1,2s+2 thc dèsmhc Nsf|Mm−1 eÐnai ω2s+1,2s+2(X) = s Xe2s+1, e2s+2 s gia X ∈ ∆(M2) kai isqÔei dω2s+1,2s+2 = −Ksω1 ∧ ω2. (2.18) Prìtash 2.3.2. 'Estw f : M2 −→ Sn sumpag c, koresmènh, elaqistik epifˆneia gènouc mhdèn. Tìte sto Mm−1, ìpou m = 1 + [n−1 2 ], isqÔoun: (i) K = 1 − 2κ2 1. (ii) Gia s ∈ {1, ..., m − 1} h kampulìthta Ks thc dèsmhc Nsf|Mm−1 eÐnai Ks = 2κ2 1 − 2 κ2 2 κ2 1 , s = 1, 2 κ2 s κ2 s−1 − 2 κ2 s+1 κ2 s , 1 < s < m − 1, κ2 m−1 κ2 m−2 , s = m − 1. (iii) ∆ log κs = (s+1)K−Ks, 1 ≤ s ≤ m−1, kai ∆ log(κ1...κm−1) = m(m+1) 2 K−1. Apìdeixh. SÔmfwna me thn Prìtash 2.3.1(i) èqoume n = 2m. Epilègoume orjomona- diaÐo plaÐsio ìpwc sthn Prìtash 2.3.1(iii). (i) Se kˆje shmeÐo tou M2 èqoume |B(e1, e1)| = |B(e1, e2)| = κ1, epomènwc apì th sqèsh (1.3) èqoume K = 1 − 2κ2 1 sto M2, ˆra kai sto Mm−1. (ii) H kampulìthta thc N1f|Mm−1 dÐnetai apì th sqèsh dω34 = −K1ω1 ∧ ω2. Apì thn exÐswsh Ricci èqoume dω34 = ω31 ∧ ω14 + ω32 ∧ ω24 + α≥3 ω3α ∧ ωα4. 'Omwc ω31 ∧ ω14(e1, e2) = −κ2 1 = ω32 ∧ ω24(e1, e2) kai apì tic sqèseic ω33 = ω44 = 0, ω35(e1) = ω54(e2) = κ2 κ1 ,
- 58. 50 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 ω3α(e1) = ω4α(e2) = 0, α > 5, ω46(e1) = ω36(e2) = κ2 κ1 , ω4α(e1) = ω3α(e2) = 0, α ≥ 5, α = 6, èqoume α≥3 ω3α ∧ ωα4(e1, e2) = 2 κ2 2 κ2 1 . Epomènwc gia thn kampulìthta thc N1f|Mm−1 isqÔei h sqèsh K1 = 2κ2 1 − 2 κ2 2 κ2 1 . H kampulìthta Ks thc Nsf|Mm−1 , ìpou s = 2, ..., m − 2, dÐnetai apì th sqèsh dω2s+1,2s+2 = −Ksω1 ∧ ω2. H exÐswsh Ricci eÐnai dω2s+1,2s+2 = ω2s+1,1 ∧ ω1,2s+2 + ω2s+1,2 ∧ ω2,2s+2 + α≥3 ω2s+1,α ∧ ωα,2s+2. SÔmfwna me thn Prìtash 2.3.1(iii), isqÔoun oi sqèseic ωrα = 0, 1 ≤ r ≤ 2s − 2, α ≥ 2s + 1. Epiplèon, isqÔoun oi sqèseic ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) = κs κs−1 , ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1, ω2s,2s+2(e1) = ω2s−1,2s+2(e2) = κs κs−1 , ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2. Akìmh, isqÔoun oi sqèseic ω2s+1,2s+3(e1) = −ω2s+2,2s+3(e2) = κs+1 κs , ω2s+1,α(e1) = ω2s+2,α(e2) = 0, α > 2s + 3, ω2s+2,2s+4(e1) = ω2s+1,2s+4(e2) = κs+1 κs , ω2s+2,α(e1) = ω2s+1,α(e2) = 0, α ≥ 2s + 3, α = 2s + 4, apì ìpou paÐrnoume ìti ω2s+1,α = ω2s+2,α = 0 gia kˆje α ≥ 2s + 5. UpologÐzoume dω2s+1,2s+2(e1, e2) = 2 j=1 ω2s+1,j ∧ ωj,2s+2(e1, e2) + α≥3 ω2s+1,α ∧ ωα,2s+2(e1, e2)
- 59. Bohjhtikˆ apotelèsmata 51 = 3≤α≤2s−2 ω2s+1,α∧ ωα,2s+2(e1, e2) + 2s−1≤α≤2s+4 ω2s+1,α ∧ ωα,2s+2(e1, e2) + α≥2s+5 ω2s+1,α ∧ ωα,2s+2(e1, e2) = 2s−1≤α≤2s+4 ω2s+1,α ∧ ωα,2s+2(e1, e2) = −2 κ2 s κ2 s−1 + 2 κ2 s+1 κ2 s . Telikˆ gia s = 2, ..., m − 2 isqÔei Ks = Ksω1 ∧ ω2(e1, e2) = −dω2s+1,2s+2(e1, e2) = 2 κ2 s κ2 s−1 − 2 κ2 s+1 κ2 s . H dianusmatik dèsmh Nm−1f|Mm−1 èqei kampulìthta Km−1 gia thn opoÐa gnw- rÐzoume ìti Km−1 = −dω2m−1,2m(e1, e2). Epeid ωrα = 0, 1 ≤ r ≤ 2m − 4, α ≥ 2m − 1, ω2m−3,2m−1(e1) = −ω2m−2,2m−1(e2) = κm−1 κm−2 , ω2m−2,2m(e1) = ω2m−3,2m(e2) = κm−1 κm−2 , ω2m−3,α(e1) = ω2m−2,α(e2) = 0, α > 2m − 1, ω2m−2,α(e1) = ω2m−3,α(e2) = 0, α ≥ 2m − 1, α = 2m, paÐrnoume apì thn exÐswsh Ricci Km−1 = −dω2m−1,2m(e1, e2) = = 2 j=1 ω2m−1,j ∧ ωj,2m(e1, e2) + α≥3 ω2m−1,α ∧ ωα,2m(e1, e2) = α≥2m−3 ω2m−1,α ∧ ωα,2m(e1, e2) = − κ2 m−1 κ2 m−2 . (iii) ParagwgÐzontac exwterikˆ th sqèsh ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs, pou apodeÐxame sthn Prìtash 2.3.1(iii) paÐrnoume dω2s+1,2s+2 − (s + 1)dω12 = d(∗d log κs). Apì to L mma 2.3.1(i), tic sqèseic (1.4) kai (2.18) èqoume sto Mm−1 th sqèsh ∆ log κs = (s + 1)K − Ks
- 60. 52 Elaqistikèc epifˆneiecsthn Sn omoiomorfikèc me thn S2 gia kˆje s ∈ {1, ..., m−1}. AjroÐzontac wc proc s, gia 1 ≤ s ≤ m−1, kai lambˆnontac upìyh to (i) kai to (ii), ftˆnoume sth sqèsh ∆ log(κ1...κm−1) = m(m + 1) 2 K − 1.
- 61. Kefˆlaio 3 KÔria apotelèsmata 3.1ApodeÐxeic twn kurÐwn apotelesmˆtwn EÐmaste plèon ètoimoi na d¸soume tic apodeÐxeic twn jewrhmˆtwn, pou èqoun a- naferjeÐ sthn eisagwg . Stic apodeÐxeic ja gÐnei qr sh, ektìc apì ta apotelèsmata tou KefalaÐou 2, apotelesmˆtwn thc Olik c Diaforik c GewmetrÐac, ìpwc tou Jew- r matoc Gauss-Bonnet, tou Jewr matoc Gauss-Green, thc Arq c MegÐstou kai tou Jewr matoc Monadikìthtac twn isometrik¸n embaptÐsewn. To apotèlesma pou akoloujeÐ ofeÐletai ston Calabi [6]. Je¸rhma 3.1.1. 'Estw f : (M, , ) −→ Sn, n ≥ 3, sumpag c, prosanatolismènh, koresmènh, elaqistik epifˆneia gènouc mhdèn. Tìte: (i) O arijmìc n eÐnai ˆrtioc (n = 2m). (ii) To embadì A(M) thc epifˆneiac eÐnai akèraio pollaplˆsio tou 2π kai isqÔei A(M) ≥ 2πm(m + 1). Apìdeixh. (i) 'Eqei apodeiqjeÐ sthn Prìtash 2.3.1(i). (ii) Lambˆnontac upìyh thn Prìtash 2.3.1(ii) kai thn Prìtash 2.3.2(iii) èqoume ∆ log u = m(m + 1)K − 2 (3.1) sto Mm−1, ìpou u := g2 1...g2 m−1. Apì thn Prìtash 2.3.1, to Mc m−1 apoteleÐtai apì peperasmèno to pl joc shmeÐwn. JewroÔme tuqìn shmeÐo tou p. GÔrw apì to p jewroÔme migadikì qˆrth (U, z) tou M ¸ste z(p) = 0, z = x + iy, , = E|dz|2 kai U ∩ Mc m−1 = {p}. Epilègoume orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√ E ∂ ∂x , e2 = 1√ E ∂ ∂y kai kˆjeto mèroc {eα} ¸ste sto U − {p} na isqÔei e2s+1 = 1 κs Bs(e1, ..., e1), e2s+2 = 1 κs Bs(e1, ..., e1, e2) gia kˆje s ∈ {1, ..., m − 1}. Lìgw thc Prìtashc 2.3.1(ii) isqÔei u = |z|2lu0 sto U me l := l1 + ... + lm−1, ìpou u0 eÐnai diaforÐsimh jetik sunˆrthsh orismènh kontˆ sto p ∈ Mc m−1. 'Estw Mc m−1 = {p1, ..., pt}. Gia kˆje s ∈ {1, ..., t} jewroÔme gewdaisiak mpˆla Bε(ps) me kèntro to ps kai aktÐna ε > 0 ètsi ¸ste Bε(ps) ∩ Mc m−1 = {ps} kai jètoume Mε := M − t s=1 Bε(ps). 53
- 62. 54 KÔria apotelèsmata ToMε eÐnai polÔptugma me sÔnoro ∂Mε = t s=1 ∂Bε(ps). ProsanatolÐzoume jetikˆ to sÔnoro ∂Mε kai sumbolÐzoume me ν to exwterikì mona- diaÐo kˆjeto tou Mε. Apì to Je¸rhma twn Gauss-Green èqoume Mε ∆ log udM = ∂Mε ν, grad log u dσ = t s=1 ∂Bε(ps) ν, grad log u dσ, ìpou dM eÐnai to stoiqeÐo embadoÔ tou M kai dσ eÐnai to stoiqeÐo m kouc tou ∂Mε. 'Eqoume dh jewr sei migadikì qˆrth (U, z) gÔrw apì to ps me z(ps) = 0. JewroÔme polikèc suntetagmènec (ρ, θ), ìpou x = ρ cos θ, y = ρ sin θ. Gia ta dianusmatikˆ pedÐa ∂ ∂ρ , ∂ ∂θ isqÔoun ∂ ∂ρ = cos θ ∂ ∂x + sin θ ∂ ∂y , ∂ ∂θ = −ρ sin θ ∂ ∂x + ρ cos θ ∂ ∂y . Epeid gÔrw apì kˆje shmeÐo tou Mc m−1 isqÔei u = |z|2lu0, an jèsoume |z| = ρ, tìte èqoume u = ρ2lu0, log u = 2l log ρ + log u0 kai u|∂Bε(ps) = ε2lu0|∂Bε(ps). EpÐshc, to exwterikì monadiaÐo kˆjeto eÐnai to ν = − ∂ ∂ρ | ∂ ∂ρ | = − 1 √ E ∂ ∂ρ kai to stoiqeÐo m kouc tou ∂Bε(ps) eÐnai to dσ = | ∂ ∂θ |dθ = ρ √ Edθ. Sunep¸c èqoume Mε ∆ log udM = t s=1 ∂Bε(ps) ν, grad log u dσ = t s=1 ∂Bε(ps) − 1 √ E ∂ ∂ρ , grad log u0 − 2l ρ √ E dσ = − t s=1 ∂Bε(ps) ρ ∂ ∂ρ , grad log u0 dθ − 2l t s=1 ∂Bε(ps) dθ = −4πlt − t s=1 ∂Bε(ps) ε ∂ ∂ρ , grad log u0 dθ. PaÐrnontac to ìrio kaj¸c to ε teÐnei sto 0, brÐskoume lim ε→0 Mε ∆ log udM = −4πlt, (3.2)
- 63. ApodeÐxeic twn kurÐwnapotelesmˆtwn 55 epeid h posìthta ∂ ∂ρ , grad log u0 eÐnai fragmènh. Oloklhr¸noume th sqèsh (3.1) sto Mε, paÐrnoume to ìrio kai lambˆnontac upìyh thn (3.2) èqoume diadoqikˆ lim ε→0 Mε ∆ log udM = m(m + 1) lim ε→0 Mε KdM − 2 lim ε→0 Mε dM, −4πlt = m(m + 1) M KdM − 2 M dM, A(M) = 2πlt + m(m + 1) 2 M KdM. Epeid to M eÐnai sumpagèc kai gènouc mhdèn, èqoume M KdM = 4π, lìgw tou Jewr matoc Gauss-Bonnet. Epomènwc isqÔei A(M) = 2π m(m + 1) + lt , dhlad to embadì tou M eÐnai akèraio pollaplˆsio tou 2π kai epiplèon A(M) ≥ 2πm(m + 1). Parat rhsh 3.1.1. AxÐzei na shmeiwjeÐ ìti gia to embadì tou M isqÔei A(M) = 2πm(m + 1) ìtan Mm−1 = M. 'Amesh apìrroia tou Jewr matoc 3.1.1 eÐnai to akìloujo [2] Pìrisma 3.1.1. 'Estw f : (M, , ) −→ S3 sumpag c, prosanatolismènh, elaqisti- k epifˆneia gènouc mhdèn. Tìte h f eÐnai olikˆ gewdaisiak . Sth sunèqeia ja apodeÐxoume to shmantikì apotèlesma akamyÐac pou ofeÐletai ston Barbosa [4], gia thn apìdeixh tou opoÐou qreiazìmaste to Je¸rhma Monadikìth- tac twn isometrik¸n embaptÐsewn sth sfaÐra [10]. Je¸rhma 3.1.2. 'Estwsan f, f : Mn −→ Sn+k isometrikèc embaptÐseic tou sune- ktikoÔ poluptÔgmatoc Riemann Mn sthn Sn+k. SumbolÐzoume me Nf, B, ⊥ kai Nf, B, ⊥ thn kˆjeth dèsmh, th deÔterh jemeli¸dh morf kai thn kˆjeth su- noq twn f kai f antÐstoiqa. Upojètoume ìti upˆrqei isomorfismìc T metaxÔ twn dianusmatik¸n desm¸n Nf, Nf, ξp = (p, ξ) −→ Tξp = (p, ξ) tètoioc ¸ste: (i) Tξp, Tηp = ξp, ηp gia kˆje ξp, ηp ∈ Nf. (ii) Oi deÔterec jemeli¸deic morfèc sundèontai mèsw tou isomorfismoÔ me th sqèsh T ◦ B = B. (iii) Gia X ∈ ∆(Mn) kai ξ ∈ ∆⊥(f) isqÔei T( ⊥ Xξ) = ⊥ X(Tξ). Tìte upˆrqei isometrÐa τ thc Sn+k ètsi ¸ste f = τ ◦ f. Je¸rhma 3.1.3. 'Estwsan f : (M, , ) −→ S2m, f : (M, , ) −→ S2m sumpageÐc, prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn. Tìte upˆrqei isometrÐa τ thc S2m tètoia ¸ste f = τ ◦ f.
- 64. 56 KÔria apotelèsmata Apìdeixh.JewroÔme topikì orjomonadiaÐo plaÐsio {e1, e2} ston prosanatolismì tou M. EpÐshc, ìpwc sthn Prìtash 2.3.1, gia kˆje s ∈ {1, ..., m − 1} jewroÔme gia thn f sto sÔnolo Mm−1 topikì orjomonadiaÐo plaÐsio e2s+1 = 1 κs Bs(e1, ..., e1), e2s+2 = 1 κs Bs(e1, ..., e1, e2) thc dianusmatik c dèsmhc Nsf|Mm−1 . AntÐstoiqa gia thn f sto sÔnolo Mm−1 jew- roÔme topikì orjomonadiaÐo plaÐsio e2s+1 = 1 κs Bs(e1, ..., e1), e2s+2 = 1 κs Bs(e1, ..., e1, e2) thc dianusmatik c dèsmhc Nsf|fMm−1 . To Mm−1 ∩ Mm−1 eÐnai anoiktì wc tom anoikt¸n. 'Estw U mia sunektik suni- st¸sa tou. OrÐzoume apeikìnish T metaxÔ twn dianusmatik¸n desm¸n Nf|U , Nf|U ¸ste gia kˆje p ∈ U kai s ∈ {1, ..., m − 1} na isqÔei Te2s+1|p = e2s+1|p, Te2s+2|p = e2s+2|p kai thn epekteÐnoume D(U)-grammikˆ. Profan¸c h T eÐnai isomorfismìc. EpÐshc, eÐnai fanerì ìti gia kˆje ξp, ηp ∈ Npf|U plhreÐtai h sqèsh Tξp, Tηp = ξp, ηp . GnwrÐzoume apì thn Prìtash 2.3.2 ìti sto U isqÔoun oi sqèseic K = 1 − 2κ2 1 = 1 − 2κ2 1. Sunep¸c eÐnai κ1 = κ1. Gia kˆje p ∈ U isqÔei T ◦ B(e1, e1)|p = κ1(p)e3|p = κ1(p)e3|p = B(e1, e1)|p kai ìmoia T ◦ B(e1, e2)|p = B(e1, e2)|p. Epeid oi B, B eÐnai D(M)-grammikèc, sunˆgoume ìti T ◦ B = B. Ja deÐxoume ìti gia tic morfèc kˆjethc sunoq c twn f, f isqÔei ωαβ = ωαβ. Sto U gia thn f apì thn Prìtash 2.3.2 èqoume tic sqèseic Ks = 2κ2 1 − 2 κ2 2 κ2 1 , s = 1, 2 κ2 s κ2 s−1 − 2 κ2 s+1 κ2 s , 1 < s < m − 1, κ2 m−1 κ2 m−2 , s = m − 1, kai ∆ log κs = (s + 1)K − Ks, ìpou Ks eÐnai h kampulìthta thc dianusmatik c dèsmhc Nsf|U kai κs eÐnai to m koc tou Bs(e1, ..., e1) gia kˆje s ∈ {1, ..., m − 1}. 'Omoia sto U gia thn f èqoume Ks = 2κ2 1 − 2 κ2 2 κ2 1 , s = 1, 2 κ2 s κ2 s−1 − 2 κ2 s+1 κ2 s , 1 < s < m − 1, 2 κ2 m−1 κ2 m−2 , s = m − 1,
- 65. ApodeÐxeic twn kurÐwnapotelesmˆtwn 57 kai ∆ log κs = (s + 1)K − Ks, ìpou Ks eÐnai h kampulìthta thc dianusmatik c dèsmhc Nsf|U kai κs eÐnai to m koc tou Bs(e1, ..., e1) gia kˆje s ∈ {1, ..., m − 1}. Sth sunèqeia ergazìmaste sto U. Epeid κ1 = κ1, apì tic proanaferjèntec sqèseic eÔkola paÐrnoume ìti κs = κs kai Ks = Ks gia 1 ≤ s ≤ m − 1. Apì thn Prìtash 2.3.1(iii) gia thn f gnwrÐzoume ìti ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs kai omoÐwc gia thn f gnwrÐzoume ìti ω2s+1,2s+2 − (s + 1)ω12 = ∗d log κs. Apì autèc su- nˆgetai ìti ω2s+1,2s+2 = ω2s+1,2s+2. Epiplèon apì thn Prìtash 2.3.1(iii) gia thn f|U èqoume ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) = κs κs−1 , ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1, ω2s,2s+2(e1) = ω2s−1,2s+2(e2) = κs κs−1 , ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2 kai gia thn f|U èqoume ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) = κs κs−1 , ω2s−1,α(e1) = ω2s,α(e2) = 0, α > 2s + 1, ω2s,2s+2(e1) = ω2s−1,2s+2(e2) = κs κs−1 , ω2s,α(e1) = ω2s−1,α(e2) = 0, α ≥ 2s + 1, α = 2s + 2, ìpou 1 ≤ s ≤ m − 1. Apì autèc tic sqèseic prokÔptei ìti ωαβ = ωαβ, afoÔ κs = κs. 'Estw X ∈ ∆(U) kai ξ ∈ Γ(Nf|U ). To ξ èqei thn anˆlush ξ = 2m α=3 ξ, eα eα. Epeid gia kˆje p ∈ U isqÔei T( ⊥ Xeα|p) = T 2m β=3 ωαβ(X)eβ |p = 2m β=3 ωαβ(X)eβ|p = ⊥ Xeα|p, paÐrnoume T( ⊥ Xξ) = ⊥ X(Tξ). SÔmfwna me to Je¸rhma 3.1.2 upˆrqei isometrÐa τ : S2m −→ S2m tètoia ¸ste f|U = τ ◦ f|U . Jètoume F := f − τ ◦ f : M −→ R2m+1. GnwrÐzoume ìti oi f, f eÐnai analutikèc (Prìtash 1.3.3). Sunep¸c h F eÐnai analutik sto M, wc diaforˆ analutik¸n kai afoÔ F|U = 0 sumperaÐnoume ìti F = 0 sto M, dhlad f = τ ◦ f. To teleutaÐo apotèlesma eÐnai h taxinìmhsh twn sumpag¸n, prosanatolismènwn, koresmènwn, elaqistik¸n epifanei¸n gènouc mhdèn sth sfaÐra me stajer kampulìthta Gauss, to opoÐo ofeÐletai ston Calabi [6]. Gia to skopì autì ja anafèroume tic epifˆneiec Veronese, oi opoÐec eÐnai sumpageÐc kai èqoun stajer kampulìthta Gauss.
- 66. 58 KÔria apotelèsmata SÔmfwname gnwstì Je¸rhma tou Takahashi [19], mia isometrik embˆptish g = (g1, ..., gn+k+1) apì èna n-diˆstato polÔptugma Riemann Mn sthn Sn+k eÐnai elaqi- stik an kai mìno an oi sunart seic suntetagmènwn thc g eÐnai idiosunart seic tou telest Laplace ∆ tou Mn me antÐstoiqh idiotim n, dhlad an ∆gi = −ngi, i ∈ {1, ..., n + k + 1}. JewroÔme th didiˆstath sfaÐra S2(R) aktÐnac R > 0, efodiasmènh me th sun jh metrik Riemann. EÐnai gnwstì ìti oi idiotimèc tou telest Laplace thc S2(R) eÐnai λκ = κ(κ+1) R2 , ìpou κ eÐnai mh arnhtikìc akèraioc. Epiprìsjeta, o idioq¸roc Vλκ pou antistoiqeÐ sthn idiotim λκ, parˆgetai apì ta armonikˆ omogen polu¸numa bajmoÔ κ tou R3 periorismèna sthn S2(R) kai èqei diˆstash 2κ + 1. H epifˆneia Veronese sthn S2κ, ìpou κ eÐnai jetikìc akèraioc, eÐnai h isometrik elaqistik embˆptish fκ : S2 (R) −→ S2κ , R = κ(κ + 1) 2 , me fκ = (g1, ..., g2κ+1) kai g2 1 + ... + g2 2κ+1 = 1, ìpou {g1, ..., g2κ+1} eÐnai bˆsh tou idioq¸rou Vλκ apoteloÔmenh apì armonikˆ omogen polu¸numa bajmoÔ κ tou R3 pe- riorismèna sthn S2(R) kai h opoÐa eÐnai orjomonadiaÐa wc proc to eswterikì ginìmeno tou dianusmatikoÔ q¸rou D S2(R) , : D S2 (R) × D S2 (R) −→ R, (g, h) −→ g, h := S2(R) ghdS2 (R), ìpou dS2(R) eÐnai to stoiqeÐo embadoÔ thc S2(R). EÐnai t¸ra fanerì ìti oi sunart seic suntetagmènwn thc fκ eÐnai idiosunart seic tou telest Laplace thc S2(R). Sunep¸c, apì to Je¸rhma Takahashi gia kˆje jetikì akèraio κ h fκ eÐnai elaqistik epifˆneia. Profan¸c oi epifˆneiec Veronese eÐnai gènouc mhdèn kai èqoun stajer kampulìthta Gauss K = 2 κ(κ+1) . Gia parˆdeigma, h epifˆneia Veronese sthn S4 eÐnai h f2 : S2 ( √ 3) −→ S4 , f2(x, y, z) = xy √ 3 , xz √ 3 , yz √ 3 , x2 − y2 2 √ 3 , x2 + y2 − 2z2 6 . To akìloujo je¸rhma dhl¸nei pwc ousiastikˆ oi epifˆneiec Veronese eÐnai oi mìnec elaqistikèc epifˆneiec me autèc tic idiìthtec. Je¸rhma 3.1.4. 'Estw f : (M, , ) −→ S2m sumpag c, prosanatolismènh, kore- smènh, elaqistik epifˆneia gènouc mhdèn. An to M èqei stajer kampulìthta Gauss K, tìte K = 2 m(m+1) kai upˆrqoun isometrÐec F : (M, , ) −→ S2 m(m+1) 2 kai τ : S2m −→ S2m ¸ste f ◦ F−1 = τ ◦ fm, ìpou fm : S2 m(m+1) 2 −→ S2m eÐnai h epifˆneia Veronese sthn S2m. Apìdeixh. Apì thn Prìtash 2.3.2 èqoume tic sqèseic K = 1 − 2κ2 1, ∆ log(κ1...κm−1)2 = m(m + 1)K − 2
- 67. EikasÐa tou U.Simon 59 kai ∆ log κs = (s + 1)K − Ks sto Mm−1 gia kˆje s ∈ {1, ..., m − 1}. AfoÔ h kampulìthta Gauss K eÐnai stajer , to κ1 eÐnai stajerì kai sunep¸c ∆ log κ1 = 0. Lìgw thc ∆ log κ1 = 2K − K1 h teleutaÐa sqèsh èqei wc sunèpeia to K1 stajerì kai epeid sthn Prìtash 2.3.2 eÐdame ìti K1 = 2κ2 1 − 2 κ2 2 κ2 1 , èqoume κ2 stajerì kai ∆ log κ2 = 0, to opoÐo me th seirˆ tou mac dÐnei to K2 stajerì. Epagwgikˆ ftˆnoume sto sumpèrasma ìti κ1, ..., κm−1 eÐnai stajerèc kai epomènwc èqoume ∆ log(κ1...κm−1)2 = 0, dhlad K = 2 m(m+1). To M eÐnai pl rec kai aplˆ sunektikì. Apì to Je¸rhma Taxinìmhshc Aplˆ Su- nektik¸n Q¸rwn Morf c [14] upˆrqei isometrÐa F : M −→ S2 m(m+1) 2 . EÐnai fanerì ìti h f ◦ F−1 : S2 m(m+1) 2 −→ S2m eÐnai sumpag c koresmènh elaqistik epifˆneia gènouc mhdèn. 'Omwc kai h epifˆneia Veronese fm : S2 m(m+1) 2 −→ S2m eÐnai sumpag c koresmènh elaqistik epifˆneia gènouc mhdèn. Apì to Je¸rhma 3.1.3 upˆrqei isometrÐa τ thc S2m ¸ste τ ◦ fm = f ◦ F−1. 3.2 EikasÐa tou U. Simon H eikasÐa pou akoloujeÐ diatup¸jhke sto [15] kai èqei epalhjeujeÐ mìno se eidikèc peript¸seic. EikasÐa 3.2.1. 'Estw f : (M, , ) −→ Sn sumpag c, prosanatolismènh, elaqistik epifˆneia me kampulìthta Gauss K. An h K plhroÐ thn anisìthta K(s + 1) ≤ K ≤ K(s), ìpou K(s) := 2 s(s+1) , s jetikìc akèraioc, tìte eÐte K = K(s) kai upˆrqoun isometrÐec F : (M, , ) −→ S2 s(s+1) 2 kai τ : S2s −→ S2s ¸ste f ◦ F−1 = τ ◦ fs, ìpou fs : S2 s(s+1) 2 −→ S2s eÐnai h epifˆneia Veronese sthn S2s, K = K(s + 1) kai upˆrqoun isometrÐec F : (M, , ) −→ S2 (s+1)(s+2) 2 kai τ : S2(s+1) −→ S2(s+1) ¸ste f ◦ F−1 = τ ◦ fs+1, ìpou fs+1 : S2 (s+1)(s+2) 2 −→ S2(s+1) eÐnai h epifˆneia Veronese sthn S2(s+1). H eikasÐa èqei apodeiqjeÐ sto [15] stic peript¸seic ìpou s = 1 kai s = 2. Ja d¸soume mia apìdeixh gia autèc tic peript¸seic me th mejodologÐa pou anaptÔqjhke sthn paroÔsa ergasÐa, dhlad ja apodeÐxoume to akìloujo Je¸rhma 3.2.1. H eikasÐa tou Simon eÐnai alhj c gia s = 1 kai s = 2. Apìdeixh. Lìgw thc upìjeshc K(s+1) ≤ K ≤ K(s), s = 1, 2, èqoume M KdM > 0, ìpou dM eÐnai to stoiqeÐo embadoÔ tou M. Apì to Je¸rhma twn Gauss-Bonnet pro- kÔptei ìti to M eÐnai omoiomorfikì me thn S2 kai epomènwc M KdM = 4π. SÔmfwna me to Je¸rhma 3.1.1 h f eÐnai koresmènh se sfaÐra S2m ˆrtiac diˆstashc. DiakrÐnoume dÔo peript¸seic: (i) PerÐptwsh s = 1.
- 68. 60 KÔria apotelèsmata Exetˆzoumesumpag , koresmènh, elaqistik epifˆneia f : M −→ S2m gènouc mhdèn me kampulìthta Gauss 1 3 ≤ K ≤ 1. Apì to Je¸rhma 3.1.1 gia to embadì tou M isqÔei A(M) ≥ 2πm(m + 1). Epiplèon, 4π = M KdM ≥ M 1 3 dM ≥ 2π 3 m(m + 1), apì ìpou sumperaÐnoume ìti m = 1 2. An m = 1, tìte h f eÐnai olikˆ gewdaisiak kai apì th sqèsh (1.2) paÐrnoume ìti K = 1. An m = 2, tìte to sÔnolo M1 = {p ∈ M : B|p = 0} eÐnai mh kenì kai anoiktì uposÔnolo tou M. Epiplèon, apì thn Prìtash 2.3.2, sto M1 isqÔoun oi sqèseic K = 1−2κ2 1, K1 = 2κ2 1 = 1−K kai ∆ log κ1 = 2K −K1. Sunduˆzontac tic parapˆnw sqèseic kai lìgw thc upìjeshc paÐrnoume ∆ log(1 − K) = 2(3K − 1) ≥ 0. An u ∈ D(M) me u > 0, tìte isqÔei ∆ log u = ∆u u − |gradu|2 u2 . Prˆgmati, an {e1, e2} topikì orjomonadiaÐo plaÐsio tou M, tìte ∆ log u = 2 j=1 ej ej(log u) − ( ej ej)(log u) = 2 j=1 ej ej(u) u − ( ej ej)(u) u = 2 j=1 ej ej(u) u − (ej u) 2 u2 − ( ej ej)(u) u = 2 j=1 ej ej(u) − ( ej ej)(u) u − 2 j=1 ej(u) 2 u2 = ∆u u − |gradu|2 u2 . Epeid sto M1 isqÔei 1 − K > 0 lìgw thc (1.2) , paÐrnoume ∆ log(1 − K) = ∆(1 − K) 1 − K − |grad(1 − K)|2 (1 − K)2 . Epomènwc, èqoume ∆(1−K) ≥ |grad(1−K)|2 1−K ≥ 0 kai katˆ sunèpeia isqÔei ∆(1−K) ≥ 0 sto M1. Epeid to M1 eÐnai puknì uposÔnolo tou M isqÔei ∆(1−K) ≥ 0 se olìklhro to M. Apì thn Arq MegÐstou sunˆgetai ìti h kampulìthta Gauss eÐnai stajer . SÔmfwna me to Je¸rhma 3.1.4 to M èqei kampulìthta Gauss K = 1 3 kai upˆrqoun isometrÐec F : M −→ S2( √ 3) kai τ : S4 −→ S4 ¸ste f ◦ F−1 = τ ◦ f2, ìpou f2 : S2( √ 3) −→ S4 eÐnai h epifˆneia Veronese sthn S4. (ii) PerÐptwsh s = 2. Se aut th perÐptwsh èqoume sumpag , koresmènh, elaqistik epifˆneia f : M −→ S2m gènouc mhdèn me kampulìthta Gauss 1 6 ≤ K ≤ 1 3. Apì to Je¸rhma 3.1.1 gnw- rÐzoume ìti gia to embadì tou M isqÔei h anisìthta A(M) ≥ 2πm(m + 1), epomènwc 4π = M KdM ≥ M 1 6 dM ≥ π 6 m(m + 1).
- 69. EikasÐa tou U.Simon 61 Sunep¸c m = 1 m = 2 m = 3. An m = 1, tìte h f ja tan olikˆ gewdaisiak kai epomènwc h kampulìthta Gauss eÐnai Ðsh me 1, ˆtopo. 'Ara m = 2 m = 3. Upojètoume ìti m = 2. Tìte èqoume sumpag , koresmènh, elaqistik epifˆneia f : M −→ S4 gènouc mhdèn, epomènwc B = 0 kai B2 = 0 (κ1 = 0, κ2 = 0). Epeid 2K = − B 2 +2 sqèsh (1.2) , lìgw thc upìjeshc 1 6 ≤ K ≤ 1 3 prokÔptei ìti M1 = {p ∈ M : B|p = 0} = {p ∈ M : K|p < 1} = M. Apì thn Prìtash 2.3.2 sto M loipìn, isqÔei K = 1 − 2κ2 1, K1 = 2κ2 1 = 1 − K kai ∆ log κ1 = 2K − K1, epomènwc ∆ log(1 − K) = 2(3K − 1) ≤ 0. Apì thn Arq MegÐstou sunˆgetai ìti h kampulìthta Gauss K eÐnai stajer . Epiplèon, apì th sqèsh ∆ log(1−K) = 2(3K −1), sumperaÐnoume ìti K = 1 3 kai apì to Je¸rhma 3.1.4 upˆrqoun isometrÐec F : M −→ S2( √ 3) kai τ : S4 −→ S4 ¸ste f ◦ F−1 = τ ◦ f2, ìpou f2 : S2( √ 3) −→ S4 eÐnai h epifˆneia Veronese sthn S4. Upojètoume ìti m = 3. Tìte èqoume sumpag , koresmènh, elaqistik epifˆneia f : M −→ S6 gènouc mhdèn. Lìgw thc Prìtashc 2.3.2 sto anoiktì kai puknì uposÔnolo M2 tou M èqoume tic sqèseic K = 1 − 2κ2 1, K1 = 2κ2 1 − 2 κ2 2 κ2 1 kai K2 = 2 κ2 2 κ2 1 , apì ìpou paÐrnoume ìti K1 = 1 − K. Epiplèon, sto M2 isqÔoun oi sqèseic ∆ log κ1 = 2K − K1, ∆ log κ2 = 3K − K2 apì ìpou brÐskoume ∆ log(κ1κ2) = 6K − 1, ∆ log(κ2 1κ2 2) = 2(6K − 1) ≥ 0. (3.3) Epomènwc ∆(κ2 1κ2 2) κ2 1κ2 2 ≥ |grad(κ2 1κ2 2)|2 κ4 1κ4 2 ≥ 0. 'Eqoume dhlad ∆(κ2 1κ2 2) ≥ 0 sto puknì uposÔnolo M2 tou M. Lambˆnontac upìyh thn Prìtash 2.3.1(ii) èqoume ∆(g1g2) ≥ 0 sto M2. Lìgw sunèqeiac isqÔei ∆(g1g2) ≥ 0 sto M. Apì thn Arq MegÐstou sumperaÐnoume ìti h sunˆrthsh g1g2 eÐnai stajer sto M. Autì shmaÐnei ìti κ2 1κ2 2 eÐnai stajer kai lìgw thc sqèshc (3.3) èqoume K = 1 6 sto M2. Lìgw sunèqeiac isqÔei K = 1 6 sto M. SÔmfwna me to Je¸rhma 3.1.4 upˆrqoun isometrÐec F : M −→ S2( √ 6) kai τ : S6 −→ S6 ¸ste na isqÔei f ◦ F−1 = τ ◦ f3, ìpou f3 : S2( √ 6) −→ S6 eÐnai h epifˆneia Veronese sthn S6.
- 70.
- 71. PerÐlhyh Sthn paroÔsa ergasÐaexetˆzoume sumpageÐc, prosanatolismènec elaqistikèc e- pifˆneiec gènouc mhdèn sthn Sn. ApodeiknÔoume ìti sthn perÐptwsh pou eÐnai kore- smènec, to n eÐnai ˆrtioc kai epiplèon dÐnoume mia ektÐmhsh tou embadoÔ twn. Parˆl- lhla apodeiknÔoume ìti autèc eÐnai ˆkamptec. An ìmwc èqoun stajer kampulìthta Gauss ousiastikˆ eÐnai oi epifˆneiec Veronese. Ta apotelèsmata autˆ ofeÐlontai stouc E. Calabi, S.S. Chern kai J.L.M. Barbosa. We study compact, oriented minimal surfaces of genus zero in the sphere Sn. We prove that when such surfaces lie fully in Sn, then n is even, and provide an estimate for their area. Moreover, we show that these minimal surfaces are rigid. Furthermore, we prove that the Veronese surfaces are actually the only compact minimal surfaces of genus zero with constant Gaussian curvature. These results are due to E. Calabi, S.S. Chern and J.L.M. Barbosa. 63
- 72.
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