Biharmonic submanifolds in_three_dimensisional sphere

PANEPISTHMIO PATRWN
TMHMA MAJHMATIKWN
DIARMONIKES UPOPOLLAPLOTHTES THS
SFAIRAS S3
STELLA SEREMETAKH
MAJHMATIKOS
ErgasÐa gia metaptuqiakì dÐplwma eidÐkeushc
sta Jewrhtikˆ Majhmatikˆ
Epiblèpwn : Lèktorac Andrèac Arbanitoge¸rgoc
PATRA 2006

EuqaristÐec
Jewr¸ kaj kon, na ekfrˆsw tic eilikrineÐc kai pio jermèc mou euqaristÐec
ston epiblèponta Lèktora Andrèa Arbanitogèwrgo kaj¸c kai kai sta ˆl-
la dÔo mèlh thc TrimeloÔc Sumbouleutik c Epitrop c, Kajhght BasÐleio
PapantwnÐou kai Kajhght Ajanˆsio Kotsi¸lh gia th sumbol touc sthn
teleiopoÐhsh aut c thc metaptuqiak c ergasÐac.
Jewr¸ epÐshc upoqrèws mou na euqarist sw thn oikogèneiˆ mou gia thn
hjik kai oikonomik upost rixh pou mou prìsferan katˆ th diˆrkeia twn
spoud¸n mou.
Stèlla Seremetˆkh
Pˆtra, Septèmbrioc 2006

Perieqìmena
Prìlogoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
1 Basikèc 'Ennoiec 1
2 Logismìc twn metabol¸n 13
2.1 Eisagwg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Armonikèc kai diarmonikèc apeikonÐseic 23
3.1 Armonikèc apeikonÐseic . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Diarmonikèc apeikonÐseic . . . . . . . . . . . . . . . . . . . . . 38
4 Diarmonikèc Upopollaplìthtec 41
4.1 Eisagwg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Diarmonikèc kampÔlec sthn S3
. . . . . . . . . . . . . . . . . . 42
4.3 Diarmonikèc epifˆneiec sthn S3
. . . . . . . . . . . . . . . . . 49
bibliografÐa 61
i

ii PERIEQŸOMENA
Prìlogoc
Skopìc thc ergasÐac aut c eÐnai h anaz thsh twn diarmonik¸n upopol-
laplot twn Mm
, m = 1, 2, thc sfaÐrac S3
.
H mèjodoc pou efarmìzoume sundèetai me thn arq logismoÔ twn metabol¸n
wc mia mèjodoc sullog c twn bèltistwn antikeimènwn apì ènan q¸ro X me
ton ex c trìpo:
(1) Sullègoume ìla ta antikeÐmena sto q¸ro X.
(2) Epilègoume mia katˆllhlh sunˆrthsh E ston X. Tìte ta mègista
elˆqista thc sunˆrthshc eÐnai ta bèltista antikeÐmena pou anazhtˆme.
Pio sugkekrimèna, ja parousiˆsoume me ekten trìpo apotelèsmata apì tic
ergasÐec [6] , [8] twn R. Caddeo, S. Montaldo, C. Oniciuc, oi opoÐec aforoÔn
diarmonikèc upopollaplìthtec thc sfaÐrac S3
.
Analutikìtera, h diˆrjrwsh thc ergasÐac èqei ¸c ex c:
Sto kefˆlaio 1 parousiˆzontai sunoptikˆ orismoÐ kai ènnoiec apì th jewrÐa
pollaplot twn pou apaitoÔntai gia thn parousÐash thc metaptuqiak c er-
gasÐac.
Sto kefˆlaio 2 parousiˆzetai o sun jhc trìpoc prosèggishc tou logismoÔ
twn metabol¸n kaj¸c kai kˆpoiec gnwstèc jewrÐec pou phgˆzoun apì tic
mejìdouc metabol¸n.
Sto kefˆlaio 3 orÐzontai oi ènnoiec thc armonik c kai diarmonik c apeikì-
nishc metaxÔ duo pollaplot twn Riemann kai dÐnontai paradeÐgmata tètoiwn
apeikonÐsewn.
Sto kefˆlaio 4 anazhtoÔme tic diarmonikèc kampÔlec kai tic diarmonikèc epifˆneiec
thc sfaÐrac S3
. Oi kentrikèc mac anaforèc eÐnai oi ergasÐec [8] , [11] twn R.
Caddeo, S. Montaldo, C. Oniciuc kai J. Eells, L. Lemaire.

Kefˆlaio 1
Basikèc 'Ennoiec
Orismìc 1.1. Onomˆzoume topologik pollaplìthta diˆstashc
n ènan sunektikì topologikì q¸ro tou Hausdorﬀ me thn idiìthta se kˆje
shmeÐo tou na upˆrqei perioq omoiomorfik me èna anoiktì uposÔnolo tou
Rn
.
Tètoia pollaplìthta gia parˆdeigma eÐnai o q¸roc Rn
.
Orismìc 1.2. Topikìc qˆrthc pˆnw se mia n-diˆstath topologik pol-
laplìthta M lègetai kˆje duˆda (U, φ) ìpou φ eÐnai h omoiomorfik apeikì-
nish φ : U ⊆ Mn
→ V ⊆ Rn
, ìpou U eÐnai èna anoiktì uposÔnolo thc Mn
kai V èna anoiktì uposÔnolo tou EukleÐdiou q¸rou Rn
.
JewroÔme èna qˆrth (U, φ) miac topologik c pollaplìthtac Mn
. Tìte kˆje
shmeÐo p ∈ U kajorÐzetai apì tic suntetagmènec {x1(p), x2(p), ..., xn(p)} tou
shmeÐou φ(p) ∈ Rn
. Dhlad , xi(p) = xi(φ(p)) = (xi ◦ φ)(p), i = 1, 2, ..., n.
An to sÔnolo U eÐnai sunektikì, tìte oi arijmoÐ xi(p) lègontai topikèc
suntetagmènec tou shmeÐou p wc proc to qˆrth (U, φ) kai h n-ˆda twn
sunart sewn
xi : U ⊆ M → R
1

2 KEFŸALAIO 1. BASIKŸES ŸENNOIES
p → xi(p) = (φ(p))i , i = 1, 2, ..., n
lègetai sÔsthma topik¸n suntetagmènwn sto U wc proc to qˆrth
(U, φ), ìpou h i-suntetagmènh tou p eÐnai h i-suntetagmènh tou φ(p).
Epomènwc kˆje topikìc qˆrthc thc M orÐzei èna topikì sÔsthma suntetag-
mènwn aut c.
Orismìc 1.3. Onomˆzoume ˆtlanta diˆstashc n kai klˆshc Cr
pˆn-
w se mia n-diˆstath topologik pollaplìthta M, mia oikogèneia topik¸n
qart¸n Uα = {, Uαφα}α∈I (ìpou I eÐnai èna sÔnolo deikt¸n), pou ikanopoieÐ
ta parakˆtw axi¸mata :
(1) Ta sÔnola Uα kalÔptoun thn topologik pollaplìthta M, dhlad
α∈I
Uα = M
(2) An Uα ∩ Uβ = ∅, oi omoiomorfismoÐ φα kai φβ eÐnai tètoioi ¸ste o o-
moiomorfismìc
φβ ◦ φ−1
α : φα(Uα ∩ Uβ) ⊆ Rn
→ φβ(Uα ∩ Uβ) ⊆ Rn
na eÐnai amfidiaforÐsimoc klˆshc Cr
.
Orismìc 1.4. Oi qˆrtec c1 = (Uα, φα) kai c2 = (Uβ, φβ) klˆshc
Cr
, pˆnw se mia n-diˆstath topologik pollaplìthta M, onomˆzonai Cr
-
sumbibastoÐ, an
(1) Uα ∩ Uβ = ∅, efìson Uα ∩ Uβ = ∅,
(2) h apeikìnish
φβ ◦ φ−1
α : φα(Uα ∩ Uβ) ⊆ Rn
→ φβ(Uα ∩ Uβ) ⊆ Rn
na eÐnai klˆshc Cr
.

3
Orismìc 1.5. DÔo Cr
-ˆtlantec U1,U2 diˆstashc n miac topologik c pol-
laplìthtac M onomˆzontai Cr
-sumbibastoÐ, an
(1) U1 ∪ U2 eÐnai pˆli ènac Cr
-ˆtlantac thc M kai
(2) An c1 ∈ U1 kai c2 ∈ U2 eÐnai dÔo tuqaÐoi qˆrtec, tìte oi qˆrtec autoÐ
eÐnai Cr-sumbibastoÐ.
Orismìc 1.6. DiaforÐsimh pollaplìthta diˆstashc n kai klˆshc
Cr
, onomˆzoume kˆje n-diˆstath topologik pollaplìthta M, efodiasmènh
me mia klˆsh isodÔnamwn Cr
-sumbibast¸n atlˆntwn pˆnw sth M.
Upojètoume ìti M eÐnai mia diaforÐsimh pollaplìthta diˆstashc n, tˆxhc
diaforisimìthtac r( klˆshc Cr
) kai ìti A eÐnai èna anoiktì uposÔnolo thc
M.
Orismìc 1.7. H sunˆrthsh f : A ⊆ M → R onomˆzetai diaforÐsimh
tˆxhc r ( klˆshc Cr
) pˆnw sto A an h sunˆrthsh
f ◦ φ−1
: φ(U ∩ A) ⊆ Rn
→ R
eÐnai diaforÐsimh gia kˆpoio qˆrth (U, φ) pˆnw sth M.
To sÔnolo twn diaforÐsimwn sunart sewn klˆshc Cr
, pou orÐzontai sth n-
diˆstath pollaplìthta M klˆshc Cr
, sumbolÐzetai me Dr
(M), en¸ to sÔnolo
twn diaforÐsimwn sunart sewn pou orÐzontai sthn pollaplìthta M, klˆshc
C∞
, sumbolÐzetai me D0
(M).

Orismìc 1.8. H apeikìnish f : A ⊆ Mn
→ Nm
onomˆzetai dia-
forÐsimh klˆshc Cr
sto shmeÐo p ∈ A, an gia kˆje qˆrth (U, φ) thc M
kai (V, ψ) thc N tètoio ¸ste p ∈ U kai f(p) ∈ V , h apeikìnish
F = ψ ◦ f ◦ φ−1
: φ(U ∩ f−1
(V )) ⊆ Rn
→ Rm
na eÐnai diaforÐsimh klˆshc Cr
sto shmeÐo φ(p) ∈ Rn
.
'Estw to sÔnolo Dr
(M, p) ìlwn twn diaforÐsimwn sunart sewn klˆshc Cr
sto shmeÐo p ∈ M. To sÔnolo Dr
(M, p) apoteleÐ dianusmatikì q¸ro, o
opoÐoc gÐnetai ˆlgebra an orÐsoume wc deÔtero nìmo eswterik c sÔnjeshc
ton pollaplasiasmì sunart sewn.
Orismìc 1.9. An p eÐnai èna tuqaÐo shmeÐo thc n-diˆstathc pollaplìthtac
M kai X = (X1
, X2
, ..., Xn
) èna diˆnusma sto shmeÐo p, onomˆzoume
Efaptìmeno diˆnusma sto shmeÐo p thc n-diˆstathc pollaplìthtac M
thn apeikìnish
Xp : Dr
(M, p) → R
me tim
Xp(φ) =
n
i
(
∂φ
∂xi
)Xi
p
pou ikanopoieÐ tic parakˆtw sunj kec :
(1) Xp(λf + µg) = λXpf + µXpg
(2) Xp(fg) = f(p)Xg(f) + g(p)Xp(f),
gia kˆje f, g ∈ Dr
(M, p), λ, µ ∈ R
To sÔnolo twn efaptìmenwn dianusmˆtwn sto shmeÐo p miac diaforÐsimhc pol-
laplìthtac M, apoteleÐ dianusmatikì q¸ro. Ton dianusmatikì autì q¸ro ton

5
lème efaptìmeno q¸ro thc M sto shmeÐo p kai ja ton sumbolÐzoume
me TpM.
Orismìc 1.10. O duikìc q¸roc tou TpM eÐnai o grammikìc q¸roc pou
apoteleÐtai apì to sÔnolo twn grammik¸n apeikonÐsewn me pedÐo orismoÔ to
q¸ro TpM kai timèc sto sÔnolo R. O q¸roc autìc sumbolÐzetai me T∗
p M,
eÐnai isomorfikìc me ton TpM kai onomˆzetai sunefaptìmenoc q¸roc thc M
sto p. H sullog ìlwn twn efaptìmenwn (sunefaptìmenwn) q¸rwn thc M
se kˆje shmeÐo aut c sumbolÐzetai me TM (T∗
M antÐstoiqa) kai lègetai e-
faptìmenh dèsmh (sunefaptìmenh dèsmh antÐstoiqa),
TM =
p∈M
TpM = (p, Xp); p ∈ M, Xp ∈ TpM
kai
T∗
M =
p∈M
T∗
p M
Orismìc 1.11. Mia diaforik morf pr¸thc tˆxhc diaforik 1-morf
epÐ thc diaforÐsimhc pollaplìthtac M onomˆzetai h apeikìnish
ω : M →
p∈M
T∗
p M
h opoÐa se kˆje shmeÐo p ∈ M antistoiqeÐ to sunefaptìmeno diˆnusma ωp
tou sunefaptìmenou q¸rou T∗
p M. Dhlad gia kˆje p ∈ M h antÐstoiqh di-
aforik 1-morf eÐnai mia grammik morf pˆnw ston TpM, (ωp : TpM → R).
Ean D1
(M) eÐnai to sÔnolo twn dianusmatik¸n pedÐwn epÐ thc M kai D1(M)
to duikì tou sÔnolo, tìte wc diaforÐsimec 1-morfèc orÐzontai ta stoiqeÐa tou

D1(M) ìpou D1(M) = ω; ω : D1
(M) → D0
(M) kai h ω eÐnai diaforÐsimh
grammik apeikìnish en¸ D0
(M) eÐnai o q¸roc twn diaforÐsimwn sunart sewn.
'Estw M, N dÔo diaforÐsimec pollaplìthtec kai φ mia apeikìnish apì th
M sth N.
Orismìc 1.12. H apeikìnish
dφp : TpM → Tφ(P)N
me tim
dφp : Xp → dφp(Xp)
onomˆzetai diaforikì thc apeikìnishc φ : M → N sto shmeÐo p. Sum-
bolÐzetai epÐshc kai me φ∗p kai eÐnai mia grammik apeikìnish tou efaptìmenou
q¸rou TpM ston efaptìmeno q¸ro Tφ(p)N, ìpou,
dφp(Xp) : D0
(N) → R
dφp(Xp) : g → dφp(Xp)g = Xp(g ◦ ϕ)
Orismìc 1.13. An φ : M → N eÐnai mÐa diaforÐsimh apeikìnish, to
diaforikì dφp thc φ sto p ∈ M eÐnai mÐa grammik apeikìnish
dφp : TpM → R
me tim pou orÐzetai mèsw thc apeikìnishc
dφp : Xp → dφp(Xp)

7
kai exaitÐac thc isomorfik c taÔtishc Tφ(p)R ≡ R èqoume
dφp(Xp) = Xp(φ)
Orismìc 1.14. 'Estw φ : M → N mia diaforÐsimh apeikìnish metaxÔ twn
pollaplot twn M kai N kai èstw ω ∈ T∗
M. Onomˆzoume apeikìnish
epistrof c (pull back) thc ω mèsou thc φ thn apeikìnish
φ∗
ω : Tφ(p)N → TpM
me tim
φ∗
ω(u1, u2, u3) = ω(φ∗(u1), ..., φ∗(un))
gia kˆje ui ∈ TpM, i = 1, 2, ..., n kai p ∈ M.
Orismìc 1.15. O metrikìc tanust c Riemann eÐnai ènac sunalloÐ-
wtoc tanust c tÔpou (0,2), tètoioc ¸ste se kˆje shmeÐo p ∈ M antistoiqeÐ
thn apeikìnish
, : TpM × TpM → R
me tic akìloujec idiìthtec:
1. (i) vp + wp, zp kai (ii) λvp, wp = λ vp, wp
2. vp, wp = wp, vp
3. vp, vp ≥ 0 me vp, vp = 0 an kai mìno an vp = 0,
gia kˆje vp, wp, zp ∈ TpM.
Orismìc 1.16. Kˆje pollaplìthta M efodiasmènh me mia metrik Riemann
, , lègetai pollaplìthta Riemann.
Orismìc 1.17. Sunoq sunalloÐwth parˆgwgo se mia C∞
−
pollaplìthta M kaloÔme thn apeikìnish :
: D1
(M) × D1
(M) → D1
(M)

(X, Y ) → XY
pou ikanopoieÐ tic akìloujec sunj kec :
(1) X(Y + Z) = XY + XZ
(2) X+Y Z = XZ + Y Z
(3) fXY = f XY
(4) X(fX) = (Xf)Y + f XY
gia kˆje f ∈ C∞
(M) kai X, Y ∈ D1
(M)
(me D1
(M) sumbolÐzoume to sÔnolo ìlwn twn dianusmatik¸n pedÐwn epÐ thc
M.)
Orismìc 1.18. Gia kˆje dianusmatikì pedÐo X, Y ∈ D1
(M) to diaforÐsimo
dianusmatikì pedÐo thc M,
[X, Y ] = XY − Y X
pou dra sto q¸ro D0
(M) twn diaforÐsimwn sunart sewn thc M me tim
[X, Y ]f = X(Y f) − Y (Xf)
pou eÐnai epÐshc mia diaforÐsimh sunˆrthsh, gia kˆje f ∈ D0
(M), lègetai
agkÔlh tou Lie twn dianusmatik¸n pedÐwn X, Y tou D1
(M).
Je¸rhma 1.1. 'Estw (M, g) mia C∞
− pollaplìthta Riemann diˆstashc
n. H sunoq pou ikanopoieÐ th sqèsh
2g( XY, Z) = X(g(Y, Z))+Y (g(Z, X))−Z(g(X, Y ))+g(Z, [X, Y ])+g(Y, [Z, X])−g(X, [Y, Z])
gia kˆje X, Y, Z ∈ D1
(M), kaleÐtai Levi-Civita
Epiplèon h sunoq Levi-Civita ikanopoieÐ tic sunj kec
(1) X(g(Y, Z)) = g( XY, Z) + g(Y, XZ)
(2) XY − Y X = [X, Y ]
Antistrìfwc, kˆje sunoq pou ikanopoieÐ tic (1) kai (2) eÐnai Levi-Civita.

9
Orismìc 1.19. Tanustikì pedÐo kampulìthtac R miac pol-
laplìthtac M efodiasmènhc me mia sÔndesh kaleÐtai to tanustikì pedÐo
tÔpou (1,3) me tim
R(X, Y )Z = X Y Z − Y XZ − [X,Y ]Z, gia kˆje X, Y, Z ∈ D1
(M).
To tanustikì pedÐo kampulìthtac ikanopoieÐ tic akìloujec sqèseic:
(1) R(X, Y )Z = −R(Y, X)Z
(2) R(X1 + X2, Y ) = R(X1, Y ) + R(X2, Y )
(3) R(X, Y1 + Y2) = R(X, Y1) + R(X, Y2)
(4) R(fX, gY )hZ = fghR(X, Y )Z
Eˆn h sunoq eÐnai summetrik , tìte isqÔoun oi tautìthtec
(5) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0
(6) ( XR)(Y, Z)W + ( Y R)(Z, X)W + ( ZR)(X, Y )W = 0
gia kˆje X, Y, Z ∈ D1
(M) kai f, g, h ∈ D0
(M).
Oi tautìthtec (5) kai (6) kaloÔntai pr¸th kai deÔterh tautìthta tou Bianchi
antÐstoiqa.
Orismìc 1.20. SunalloÐwto tanustikì pedÐo twn Cristoﬀel-
Riemann lègetai h tetragrammik apeikìnish
R : D1
(M) × D1
(M) × D1
(M) × D1
(M) → D0
(M) me tim
R(X, Y, Z, W) = g(R(X, Y )Z, W) gia kˆje X, Y, Z, W ∈ D1
(M).
To sunalloÐwto tanustikì pedÐo twn Cristoﬀel-Riemann ikanopoeÐ tic akìlou-
jèc idiìthtec:
(1) R(X, Y, Z, W) = −R(Y, X, Z, W) = −R(X, Y, W, Z) = R(Y, X, Z, W)

(2) R(X, Y, Z, W) + R(X, Z, W, Y ) + R(X, W, Y, Z) = 0
(3) ( XR)(Y, Z, W, V ) + ( Y R)(Z, X, W, V ) + ( ZR)(X, Y, W, V ) = 0
'Ena ˆllo tanustikì pedÐo pou orÐzetai apì to tanustikì pedÐo twn Cristoﬀel-
Riemann , eÐnai to tanustikì pedÐo tou Ricci kai gia kˆje shmeÐo thc
pollaplìthtac M o antÐstoiqoc tanust c tou Ricci.
'Estw p ∈ M tuqaÐo shmeÐo thc pollaplìthtac M kai TpM o efaptìmenoc
q¸roc aut c sto shmeÐo p. JewroÔme thn apeikìnish
R(−, X)Y : TpM → TpM
me tim
R(−, X)Y : Z → R(Z, X)Y
Orismìc 1.20. O tanust c tou Ricci orÐzetai wc to Ðqnoc thc apeikìni-
shc, R(−, X)Y , gia kˆje X, Y ∈ TpM, kai sumbolÐzetai me S(X, Y ) Ric(X, Y ).
Me th bo jeia topikoÔ sust matoc suntetagmènwn {xi}n
i=1, o tanust c tou
Ricci grˆfetai wc èxhc:
S(X, Y ) = Ric(X, Y ) =
n
i=1
g(R(ei, X)Y, ei) =
n
i=1
R(ei, X, Y, ei)
ìpou X, Y ∈ TpM kai {e1, e2, ..., en} eÐnai mia orjokanonik bˆsh tou TpM wc
proc to topikì sÔsthma suntetagmènwn {xi}n
i=1.
Orismìc 1.21. Metasqhmatismìc tou Ricci telest c kampu-
lìthtac tou Ricci sto shmeÐo p ∈ M, wc proc to efaptìmeno diˆnusma
X ∈ TpM, lègetai h apeikìnish
Sx = R(−, X)X : TpM → TpM
me tim
Sx : Y → Sx(Y ) = R(Y, X)X

11
Orismìc 1.22. Bajmwt kampulìthta thc pollaplìthtac (M, g)
lègetai h sunˆrthsh sg h opoÐa orÐzetai apì th sustol twn deikt¸n tou
tanustikoÔ pedÐou tou Ricci kai dÐnetai apì th sqèsh
sg =
n
i,j=1
gij
Sij =
n
i=1
Ric(ei, ei) =
n
i=1
g(Qei, ei)
ìpou Q = n
i=1 R(−, ei)ei eÐnai o metasqhmatismìc tou Ricci kai {e1, e2, ..., en}
eÐnai mia orjokanonik bˆsh tou TpM.
Orismìc 1.23. Gia dÔo grammik¸c anexˆrthta dianÔsmata u, v tou
efaptìmenou q¸rou TpM sto p thc pollaplìthtac Riemann (M, g) o arijmìc
K(u, v) =
g(R(u, v)v, u)
g(u, u)g(v, v) − g(u, v)2
lègetai kampulìthta tom c thc (M, g) wc proc to zeÔgoc u, v . H
(M, g) èqei jetik (arnhtik ) kampulìthta tom c eˆn gia kˆje p ∈ M kai
gia duo grammik¸c anexˆrthta dianÔsmata u, v tou TpM, K(u, v) ≥ 0,
(K(u, v) ≤ 0).

Kefˆlaio 2
Logismìc twn metabol¸n
2.1 Eisagwg
O logismìc twn metabol¸n eÐnai mia jewrÐa pou basÐzetai sthn idèa ìti eÐnai
dunatìn na ermhneujoÔn pollˆ fainìmena sta majhmatikˆ kai sth fusik wc
krÐsima shmeÐa sunarthsoeid¸n. Sto kefˆlaio autì ja parousiˆsoume merikèc
shmantikèc jewrÐec twn majhmatik¸n kai thc fusik c, oi opoÐec proèrqontai
apì mejìdouc tou logismoÔ metabol¸n (variational methods) kaj¸c kai pa-
radeÐgmata twn mejìdwn aut¸n. H arq tou logismoÔ twn metabol¸n eÐnai mia
mèjodoc sullog c twn bèltistwn apì mia sullog majhmatik¸n antikeimènwn
me ton ex c trìpo:
(1) Sullègoume ìla ta antikeÐmena apì èna q¸ro X.
(2) Epilègoume mia katˆllhlh sunˆrthsh E ston X. Ta mègista ta
elˆqista thc sunˆrthshc aut c eÐnai ta bèltista antikeÐmena pou anazhtoÔme.
ArketoÐ epist monec, ìpwc oi I. Newton, G.W. Leibnitz, P.L. Maupertuis, L.
Euler kai J.L. Lagrange asqol jhkan me to logismì metabol¸n.
13

14 KEFŸALAIO 2. LOGISMŸOS TWN METABOLŸWN
O sunhjismènoc trìpoc prosèggishc tou logismoÔ twn metabol¸n eÐnai o
ex c:
(1) Sto q¸ro X jewr¸ to diaforikì E thc sunˆrthshc E.
(2) Eˆn to x ∈ X eÐnai èna apì ta bèltista majhmatikˆ antikeÐmena tìte
autì epitugqˆnei thn elaqistopoÐhsh th megistopoÐhsh thc sunˆrthshc E.
Epomènwc h parˆgwgoc thc E mhdenÐzetai sto x, dhlad E (x) = 0.
(3) To shmeÐo x pou ikanopoieÐ th sqèsh E (x) = 0 kaleÐtai krÐsimo
shmeÐo. H parapˆnw sqèsh antistoiqeÐ sthn exÐswsh twn Euler-Lagrange.
(4) Skopìc eÐnai na lujeÐ h exÐswsh aut .
Kˆpoiec forèc sqediˆzoume thn antÐstrofh diadikasÐa:
(1) Jèloume na lÔsoume tic diaforikèc exis¸seic kˆpoiou problhmatìc mac
(2) Gia na pragmatopoihjeÐ autì, jewroÔme ènan q¸ro X kai mia sunˆrthsh
E ston X ètsi ¸ste h exÐswsh twn Euler-Lagrange na antistoiqeÐ sthn exÐsw-
sh tou problhmatìc mac.
(3) ArkeÐ tìte na brejeÐ èna elˆqisto mègisto thc sunˆrthshc E ston X.
Sto mèso thc dekaetÐac tou 1960, oi R. Palais kai S. Smale dieukrÐnhsan kˆtw
apì poièc sunjhkèc h sunˆrthsh E èqei elˆqista. H sunj kh aut kaleÐtai
sunj kh twn Palais-Smale (P-S) kai perigrˆfetai wc ex c : Upojètoume ìti
(M, g) eÐnai mia Ck+1
-pollaplìthta Riemann kai f : M → N mia Ck+1
-
sunˆrthsh ( k ≥ 1) kai èstw S èna uposÔnolo thc M. H f ikanopoieÐ th
sunj kh (P-S) eˆn isqÔoun ta ex c:
(1) H f eÐnai fragmènh sto S kai
(2) inf f(x) : x ∈ S = 0
Tìte upˆrqei shmeÐo x sth j kh ¯S tou S, ètsi ¸ste to x na eÐnai krÐsimo
shmeÐo thc f, dhlad fx= 0. ( f : M → fx ∈ TxM gia kˆje x ∈ M).

2.1. EISAGWGŸH 15
Gia na exhg soume th sunj kh (P-S) jewroÔme to ex c parˆdeigma :
'Estw duo sunart seic f kai g ston M = R me tÔpouc,
(1) f(x) = x2
, −∞ < x < ∞
(2) g(x) = ex3
, −∞ < x < ∞
Kai oi duo sunart seic èqoun inﬁma mhdèn. H pr¸th èqei elˆqisto sto
shmeÐo (0, 0), en¸ h deÔterh den èqei elˆqisto.
PoÔ ofeÐletai to parapˆnw fainìmeno;
H apˆnthsh eÐnai ìti h sunˆrthsh f(x) ikanopoieÐ th sunj kh (P-S), en¸
h sunˆrthsh g(x) ìqi. SumbaÐnei wstìso, gia kˆpoia probl mata pou den
ikanopoioÔn th sunj kh (P-S) h sunˆrthsh E na èqei elˆqisto.
1.1. Mèjodoc twn metabol¸n kai jewrÐec pedÐou
H mèjodoc twn metabol¸n brÐskei efarmog sth fusik , kurÐwc stic jew-
rÐec pedÐou (ﬁeld theories). Se aut n th parˆgrafo ja d¸soume mia eikìna
twn armonik¸n apeikonÐsewn kai ˆllwn jewri¸n pedÐou. EÐnai gnwstì ìti
sth fÔsh upˆrqoun tessˆrwn eid¸n dunˆmeic, h barÔthta (gravitation), h h-
lektromagnhtik dÔnamh (electromagnetism), h asjènhc allhlepÐdrash (weak
interaction) kai h isqur allhlepÐdrash (strong interaction). Eqoun gÐnei
prospˆjeiec na sumperilhfjoÔn oi dunˆmeic autèc se mia enwpoihmènh jewrÐa
pedÐou. H barÔthta èqei perigrafeÐ apì th jewrÐa sqetikìthtac tou Einstein
kai o hlektromagnhtismìc apì th jewrÐa tou Maxwell. Autèc oi tèsseric
dunˆmeic èqoun katagrafeÐ apì touc fusikoÔc wc jewrÐec bajmÐdac.
Ja perigrˆyoume tic jewrÐec autèc ìpwc phgˆzoun apì tic mejìdouc metabol¸n.
Metrikèc tou Einstein
'Estw M mia pollaplìthta diˆstas c m kai X o q¸roc ìlwn twn metrik¸n
Riemann g sth M pou èqoun ìgko monˆda. 'Estw E h sunˆrthsh ston X,

pou dÐdetai apì th sqèsh
E(g) =
M
Sgvg, g ∈ X,
ìpou Sg h bajmwt kampulìthta thc g kai vg to stoiqeÐo ìgkou pou dÐnetai
apì th sqèsh vg = det(gij).dx1...dxm
H sunˆrthsh E onomˆzetai sunarthsoeidèc olik c kampulìthtac.
JewroÔme mia tuqaÐa metabol (deformation) gt , (− < t < ) , g0 = g thc g.
Tìte h g eÐnai krÐsimo shmeÐo thc E ston X an kai mìno an
d
dt t=0
E(gt) = 0
to opoÐo apodeiknÔetai ìti isodunameÐ me thn exÐswsh
Ric(g) = cg
ìpou Ric(g) eÐnai o tanust c Ricci thc g kai c mia stajerˆ.
Mia metrik g pou ikanopoieÐ th parapˆnw exÐswsh kaleÐtai metrik tou Ein-
stein.
Sunoqèc Yang - Mills (Yang - Mills Connections)
Estw E mia dianusmatik dèsmh se mia sumpag pollaplìthta Riemann
(M, g). JewroÔme to q¸ro X ìlwn twn sunoq¸n thc dianusmatik c dèsmhc
E kai th sunˆrthsh E ston X me tÔpo
E( ) =
1
2 M
R
2
vg, ∈ X
O R eÐnai o tanust c kampulìthtac thc sunoq c sth dianusmatik
dèsmh E. JewroÔme mia metabol (deformation) t , (− < t < ), 0 =
thc .
Tìte h sunoq apoteleÐ krÐsimo shmeÐo thc E an kai mìno an
d
dt t=o
E( t) = 0

2.1. EISAGWGŸH 17
Ta krÐsima shmeÐa tou parapˆnw sunarthsoeidoÔc kaloÔntai sunoqèc Yang-
Mills.
Armonikèc apeikonÐseic
'Estw dÔo sumpageÐc pollaplìthtec Riemann (M, g) kai (N, h) kai èstw
to sÔnolo X ìlwn twn leÐwn apeikonÐsewn apì th M sth N, dhlad X =
C∞
(M, N). 'Estw h sunˆrthsh E ston X pou dÐnetai apì th sqèsh
E(φ) =
1
2 M
|dφ|2
vg, φ ∈ X
ìpou h apeikìnish dφ : TM → TN eÐnai to diaforikì thc φ.
'Estw mia tuqoÔsa metabol φt , (− < t < ) , φ0 = φ , thc φ.
(Bl. sq ma 2.1)
Tìte, h φ eÐnai armìnikh apeikìnish an kai mìno an eÐnai krÐsimo shmeÐo
thc E, dhlad an kai mìno an
d
dt t=0
E(φt) = 0
Parˆdeigma : Kleistèc gewdaisiakèc sth sfaÐra
'Estw mia kleist diaforÐsimh kampÔlh φ(x) = (φ1(x), φ2(x), φ3(x)), x ∈
[0, 2π] ston R3
me perÐodo 2π. (Periodikìthta shmaÐnei ìti: φ(x+2π) = φ(x),
dhlad φi(x + 2π) = φi(x), i = 1, 2, 3). AnazhtoÔme tic kampÔlec ekeÐnec pou
apoteloÔn krÐsima shmeÐa tou sunarthsoeidoÔc thc enèrgeiac
E(φ) =
1
2
2π
0
3
i=1
dφi
dx
2
dx
'Estw φε(x) = (φε,1(x), φε,2(x), φε,3(x)), x ∈ [0, 2π] mia metabol thc φ me
φ0 = φ kai φε(x + 2π) = φε(x), x ∈ [0, 2π]
'Eqoume ìti
d
dε ε=0
E(φε) =
1
2
2π
0
d
dε ε=0
3
i=1
dφε,i
dx
2
dx =
2π
0
3
i=1
d
dε ε=0
dφε,i(x)
dx
dφi(x)
dx
dx

=
3
i=1
d
dε ε=0
φε,i(x)
dφi(x)
dx
x=2π
x=0
−
2π
0
3
i=1
d
dε ε=0
φε,i(x)
d2
φi(x)
dx2
dx
Epeid oi φε,i kai φi eÐnai periodikèc me perÐodo 2π o pr¸toc ìroc tou deÔterou
mèlouc mhdenÐzetai, opìte prokÔptei ìti
d
dε ε=0
E(φε) =
2π
0
3
i=1
d
dε ε=0
φε,i(x)
d2
φε,i(x)
dx2
dx
Epiplèon, epeid h φε(x) = (φε,1(x), φε,2(x), φε,3(x)) eÐnai mia leÐa metabol
thc φ tìte kai h
d
dε ε=0
φε(x) =
d
dε ε=0
φε,1(x),
d
dε ε=0
φε,2(x),
d
dε ε=0
φε,3(x)
eÐnai leÐa periodik apeikìnish .
Epomènwc h φ eÐnai krÐsimo shmeÐo thc enèrgeiac an kai mìno an
d
dε ε=0
E(φε) = 0,
isodÔnama
d2
φi(x)
dx2
= 0, i = 1, 2, 3
H lÔsh twn exis¸sewn eÐnai
φi(x) = Bix + Ai, i = 1, 2, 3
ìpou ta Ai, Bi eÐnai stajerèc. ExaitÐac thc periodikìthtac twn φi(x) èqoume
ìti (x + 2π)Bi + Ai = xBi + Ai,dhlad Bi = 0, opìte φi(x) = Ai gia kˆje
x ∈ [0, 2π]. Epeid oi lÔseic pou lambˆnoume sth perÐptwsh aut eÐnai mìno
oi tetrimmènec, eisˆgoume ton ex c periorismì: ApaitoÔme oi kampÔlec φ na
brÐskontai sth monadiaÐa sfaÐra S2
= (y1, y2, y3) ∈ R3
; y2
1 + y2
2 + y2
3 = 1 kai
anazhtoÔme ta krÐsima shmeÐa thc E, metaxÔ twn kampul¸n aut¸n.

2.1. EISAGWGŸH 19
Me ton Ðdio trìpo pou perigrˆyame parapˆnw, jewroÔme mia metabol φε(x)
thc φ , x ∈ [0, 2π] . Tìte h φ ∈ S2
eÐnai krÐsimo shmeÐo an kai mìno an
d
dε ε=0
E(φε) = 0
isodÔnama
2π
0
3
i=1
d
dε ε=0
φε,i(x)
d2
φi(x)
dx2
= 0
Sto shmeÐo autì prèpei na lˆboume upìyhn to periorismì φε(x) ∈ S2
, x ∈
[0, 2π]. Gia to lìgo autì, jewroÔme ton efaptìmeno q¸ro
TyS2
= V ∈ R3
; V, y = 0 thc S2
se èna y ∈ S2
, pou eÐnai to kˆjeto
epÐpedo sto diˆnusma y.
Kˆje diˆnusma V ∈ R3
mporeÐ na analujeÐ se duo sunist¸sec, mia sto kˆ-
jeto q¸ro (TyS2
)
⊥
kai mia ston TyS2
, dhlad V = V, y y + (V − V, y y)
ExaitÐac thc sunj khc φε(x) ∈ S2
gia kˆje x ∈ [0, 2π], to φε(x), φε(x) = 1.
ParagwgÐzontac th teleutaÐa sqèsh sto ε = 0 kai lambˆnontac upìyh ìti
φ0(x) = φ(x) èqoume ìti
(
d
dε
)
ε=0
φε(x), φε(x) = 0
dhlad
(
d
dε
)
ε=0
φε(x) ∈ Tφ(x)S2
Lìgw thc sqèshc V = V, y y + (V − V, y y) to diˆnusma
d2
φ
dx
=
d2
φ1
dx2
,
d2
φ2
dx2
,
d2
φ3
dx2
analÔetai wc ex c :
d2
φ
dx2
=
d2
φ(x)
dx2
, φ(x) φ(x) +
d2
φ(x)
dx2
−
d2
φ(x)
dx2
, φ(x) φ(x)
kai epeid o deÔteroc ìroc an kei ston Tφ(x)S2
autìc eÐnai mhdèn.
'Ara,
d2
φ(x)
dx2
=
d2
φ(x)
dx2
, φ(x) φ(x)

ParagwgÐzoume th sqèsh φ(x), φ(x) = 1 gia kˆje x sto [0, 2π] kai èqoume
dφ(x)
dx
, φ(x) = 0
ParagwgÐzontac xanˆ paÐrnoume
d2
φ(x)
d2(x)
, φ(x) +
dφ(x)
dx
,
dφ(x)
dx
= 0
isodÔnama
d2
φ(x)
dx2
, φ(x) = −
dφ(x)
dx
,
dφ(x)
dx
ExaitÐac thc teleutaÐac sqèshc h
d2
φ(x)
dx2
=
d2
φ(x)
dx2
, φ(x) φ(x)
paÐrnei th morf
d2
φ(x)
d2(x)
+
dφ(x)
dx
,
dφ(x)
dx
φ(x) = 0
Sth sunèqeia paragwgÐzoume to eswterikì ginìmeno
dφ(x)
dx
,
dφ(x)
dx
kai èqoume
d
dx
dφ(x)
dx
,
dφ(x)
dx
= 2
d2
φ(x)
dx2
,
dφ(x)
dx
.
Lìgw thc
d2
φ(x)
dx2
=
d2
φ(x)
dx2
, φ(x) φ(x)
h parapˆnw sqèsh gÐnetai
d
dx
dφ(x)
dx
,
dφ(x)
dx
= 2
d2
φ(x)
dx2
,
dφ(x)
dx
= −2
dφ(x)
dx
,
dφ(x)
dx
φ(x),
dφ(x)
dx
kai lìgw thc
dφ(x)
dx
, φ(x) = 0

2.1. EISAGWGŸH 21
èqoume telikˆ ìti
d
dx
dφ(x)
dx
,
dφ(x)
dx
= 2
d2
φ(x)
dx2
,
dφ(x)
dx
= −2
dφ(x)
dx
,
dφ(x)
dx
φ(x),
dφ(x)
dx
=
−2
dφ(x)
dx
,
dφ(x)
dx
φ(x),
dφ(x)
dx
= 0
Epomènwc to eswterikì ginìmeno
dφ(x)
dx
,
dφ(x)
dx
eÐnai stajerì gia kˆje x ∈ [0, 2π]. Jètoume
dφ(x)
dx
,
dφ(x)
dx
= c2
, c > 0
kai h sqèsh
d2
φ(x)
d2(x)
+
dφ(x)
dx
,
dφ(x)
dx
φ(x) = 0
gÐnetai
d2
φ(x)
dx2
+
dφ(x)
dx
,
dφ(x)
dx
φ(x) = 0
IsodÔnama
d2
φi(x)
dx2
+ c2
φi = 0, i = 1, 2, 3
H genik lÔsh tou sust matoc eÐnai
φi(x) = Ai cos(cx) + Bi sin(cx) ⇔ φ(x) = A cos(cx) + B sin(cx)
ìpou ta A kai B eÐnai dianÔsmata ston R3
.
Ikan kai anagkaÐa sunj kh ¸ste h kampÔlh φ(x), x ∈ [0, 2π] na eÐnai
periodik me perÐodo 2π, na keÐtai sth sfaÐra S2
kai na apoteleÐ krÐsimo shmeÐo
thc E eÐnai : A, A = B, B = 1, A, B = 0 kai c = m (akèraioc) Mia tètoia
kampÔlh eÐnai ènac mègistoc kÔkloc thc sfaÐrac S2
kai diagrˆfetai m forèc
kaj¸c to x metabˆletai apì to 0 èwc to 2π. (Eˆn to m eÐnai arnhtikì o kÔkloc
diagrˆfetai sthn antÐjeth kateÔjunsh ).

Sumpèrasma : Apì ìlec tic leÐec periodikèc kampÔlec
φ(x) = (φ1(x), φ2(x), φ3(x)), x ∈ [0, 2π] me perÐodo 2π, oi opoÐec brÐskontai
sth sfaÐra
S2
= (y1, y2, y3) ∈ R3
; y2
1 + y2
2 + y2
3 = 1 ta krÐsima shmeÐa thc
E(φ) =
1
2
2π
0
3
i=1
dφi
dx
2
dx
eÐnai oi lÔseic thc diaforik c exÐswshc
d2
φ(x)
dx
+
dφ(x)
dx
,
dφ(x)
dx
φ(x) = 0
Autèc oi lÔseic eÐnai mègistoi kÔkloi thc S2
pou diagrˆfontai m forèc kaj¸c
to x metabˆletai apì to 0 èwc to 2π.

Kefˆlaio 3
Armonikèc kai diarmonikèc
apeikonÐseic
3.1 Armonikèc apeikonÐseic
Orismìc 3.1.1. Mia leÐa apeikìnish φ ∈ C∞
(M, N) metaxÔ duo pol-
laplot twn Riemann (M, g) kai (N, h) kaleÐtai armonik an kai mìno an
eÐnai krÐsimo shmeÐo tou sunarthsoeidoÔc thc enèrgeiac
E(φ) =
1
2 M
|dφ|2
vg
H apeikìnish dφ : TM → TN eÐnai to diaforikì thc φ ∈ C∞
(M, N) kai
vg = det(gij)dx1dx2...dxm to stoiqeÐo ìgkou thc metrik c g.
H φ eÐnai krÐsimo shmeÐo thc E eˆn gia opoiad pote leÐa apeikìnish
F : (−ε, ε) × M → N me tim F(t, x) = φt(x), gia kˆje t ∈ (−ε, ε) kai gia
kˆje x ∈ M me F(0, x) = φ0(x) = φ(x) isqÔei h sqèsh
d
dt t=0
E(φt) = 0
23

24 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEIS
Orismìc 3.1.2. Mia C1
−kampÔlh γ : I → M thc pollapìthtac M
onomˆzetai gewdaisiak an γ γ = 0, gia kˆje shmeÐo tou anoiqtoÔ di-
ast matoc I.
'Estw èna topikì sÔsthma suntetagmènwn {xi}n
i=1 thc M.
Tìte, γ(t) = (γ1(t), γ2(t), ..., γn(t)) kai γ (t) = n
i=1 γi(t) ∂
∂xi γ(t)
Epomènwc h sqèsh γ γ = 0 isodÔnama gÐnetai
d2
γi
dt2
+
n
j,k=1
Γi
jk
dγj
dt
dγk
dt
= 0, i = 1, 2, ..., n.
Jètoume ξi = dγi
dt
kai katal goume sto ex c sÔsthma diaforik¸n exis¸sewn:
dξi
dt
= −
n
j,k=1
Γi
jkξjξk, i = 1, 2, ..., n.
Eˆn dojoÔn oi arqikèc timèc γ(0) = (γ1(0), γ2(0), ..., γn(0)) kai
dγ
dt
(0) = dγ1
dt
(0), dγ2
dt
(0), ..., dγn
dt
(0) gia t = 0 to sÔsthma èqei monadik lÔsh
gia ìla ta t sthn perioq tou mhdenìc. Autì shmaÐnei ìti gia opoiod pote
shmeÐo p thc M kai gia opoiod pote efaptìmeno diˆnusma u sto shmeÐo p tou
efaptìmenou q¸rou TM pou ikanopoioÔn tic sunj kec
(1) γ(0) = p kai
(2)γ (0) = u
upˆrqei monadik gewdaisiak γ(t) gia t kontˆ sto mhdèn.
SumbolÐzoume γ(t) = expp(tu) kai dÐnoume ton parakˆtw orismì.
Orismìc 3.1.3. Ekjetik apeikìnish sto shmeÐo p miac pollaplìthtac
M, lègetai h apeikìnish expp : TpM → M me tim ekeÐno to shmeÐo thc M
pou orÐzetai apì to γ(1), dhlad
γ(1) = expp u
gia kˆje u ∈ TpM kai tètoio ¸ste na orÐzetai to γ(1). Autì shmaÐnei ìti to
mètro tou efaptìmenou dianusmatoc u prèpei na eÐnai arketˆ mikrì, dhlad to

3.1. ARMONIKŸES APEIKONŸISEIS 25
t na paÐrnei timèc se mia perioq tou mhdenìc sto q¸ro TpM.
Orismìc 3.1.4. 'Estw mia tuqoÔsa C∞
− apeikìnish V : M → TN
me V (x) ∈ Tφ(x)N, x ∈ M kai φt : M → N h ekjetik C∞
− apeikìnish
me tim φt(x) = expφ(x)(tV (x)), x ∈ M. Onomˆzoume to dianusmatikì pedÐo
V (x) = d
dt t=0
φt(x) dianusmatikì pedÐo metabol c katˆ m koc thc φ
(variation vector ﬁeld along φ).
Antistrìfwc eˆn jewr soume mia tuqoÔsa leÐa metabol φt ∈ C∞
(M, N) thc
φ, ( < t < ) kai φ0 = φ, jètontac V (x) = d
dt t=0
φt(x) orÐzetai mia C∞
−
apeikìnish V apì thn pollaplìthta M sthn efaptìmenh dèsmh TN me tim
V (x) ∈ Tφ(x)N, x ∈ M.
Orismìc 3.1.5. 'Estw duo Ck
− pollaplìthtec E kai N kai π : E → N
mia Ck
− apeikìnish. H π : E → N onomˆzetai Ck
−dianusmatik dèsmh
epÐ thc N eˆn :
(1) Gia kˆje x ∈ N o q¸roc π−1
(x) = Ex o kaloÔmenoc n ma epÐ tou x eÐnai
dianusmatikìc q¸roc diˆstashc k
(2) upˆrqei anoiqt geitoniˆ U thc N sto x, kai ènac diaforomorfismìc
φ : π−1
(U) → U × Rk
tou opoÐou o periorismìc sto π−1
(ψ) eÐnai ènac i-
somorfismìc epÐ tou ψ × Rk
gia kˆje ψ ∈ U.
Orismìc 3.1.6. DÐnetai mia Ck
dianusmatik dèsmh p : E → N kai mi-
a Ck
− apeikìnish φ : M → N metaxÔ duo Ck
− pollaplot twn M kai N.
Kataskeuˆzoume thn dianusmatik dèsmh π : E → M, ìpou E = (p , u) ∈
M × E; φ(p ) = π(u) , π ((p , u)) = p . SumbolÐzw th dianusmatik dèsmh E
me φ∗
E φ−1
E kai thn onomˆzw epag¸menh dianusmatik dèsmh thc
dianusmatik c dèsmhc E mèsw thc φ.

Sqhmatikˆ èqoume to diˆgramma:
φ−1
E //
π

E
π

M
φ
// N
'Estw h C∞
− dianusmatik dèsmh π : TN → N me π(u) = φ(x) gia kˆje
x ∈ M. OrÐzoume thn epag¸menh dèsmh φ−1
TN thc efaptìmenhc dèsmhc TN
mèsw thc φ wc to sÔnolo
φ−1
TN = (x, u) ∈ M × TN; π(u) = φ(x), x ∈ M = x∈M Tφ(x)N
H sqhmatik parˆstash èqei wc ex c:
φ−1
TN //
π

TN
π

M
φ
// N
ìpou π : (M, TN) → M eÐnai h C∞
− dianusmatik dèsmh me π (x, u) = x, x ∈
M.
Orismìc 3.1.7. Mia C∞
−tom (section) thc epag¸menhc dèsmhc φ−1
TN
mèsw thc φ : M → N eÐnai h C∞
− apeikìnish V : M → TN me V (x) ∈
Tφ(x)N, x ∈ M.
SumbolÐzoume to sÔnolo ìlwn twn C∞
−tom¸n me
Γ(φ−1
TN) = V ∈ C∞
(M, TN), V (x) ∈ Tφ(x)N, x ∈ M .
ParathroÔme ìti to sÔnolo Γ(φ−1
TN) eÐnai to sÔnolo ìlwn twn dianus-
matik¸n pedÐwn metabol c katˆ m koc thc φ.
Gia kˆje f ∈ C∞
(M) ,V, V1, V2 ∈ Γ(φ−1
TN) kai x ∈ M orÐzoume sto sÔnolo
φ−1
TN touc ex c nìmouc :
+ : Γ(E) × Γ(E) → Γ(E)
(V1, V2) → V1 + V2

me tim
(V1 + V2)(x) = V1(x) + V2(x)
ìpou E = φ−1
TN kai
· : C∞
(M) × Γ(E) → Γ(E)
(f, V ) → f.V
me tim
(f.V )(x) = f(x).V (x)
Me ton prosjetikì nìmo (+) to Γ(E) kajÐstatai abelian omˆda. Epiplèon,
isqÔoun oi ex c idiìthtec :
(1) ((f + g)V )(x) = (fV )(x) + (gV )(x)
(2) ((f.g)V )(x) = (f.(g.V ))(x)
(3) (f.(V1 + V2))(x) = (f.V1)(x) + (f.V2)(x)
Me tic parapˆnw idiìthtec h abelian omˆda (Γ(E), +) kajÐstatai èna prìtupo
(module) epÐ thc C∞
(M).
Prin d¸soume ton orismì thc epag¸menhc sunoq c sthn epag¸menh dèsmh
φ−1
TN thc efaptìmenhc dèsmhc TN mèsw thc φ, dÐnoume touc epìmenouc
orismoÔc.
Orismìc 3.1.8. H apeikìnish σ : R → M me tim σ(t) ∈ M gia kˆje t ∈ R
eÐnai mia C1
− kampÔlh thc M. Gia t = 0 èqoume
(1) σ(0) = x kai
(2) σ (0) = Xx ⇔ d
dt t=0
σ(t) = Xx
ìpou Xx ∈ TxM kai h kampÔlh σt me tim σt(s) = σ(s) eÐnai o periorismìc thc
σ ìtan to 0 ≤ s ≤ t.
Orismìc 3.1.9. To dianusmatikì pedÐo X lègetai parˆllhlo katˆ m koc
thc C1
− kampÔlhc γ : [a, b] ⊂ R → M an ta dianÔsmata tou pedÐou X se

opoiad pote dÔo diaforetikˆ shmeÐa thc kampÔlhc eÐnai parˆllhla metaxÔ touc,
dhlad
γ X = 0
'Estw èna topikì sÔsthma suntetagmènwn {xi}n
i=1 se mia perioq U thc M.
Tìte, grˆfoume X(t) = n
i=1 ξi(t) ∂
∂xi γ(t)
, ìpou X(t) ∈ Tγ(t)M, gia kˆje
t ∈ [a, b] kai γ(t) = (γ1(t), γ2(t), ..., γn(t)), opìte γ (t) = n
i=1 γi(t) ∂
∂xi γ(t)
.
Epomènwc, apì th sqèsh
γ X = 0
isodÔnama èqoume
dξi(t)
dt
+
n
j,k=1
Γi
jk(γ(t))
dγj(t)
dt
ξk(t) = 0, i = 1, 2, ...n.
Eˆn dojeÐ h kampÔlh γ(t) kai dojeÐ h arqik tim (ξ1(α), ξ2(α), ..., ξn(α)) sto
shmeÐo p = γ(α) tìte ta ξi eÐnai monadikˆ orismèna, efìson to sÔsthma twn
diaforik¸n exis¸sewn èqei monadik lÔsh.
Epomènwc, h tim (ξ1(b), ξ2(b), ..., ξn(b)) sto q = γ(b) kai katˆ sunèpeia to
X(b) orÐzontai monadikˆ. 'Eqoume dhlad thn antistoiqÐa
Tγ(α)M X(α) → X(b) ∈ Tγ(b)M.
Orismìc 3.1.10. Onomˆzoume thn apeikìnish
Pγ : Tγ(α)M → Tγ(b)M
parˆllhlh metaforˆ katˆ m koc thc kampÔlhc γ wc proc th Levi-Civita
sunoq sth pollaplìthta (M, g).
H apeikìnish Pγ eÐnai ènac grammikìc isomorfismìc kai epiplèon,
gγ(b) (Pγ(u), Pγ(v)) = gγ(α) (u, v) , u, v ∈ Tγ(α)M.

SumbolÐzoume me kai N
tic sunoqèc Levi-Civita stic pollaplìthtec
(M, g) kai (N, h) antÐstoiqa, kai dÐnoume ton akìloujo orismì.
Orismìc 3.1.11. Gia kˆje C∞
− dianusmatikì pedÐo X thc M onomˆzoume
epag¸menh sunoq sthn epag¸menh dèsmh φ−1
TN thc efaptìmenhc
dèsmhc TN mèsw thc f, thn apeikìnish
X : N
P−1
φ◦σt
V (σ(t))Γ(φ−1
TN) → Γ(φ−1
TN)
V → XV
gia kˆje V ∈ Γ(φ−1
TN), me tim
XV (x) = N
φ∗X
V =
d
dt t=0
, x ∈ M
H ikanopoieÐ tic ex c idiìthtec :
(1) fX+gY V = f XV + g Y V
(2) X(V1 + V2) = XV1 + 2V2
(3) X(fV ) = X(f)V + f XV
gia kˆje f, g ∈ C∞
(M) gia kˆje X, Y ∈ D1
(M) kai V, V1, V2 ∈ Γ(φ−1
TN).
H apeikìnish N
Pφ◦σt : Tφ(x)N → Tφ(σ(t))N eÐnai h kaloÔmenh parˆllhlh
metaforˆ katˆ m koc thc C1
−kampÔlhc φ ◦ σt wc proc th Levi-Civita
sunoq N sth pollaplìthta (N, h).
Sth sunèqeia ja apodeÐxoume endeiktikˆ thn trÐth katˆ seirˆ apì tic idiìthtec
thc epag¸menhc sÔnoq c . Ja apodeÐxoume dhlad ìti
X(fV ) = X(f)V + f XV

Apìdeixh
Gia kˆje x ∈ M èqoume
X(fV )(x) =
d
dt t=0
N
P−1
φ◦σt
(f(σ(t))V (σ(t)))
=
d
dt t=0
f(σ(t)) N
P−1
φ◦σt
(V (σ(t))) + f(x)
d
dt
N
P−1
φ◦σt
V (σ(t))
= Xx(f)V (x) + f(x)( XV )(x)
H epag¸menh dèsmh φ−1
TN epidèqetai èna eswterikì ginìmeno proerqìmeno
apì th metrik h sth pollaplìthta N pou sumbolÐzetai me hφ(x) kai eÐnai h
apeikìnish
hφ(x) : Tφ(x)N × Tφ(x)N → R
Lambˆnontac upìyh thn isometrik diìthta :
hϕ(σ(t))(V1(σ(t)), V2(σ(t)) = hφ(x)(N
P−1
φ◦σt
V1(σ(t)), N
P−1
φ◦σt
V2(σ(t)) = hφ(x)(V1(x), V2(x))
thc apeikìnishc
N
P−1
φ◦σt
V (σ(t)) : Tϕ(σ(t))N → Tφ(x)N
ja deÐxoume ìti h epag¸menh sunoq eÐnai sumbat me th metrik hφ(x) ìpwc
thn orÐsame parapˆnw.
Prˆgmati,
Xxhφ(x)(V1, V2) =
d
dt t=0
hφ(σ(t))(V1(σ(t)), V2(σ(t))
=
d
dt t=o
hφ(x)(N
P−1
φ◦σt
V1(σ(t)), N
P−1
φ◦σt
V2(σ(t))
= hφ(x)
d
dt t=0
N
P−1
φ◦σt
V1(σ(t)), V2(x) +hφ(x) V1(x),
d
dt t=o
N
P−1
φ◦σt
V2(σ(t))
= hφ(x)( Xx V1, V2) + hφ(x)(V1, Xx V2)
gia kˆje X ∈ D1
(M), V1, V2 ∈ Γ(φ−1
TN) kai x ∈ M.

O H. Urakawa sto biblÐo tou [25] anafèrei to parakˆtw je¸rhma metabo-
l c:
Je¸rhma 3.1.1.
'Estw φ ∈ C∞
(M, N) kai φt mia tuqaÐa leÐa metabol thc φ, ìpou − t
, φ0 = φ kai V (x) = d
dt t=0
φt(x), x ∈ M to C∞
− dianusmatikì pedÐo
metabol c katˆ m koc thc φ.
Tìte
d
dt t=0
E(φt) = −
M
h(V, τ(φ))vg
ìpou to τ(φ) eÐnai stoiqeÐo tou Γ(φ−1
TN) pou kaleÐtai pedÐo èntashc thc
φ (tension ﬁeld) kai dÐdetai apì th sqèsh
τ(φ) =
m
i=1
( ei
dφ(ei) − dφ( ei
ei)
Sumpèrasma: h φ ∈ C∞
(M, N) eÐnai armonik an kai mìno an
d
dt t=0
E(φt) = 0 ⇔ τ(φ) = 0
H exÐswsh τ(φ) = 0 kaleÐtai exÐswsh twn Euler-Lagrange.
ParadeÐgmata armonik¸n apeikonÐsewn
(1) Stajerèc apeikonÐseic
'Estw dÔo sumpageÐc pollaplìthtec Riemann (M, g) kai (N, h) kai q ∈ N
èna stajerì shmeÐo. Kˆje stajer apeikìnish φ : M → N me tim φ(x) =
q, x ∈ M, eÐnai armonik kai antistrìfwc.
Apìdeixh
H φ eÐnai stajer an kai mìno an to sunarthsoeidèc thc puknìthtac
thc enèrgeiac e(φ) = 1
2
|dφ|2
thc φ eÐnai mhdèn, dhlad an kai mìno an
e(φ) = 0. 'Omwc to sunarthsoeidèc thc enèrgeiac thc φ dÐnetai apì th sqèsh
E(φ) = M
e(φ)vg.
Epomènwc, e(φ) = 0 ⇔ E(φ) = 0 ⇔ d
dt t=o
E(φ) = 0 ⇔ τ(φ) = 0

Dhlad h φ eÐnai stajer an kai mìno an eÐnai armonik .
(2) Kleistèc gewdaisiakèc sth sfaÐra S2
'Estw mia kleist diaforÐsimh kampÔlh φ(x) = (φ1(x), φ2(x), φ3(x)), x ∈
[0, 2π] ston R3
me perÐodo 2π, dhlad φ(x + 2π) = φ(x) φi(x + 2π) =
φi(x), i = 1, 2, 3. AnazhtoÔme tic kampÔlec pou apoteloÔn krÐsima shmeÐa thc
enèrgeiac
E(φ) =
1
2
2π
0
3
i=1
(
dφi
dx
)2
dx
kai brÐskontai sth monadiaÐa sfaÐra
S2
= (y1, y2, y3); y2
1 + y2
2 + y2
3 = 1
Autèc oi kampÔlec eÐnai mègistoi kÔkloi thc sfaÐrac S2
pou strèfontai m
forèc kaj¸c to x metabˆletai apì to 0 èwc to 2π.
(Analutik parousÐash ègine sthn parˆgrafo 2.2.)
Sth sunèqeia ja sundèsoume thn armonikìthta me tic pollaplìthtec elˆqi-
sthc èktashc.

Orismìc 3.1.12. 'Estw duo diaforÐsimec pollaplìthtec (M, g) kai
(N, h). Mia leÐa apeikìnish φ : M → N onomˆzetai isometrik embˆ-
ptish (isometric immersion) eˆn :
(1) to diaforikì thc φ sto p ∈ M, dhlad h apeikìnish dφp : TpM → Tφ(p)N
eÐnai 1 − 1 gia kˆje x ∈ M,
(2) xp, yp M = dφ(xp), dφ(yp) N gia kˆje xp, yp ∈ TpM.
Orismìc 3.1.13. 'Otan mia embˆptish φ : M → N eÐnai 1 − 1, tìte h
φ lègetai emfÔteush thc M sthn N . Sthn perÐptwsh aut lème ìti h
pollaplìthta M eÐnai emfuteumènh mèsa sth N mèsou thc φ, ìti h M eÐnai
mia emfuteumènh upopollaplìthta thc N.
Orismìc 3.1.14. Mia m−diˆstath pollaplìthta M onomˆzetai upopol-
laplìthta thc n−diˆstathc pollaplìthtac N ìtan :
(1) M ⊂ N (h M eÐnai topologikìc upìqwroc thc N.)
(2) H tautotik apeikìnish i : M → N eÐnai mia emfÔteush thc pollaplìthtac
M sthn pollaplìthta N.
Eˆn dimN − dimM = 1, tìte h M lègetai uperepifˆneia thc N.
'Estw M mia m− diˆstath upopollaplìthta thc n− diˆstathc pollaplìth-
tac Riemann N (m n).
An h eÐnai h metrik Riemann thc N, tìte h epag¸menh metrik thc M
eÐnai h g = i∗
h, ìpou
(1) h i : M → N eÐnai leÐa
(2) h i : M → N eÐnai tautotik me tim i(x) = x
(3) h i : M → N eÐnai 1-1
(4) h di : TpM → Ti(p)N eÐnai èna proc èna kai tautotik .
H M efodiasmènh me th g kajistˆ thn i : M → N isometrik : g(xp, yp) =

i∗
h(x, y) = h(di(xp), di(yp)) gia kˆje xp, yp ∈ TpM.
'Ena diˆnusma ξp ∈ TpN, x ∈ M lègetai kˆjeto sthn M sto shmeÐo p an
h(ξp, xp) = 0 gia kˆje xp ∈ TpM. An TM⊥
eÐnai to sÔnolo ìlwn twn kˆjetwn
dianusmˆtwn se kˆje shmeÐo p ∈ M tìte
TN = TM ⊕ TM⊥
'Estw X, Y duo dianusmatikˆ pedÐa thc M kai X, Y oi epektˆseic aut¸n sthn
pollaplìthta N, dhlad ta dianusmatikˆ pedÐa thc N ta opoÐa ìtan perior-
isjoÔn sthn pollaplìthta M eÐnai ta dianusmatikˆ pedÐa X, Y antÐstoiqa.
'Estw h sunoq thc pollaplìthtac Riemann N. Tìte h tim tou dianus-
matikoÔ pedÐou XY sto p ∈ M den exartˆtai apì tic epektˆseic X, Y twn
X, Y antÐstoiqa kai to dianusmatikì pedÐo [X, Y ] thc N eÐnai epèktash tou
dianusmatikoÔ pedÐou [X, Y ] thc M. 'Etsi grˆfoume XY antÐ XY kai
analÔoume autì to dianusmatikì pedÐo thc N se duo sunist¸sec, mia efap-
tìmenh thc M, thn XY kai mia kˆjeth sth M, thn B(X, Y ). 'Epomènwc,
XY = XY + B(X, Y )
O tÔpoc autìc onomˆzetai tÔpoc tou Gauss. H apeikìnish
: TM × TM → TM
(X, Y ) → XY
orÐzei mia sunoq sth M pou lègetai epag¸menh sunoq sthn upopol-
laplìthta M. EpÐshc h apeikìnish
B : TM × TM → TM⊥
(X, Y ) → B(X, Y )
eÐnai summetrik , digrammik kai legetai deÔterh jemeli¸dhc morf
(second fundamental form) thc upopollaplìthtac M.

'Estw ξ èna dianusmatikì pedÐo thc N kˆjeto sth M. To dianusmatikì pedÐo
Xξ analÔetai se mia efaptìmenh sunist¸sa thn −AξX kai mia kˆjeth thn
⊥
Xξ opìte isqÔei o akìloujoc tÔpoc tou Weingarten
Xξ = −AξX + ⊥
Xξ
H apeikìnish
⊥
: TM × TM⊥
→ TM⊥
(X, ξ) → ⊥
Xξ
èqei tic idiìthtec miac sunoq c kai lègetai kˆjeth sunoq (normal conne-
ction) thc upopollaplìthtac M.
H apeikìnish
Aξ : TM → TM
X → AξX
eÐnai grammik wc proc X kai ξ kai autosuzug c, dhlad , gia kˆje X, Y ∈ TM
isqÔei: AξX, Y M = X, AξY M kai kaleÐtai telest c sq matoc (shape
operator) deÔterh jemeli¸dhc morf sth kˆjeth dieÔjunsh
ξ ∈ TM⊥
(the second fundamental form in the normal direction ξ).
Jewr¸ th diaforÐsimh kampÔlh a : I ⊂ R → M sthn pollaplìthta M me
tim a(t) ∈ M pou ikanopoieÐ tic sunj kec a(t0) = p kai a (t0) = xp ∈ TpM.
To AξX = −( xp ξ) = −(ξ ◦ a) (t0) metrˆei thn allag kateÔjunshc tou ξ
kaj¸c autì dièrqetai apì to p katˆ m koc thc kampÔlhc a. O efaptìmenoc
q¸roc Ta(t)M thc M sto a(t) strèfetai kaj¸c to kˆjeto diˆnusma ξ strèfe-
tai. 'Epomènwc to AξX ekfrˆzei èna mètro strof c tou efaptìmenou q¸rou
thc M sto p kaj¸c to ξ dièrqetai apì to p katˆ m koc thc a. 'Ara o telest c
sq matoc mac dÐnei plhroforÐec gia to sq ma thc M.

Prìtash 3.1.1. Gia kˆje dianusmatikì pedÐo ξ thc N kˆjeto sth M
kai gia X, Y ∈ TM èqoume
AξX, Y M = B(X, Y ), ξ M
Apìdeixh
X ξ, Y M = Xξ, Y M + ξ, XY M ⇔
0 = Xξ + ⊥
Xξ, Y M + ξ, XY + B(X, Y ) M ⇔
0 = −AξX, Y M + ⊥
Xξ, Y M + ξ, XY )M + ξ, B(X, Y ) M ⇔
AξX, Y M = B(X, Y ), ξ M
Gia èna monadiaÐo kˆjeto diˆnusma ξ thc M sto p o telest c sq matoc Aξ
eÐnai grammikìc kai autosuzug c opìte mporoÔme na epilèxoume orjokanonik
bˆsh e1, e2, ..., em thc M ìpou ta stoiqeÐa thc na apoteloÔn idiodianÔsmata
tou Aξ, dhladh Aξ(ei) = λiei, i = 1, 2, ..., m. Ta λi ∈ R kaloÔntai kÔriec
kampulìthtec (principal curvatures) thc M wc proc thn kˆjeth dieÔjunsh
ξ kai ta idiodianÔsmata ei kaloÔntai kÔriec dieujÔnseic (principal directions).
Oi kÔriec kampulìthtec mac dÐnoun mia perigraf tou topikoÔ telest sq -
matoc thc emfuteumènhc pollaplìthtac M.
Orismìc 3.1.15. 'Estw φ : Mm
→ Nn
mia isometrik embˆptish metaxÔ
duo pollaplot twn M kai N. To dianusmatikì pedÐo mèshc kampulìth-
tac H thc φ eÐnai h apeikìnish
H : M → TM⊥
x → H(x) ∈ TxM⊥
me tim
H(x) =
1
m
m
i=1
B(ei, ei) ⇔ H(x) =
1
m
traceB

ìpou ei
m
i=1
mia orjokanonik bˆsh tou q¸rou TxM.
'Estw ξa
m
a=1
mia orjokanonik bˆsh tou TM⊥
sto x. Tìte
traceB =
a,i
B(ei, ei), ξa M
kai lìgw thc sqèshc AξX, Y M = B(X.Y ), ξ M èqoume
traceB =
a,i
Aξa (ei), ei M =
a
traceAξa
'Ara to dianusmatikì pedÐo mèshc kampulìthtac gÐnetai wc ex c:
H(x) =
1
m a
traceAξa ⇔ H(x) =
1
m
(traceA)ξ
Orismìc 3.1.16. H φ kaleÐtai elˆqisth isometrik embˆptish kai
h upopollaplìthta M elaqÐsthc èktashc (minimal submanifold) eˆn
H = 0.
Apì to tÔpo tou Gauss èqoume
XY = XY + B(X, Y )
gia kˆje X, Y ∈ TM. Gia X = Y = ei ∈ TM, i = 1, 2, ..., m o tÔpoc gÐnetai
wc ex c:
ei
ei = ei
ei + B(ei, ei) ⇔
B(ei, ei) = ei
ei − ei
ei
ìpou eÐnai h sÔndesh sthn epag¸menh dèsmh φ−1
TN thc efaptìmenhc dèsmh-
c TN.
Epomènwc to dianusmatikì pedÐo mèshc kampulìthtac
H =
1
m
m
i=1
B(ei, ei)

gÐnetai wc ex c:
H =
1
m
m
i=1
( ei
ei − ei
ei)
'Omwc
τ(φ) =
m
i=1
( ei
dφ(ei) − dφ( ei
ei)
Lìgw tou tautotikoÔ isomorfismoÔ Tφ(x)N ∼= Nφ(x) èqoume
H =
1
m
τ(φ)
Sunep¸c H = 0 an kai mìno an τ(φ) = 0
Prìtash 3.1.2. An h φ : M → N eÐnai isometrik embˆptish tìte h M eÐnai
elˆqisthc èktashc an kai mìno an to pedÐo èntashc τ(φ) thc φ mhdenÐzetai.
3.2 Diarmonikèc apeikonÐseic
Orismìc 3.2.1. MÐa leÐa apeikìnish φ ∈ C∞
(M, N) metaxÔ duo pol-
laplot twn Riemann (M, g) kai (N, h) kaleÐtai diarmonik an kai mìno an
eÐnai krÐsimo shmeÐo tou sunarthsoeidoÔc thc enèrgeiac deÔterhc tˆxhc
(bienergy)
E2(φ) =
1
2 M
|τ(φ)|2
vg
H φ eÐnai krÐsimo shmeÐo thc E2 an gia opoiad pote metabol φt ∈ C∞
(M, N)
(− t ), φ0 = φ thc φ isqÔei h sunj kh
d
dt t=0
E2(φt) = 0
Stic ergasÐec [14] , [15] o J. Jiang èdwse gia thn pr¸th metabol tou sunarth-
soeidoÔc E2 ton akìloujo tÔpo
d
dt t=0
E2(φt) = −
M
h(τ2(φ), V )vg

3.2. DIARMONIKŸES APEIKONŸISEIS 39
Je¸rhma 3.2.1. 'Estw φ ∈ C∞
(M, N) kai φt mia tuqaÐa leÐa metabol
thc φ, ìpou (− t ), φ0 = φ kai V (x) = d
dt
|t=0φt(x), x ∈ M to C∞
−
dianusmatikì pedÐo metabol c katˆ m koc thc φ.
Tìte,
d
dt t=0
E2(φt) = −
M
h(τ2(φ), V )vg
ìpou τ2(φ) = Jφ(τ(φ)) eÐnai to pedÐo tˆshc deÔterhc tˆxhc kai Jφ eÐnai ènac au-
tosuzug c, diaforikìc telest c pou dra sto sÔnolo twn dianusmatik¸n pedÐ-
wn metabol c katˆ m koc thc φ, onomˆzetai telest c tou Jacobi(Jacobi
operator) kai orÐzetai wc ex c :
Jφ = ¯ φ − Rφ
O diaforikìc telest c ¯ φ onomˆzetai Laplasian (rough Laplacian), dra
sta dianusmatikˆ pedÐa metabol c katˆ m koc thc φ kai orÐzetai wc ex c :
¯ φV = −
m
i=1
( ei ei
− ei ei
)V
ìpou V ∈ Γ(φ−1
TN), ei
m
i=1
orjokanonik bˆsh wc proc th metrik g sth
M kai (m = dimM).
Tèloc o diaforikìc telest c Rφ dra epÐshc sta dianusmatikˆ pedÐa metabol c
katˆ m koc thc φ kai dÐnetai apì th sqèsh
RφV =
m
i=1
N
R(V, dφ(ei))dφ(ei)
ìpou V ∈ Γ(φ−1
TN) kai N
R eÐnai to tanustikì pedÐo kampulìthtac thc suno-
q c N
sthn pollaplìthta (N, h).
Sumpèrasma : H φ ∈ C∞
(M, N) eÐnai diarmonik an kai mìno an
d
dt t=0
E(φt) = 0 ⇔ τ2(φ) = 0 ⇔ J(τ(φ)) = 0
H exÐswsh τ2(φ) = 0 kaleÐtai diarmonik exÐswsh.

Idiìthtec twn diarmonik¸n apeikonÐsewn
Prìtash 3.2.1. H φ eÐnai diarmonik an kai mìno an to τ(φ) an kei ston
pur na tou telest Jφ, dhlad an kai mìno an τ(φ) ∈ KerJφ.
Apìdeixh
KerJφ = V ∈ Γ(φ−1
TN); Jφ(V ) = 0
τ(φ) ∈ KerJφ ⇔ Jφ(τ(φ)) = 0 ⇔ τ2(φ) = 0, dhlad h φ eÐnai diarmonik .
Prìtash 3.2.2. Eˆn h φ ∈ C∞
(M, N) eÐnai armonik tìte eÐnai kai diar-
monik .
Apìdeixh
Jèlw na deÐxw ìti h φ eÐnai diarmonik , dhlad ìti d
dt t=0
E2(φt) = 0 gia kˆje
leÐa metabol φt, (− t ), φ0 = φ thc φ, ìpou E2(φ) = 1
2 M
|τφ)|2
vg
to sunarthsoeidèc thc enèrgeiac deÔterhc tˆxhc (bienergy). Apì thn upì-
jesh èqw pwc h φ eÐnai diarmonik , dhlad τ(φ) = 0, ˆra E2(φ) = 0, ˆra
d
dt t=0
E2(φt) = 0, ˆra h φ eÐnai diarmonik .
Prìtash 3.2.3. Mia armonik apeikìnish elaqistopoieÐ to sunarthsoeidèc
E2(φ) = 1
2 M
|τ(φ)|2
vg .
Apìdeixh
H φ eÐnai armonik , dhlad τ(φ) = 0. Epomènwc E2(φ) = 0.

Kefˆlaio 4
Diarmonikèc
Upopollaplìthtec
4.1 Eisagwg
O B.Y. Chen sthn ergasÐa tou [4] anafèrei thn ex c eikasÐa:
EikasÐa tou Chen
Kˆje diarmonik upopollaplìthta tou eukleÐdeiou q¸rou En
eÐnai armonik ,
dhlad eÐnai elˆqisthc èktashc.
Eˆn o q¸roc den eÐnai eukleÐdeioc h eikasÐa tou Chen genikˆ den epalhjeÔetai.
'Ena antiparˆdeigma anafèrei o G.Y. Jiang sthn ergasÐa tou [15] kai prìkeitai
gia to genikeumèno tìro tou Cliﬀord Sp
( 1√
2
) × Sq
( 1√
2
) ⊂ Sm+1
me p + q = m
kai p = q.
Orismìc 4.1.1. Tìroc tou Cliﬀord lègetai h eikìna f(S1
× S1
) thc
apeikìnishc f : S1
× S1
→ R4
me tim f(u, v) = (cosu, sinu, cosv, sinv). O
tìroc T2
= S1
×S1
diagrˆfetai apì thn peristrof tou kÔklou S1
me exÐswsh
(x1 − a)2
+ x2
3 = r2
, r a sto epÐpedo x10x3 gÔrw apì ton ˆxona x3.
41

42 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTES
Sth sunèqeia oi G.Y. Jiang kai C. Oniciuc stic ergasÐec touc [7] kai [21]
apèdeixan tic parakˆtw protˆseic:
Prìtash 4.1.1. Eˆn M eÐnai mia sumpag c upopollaplìthta Riemann
miac pollaplìthtac N me kampulìthta tom c RiemN
≤ 0, tìte h φ : M → N
eÐnai diarmonik an kai mìno an eÐnai armonik , dhlad elˆqisthc èktashc.
Prìtash 4.1.2. Eˆn h φ : M → N eÐnai isometrik embˆptish me |τ(φ)|
stajerì kai h kampulìthta tom c thc pollaplìthtac N eÐnai RiemN
≤ 0,
tìte h φ eÐnai diarmonik an kai mìno an eÐnai armonik , dhlad elˆqisthc èk-
tashc.
Oi parapˆnw protˆseic mac odhgoÔn sth genikeumènh eikasÐa tou Chen.
Genikeumènh eikasÐa tou Chen
Oi mìnec diarmonikèc upopollaplìthtec miac pollaplìthtac N me kampulìth-
ta tom c RiemN
≤ 0 eÐnai oi elˆqisthc èktashc, dhlad oi armonikèc.
Stìqoc mac eÐnai na anazht soume tic diarmonikèc kampÔlec kai tic diarmonikèc
epifˆneiec thc sfaÐrac S3
. Oi kentrikèc mac anaforèc eÐnai oi ergasÐec [3],[4]
twn R. Caddeo, S. Montaldo, C. Oniciuc kai h ergasÐa [11] twn J. Eells, L.
Lemaire.
4.2 Diarmonikèc kampÔlec sthn S3
Arqikˆ ja anazht soume tic diarmonikèc kampÔlec miac trisdiˆstathc pol-
laplìthtac M.
Jewr¸ (M3
, g) mia tridiˆstath pollaplìthta Riemann me stajer kampulìth-
ta tom c K kai mia diaforÐsimh kampÔlh γ : I ⊂ R → (M3
, g) parametrikopoih-
mènh wc proc to m koc tìxou thc. 'Estw T, N, B èna orjokanonikì pedÐo

4.2. DIARMONIKŸES KAMPŸULES STHN S3
43
plaisÐwn efaptìmeno sthn M3
katˆ m koc thc γ, ìpou :
• T = γ eÐnai to monadiaÐo dianusmatikì pedÐo efaptìmeno sth γ
• N to monadiaÐo kˆjeto dianusmatikì pedÐo sth dieÔjunsh tou T T
• B to dianusmatikì pedÐo kˆjeto sta T kai N epÐ thc γ epilegmèno ¸ste
h T, N, B na apoteleÐ jetikˆ prosanatolismènh bˆsh.
KaloÔme to T, N, B paidÐo plaisÐwn tou Frenet epÐ thc γ. Eˆn h
kampÔlh γ eÐnai monadiaÐac taqÔthtac, dhlad |γ (t)| = 1,tìte kg = | T T| =
|τ(γ)|. H kg onomˆzetai gewdaisiak kampulìthta kai ekfrˆzei thn
taqÔthta metabol c thc dieÔjunshc tou efaptomenikoÔ pedÐou sth kampÔlh
anˆ monˆda m kouc tìxou. H sunˆrthsh τg pou perilambˆnetai stouc parakˆtw
tÔpouc onomˆzetai gewdaisiak strèyh kai ekfrˆzei thn taqÔthta metabo-
l c thc dieÔjunshc tou dianusmatikoÔ pedÐou B. IsqÔoun oi parakˆtw exis¸-
seic tou Frenet :
T T = kgN
T N = −kgT + τgB
T B = −τgN
H kampÔlh γ eÐnai diarmonik an kai mìno an
τ2(γ) = 0 ⇔ 3
T T − R(T, kgN)T = 0 ⇔
(−3kgkg)T + (kg − k3
g − kgτ2
g + kgK)N + (2kgτg + kgτg)B = 0
ìpou
K = K(T, N) =
g(R(T, N)N, T)
g(T, T)g(N, N) − g(T, N)2
=
g(R(T, N)N, T) = R(T, N, N, T) = −R(T, N, T, N)

eÐnai h kampulìthta tom c thc M3
wc proc to zeÔgoc T, N , h opoÐa èqoume
upojèsei ìti eÐnai stajer .
Epomènwc h kampÔlh γ eÐnai diarmonik an kai mìno an τ2(γ) = 0.
IsodÔnama
(1) kgkg = 0
(2) kg − k3
g − kgτ2
g + kgK = 0
(3) 2kgτg + kgτg = 0
AnazhtoÔme diarmonikèc mh gewdaisiakèc kampÔlec, dhlad diarmonikèc
kampÔlec me gewdaisiak kampulìthta kg = 0. 'Eqontac wc upìjesh ìti
kg = 0 èqw ta ex c :
Apì thn exÐswsh (1) sunepˆgetai ìti kg = c1, ìpou h c1 eÐnai mia mh mhdenik
pragmatik stajerˆ.
Apì thn (2) sunepˆgetai ìti k2
g + τ2
g = K.
Apì thn (3) sunepˆgetai ìti τg = c2, ìpou h c2 eÐnai mia mh mhdenik prag-
matik stajerˆ.
Epomènwc katal goume sth parakˆtw prìtash.
Prìtash 4.2.1 Oi diarmonikèc mh gewdaisiakèc kampÔlec thc pollaplìth-
tac M eÐnai ekeÐnec pou èqoun stajer gewdaisiak kampulìthta kai strèyh
kai pou ikanopoioÔn th sunj kh k2
g + τ2
g = K.
Sthn perÐptwsh pou h kampulìthta tom c eÐnai mikrìterh Ðsh tou mhdenìc
(K ≤ 0) h sunj kh k2
g + τ2
g = K den mporeÐ na isqÔei parˆ mìno ìtan
kg = τg = 0. Tìte h γ eÐnai gewdaisiak , dhlad elˆqisthc èktashc (mini-
mal). Epomènwc epibebai¸netai h genikeumènh eikasÐa tou Chen .

45
Sth sunèqeia ja anazht soume diarmonikèc mh gewdaisiakèc kampÔlec sth
sfaÐra S3
. Oi kentrik mac anaforˆ eÐnai h ergasÐa [6].
Prìtash 4.2.2. 'Estw γ : I → S3
⊂ R4
mia mh gewdaisiak diarmonik
kampÔlh parametrikopoihmènh wc proc to m koc tìxou thc. Tìte isqÔei h
exÐswsh
γIV
+ 2γ + (1 − k2
g)γ = 0
Apìdeixh
PaÐrnoume th sunalloÐwth parˆgwgo wc proc T thc exÐswshc
T N = −kgT + τgB
tou Frenet kai èqoume
2
T N = T ( T N) = −kg T T + τg T B
ExaitÐac kai twn upoloÐpwn exis¸sewn tou Frenet, h parapˆnw sqèsh gÐnetai
2
T N = −kg(kgN) + τg(−τgN) = −(k2
g + τ2
g )N = −KN
Epeid h kampulìthta tom c thc sfaÐrac eÐnai K = 1, h parapˆnw sqèsh
gÐnetai
2
T N = −N ⇔ 2
T + N = 0
H exÐswsh tou Gauss gia tuqaÐo dianusmatikì pedÐo X thc S3
katˆ m koc thc
γ èqei wc ex c :
T X = X + T, X γ
Efarmìzontac thn parapˆnw sqèsh gia to dianusmatikì pedÐo N èqoume
T N = N + T, N γ

Epomènwc,
2
T N = T ( T N) = T (N + T, N γ) = T N =
N + T, N γ = N + T, T N γ = N + T, −kgT + τgB γ =
N + (−kg T, T + τg T, B )γ = N − kgγ ⇔ 2
T N = N − kgγ
'Omwc
N =
T
kg
ìpou
T = T T = T γ = γ + T, T γ γ ⇔
T = γ + γ , γ γ
Epeid h kampÔlh γ èqei monadiaÐa taqÔthta, dhlad |γ | = 1 ⇔ γ , γ = 1
èqoume telikˆ ìti
T = γ + γ
Opìte
N =
γ + γ
kg
ParagwgÐzoume thn parapˆnw sqèsh kai paÐrnoume
N =
γ + γ
kg
ParagwgÐzoume xanˆ
N =
γIV
+ γ
kg
Apì tic sqèseic
2
T N + N = 0
2
T N = N − kgγ

47
N =
γIV
+ γ
kg
èqoume telikˆ
γIV
+ 2γ + (1 − k2
g)γ = 0.
'Ara oi mh gewdaisiakèc diarmonikèc kampÔlec thc S3
eÐnai lÔseic thc diafori-
k c exÐswshc γIV
+ 2γ + (1 − kg)γ = 0.
ApodeÐxame prohgoumènwc ìti isqÔoun oi sunj kec
kg = σταθ. = 0
τg = σταθ. = 0
k2
g + τ2
g = K
gia tic mh gewdaisiakèc diarmonikèc kampÔlec γ : I → (M3
, g).
Ean h pollaplìthta M eÐnai h sfaÐra S3
tìte h kampulìthta tom c isoÔtai
me th monˆda, dhlad K = 1, opìte h trÐth sunj kh gÐnetai
k2
g + τ2
g = 1
Apì th teleutaÐa sqèsh sunepˆgetai ìti kg ≤ 1.
Je¸rhma 4.2.1. 'Estw mia mh gewdaisiak diarmonik kampÔlh γ : I → S3
parametrikopoihmènh wc proc to m koc tìxou thc. IsqÔoun ta ex c :
(1) Eˆn kg = 1, tìte h γ eÐnai kÔkloc aktÐnac 1√
2
(2) Eˆn 0 kg 1, tìte h γ eÐnai gewdaisiak tou tìrou tou Cliﬀord
S1
( 1√
2
) × S1
( 1√
2
)
Apìdeixh
Pr¸th perÐptwsh :
H sqèsh γIV
+ 2g + (1 − k2
g)γ = 0 ìtan to kg = 1 gÐnetai
γIV
+ 2γ = 0

H exÐswsh eÐnai grammik , tètarthc tˆxhc, omogen c, me stajeroÔc sunte-
lestèc. To qarakthristikì polu¸numo eÐnai f(ρ) = ρ4
+ 2ρ2
kai oi rÐzec tou
eÐnai oi ρ = 0 (dipl pragmatik ), ρ1 = i
√
2, ρ2 = −i
√
2. 'Ara h genik lÔsh
thc diaforik c exÐswshc eÐnai h
γ(t) = e0t
(c1 cos(
√
2t) + c2 sin(
√
2t) + c3t + c4 ⇔
γ(t) = c1 cos(
√
2t) + c2 sin(
√
2t) + c3t + c4
GnwrÐzontac ìti |γ|2
= 1 kai |γ |2
= 1 kai me efarmog twn sqèsewn tou
Frenet, èqoume c3 = 0, |c1|2
= |c2|2
= |c4|2
= 1
2
.
Epomènwc, h genik lÔsh eÐnai h
γ(t) =
1
√
2
cos(
√
2t),
1
√
2
sin(
√
2t), 0,
1
√
2
isodÔnama
γ(t) =
1
√
2
cos(
√
2t),
1
√
2
sin(
√
2t), d1, d2
ìpou d2
1 + d2
2 = 1
2
'Ara h kampÔlh γ eÐnai kÔkloc aktÐnac ρ = 1√
2
DeÔterh perÐptwsh :
LÔnoume th diaforik exÐswsh γIV
+ 2γ + (1 − k2
g)γ = 0 ìtan to 0 kg 1
Aut eÐnai grammik , tètarthc tax c, omogen c, me stajeroÔc suntelestèc.
To qarakthristikì polu¸numo eÐnai f(ρ) = ρ4
+ 2ρ2
+ (1 − k2
g). Oi rÐzec
tou eÐnai oi ρ1 = i 1 + kg, ρ2 = −i 1 + kg kai oi ρ3 = i 1 − kg, ρ4 =
−i 1 − kg.
Epomènwc, h genik lÔsh thc diaforik c exÐswshc eÐnai h
γ(t) = e0t
c1 cos( 1 + kg)t+c2 sin( 1 + kg)t +e0t
c3 cos( 1 − kg)t+c4 sin( 1 − kg)t

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3
49
isodÔnama
γ(t) = c1 cos( 1 + kg)t+c2 sin( 1 + kg)t+c3 cos( 1 − kg)t+c4 sin( 1 − kg)t
GnwrÐzontac ìti |γ|2
= |γ |2
= 1 kai me efarmog twn tÔpwn tou Frenet, èqw
ìti |ci|2
= 1
2
gia kˆje i = 1, 2, 3, 4.
Epomènwc, h lÔsh eÐnai h
γ(t) =
1
√
2
cos(At),
1
√
2
sin(At),
1
√
2
cos(Bt),
1
√
2
sin(Bt)
ìpou A = 1 + kg kai B = 1 − kg
H parapˆnw kampÔlh γ eÐnai gewdaisiak tou tìrou tou Cliﬀord
S1
( 1√
2
) × S1
( 1√
2
) ⊂ S3
⊂ R4
4.3 Diarmonikèc epifˆneiec sthn S3
Prin anaferjoÔme stic diarmonikèc epifˆneiec thc sfaÐrac S3
ja parousiˆ-
soume kˆpoia genikˆ apotelèsmata pou aforoÔn upopollaplìthtec sth sfaÐra
Sn
.
'Estw (M, , ) mia upopollaplìthta diˆstashc m thc Sn
kai i : M → Sn
h
apeikìnish ègklishc. SumbolÐzoume me:
• B th deÔterh jemeli¸dh morf thc M
• A to telest sq matoc thc M
• H to dianusmatikì pedÐo mèshc kampulìthtac thc M
• ⊥
thn orjog¸nia sÔndesh, dhlad th sÔndesh sthn orjog¸nia dèsmh
TM⊥
thc M

• ⊥
th Laplasian sthn orjog¸nia dèsmh TM⊥
thc M
Je¸rhma 4.3.1. H apeikìnish ègklishc i : M → Sn
eÐnai diarmonik an kai
mìno an
(1) − ⊥
H − traceB(−, AH−) + mH = 0
(2) 2traceA ⊥
(−)
H(−) + m
2
grad(|H|2
) = 0
Apìdeixh
GnwrÐzoume ìti
traceRSn
(di, τ(i))di = trace Sn
di , Sn
τ(i) di − Sn
[di,τ(i)]di = −mτ(i)
H apeikìnish i eÐnai diarmonik an kai mìno an τ2(i) = 0. 'Omwc
τ2(i) = J(τ(i)) = − (τ(i)) − traceRSn
(di, τ(i))di = trace dτ(i) + mτ(i)
'Ara h i eÐnai diarmonik an kai mìno an
τ2(i) = trace dτ(i) + mτ(i) = 0.
Gia mia isometrik emfÔteush i èqoume
H =
1
m
τ(i) ⇒
1
m
dτ(i) = dH ⇒
1
m
dτ(i) = dH ⇒
1
m
trace dτ(i) = trace dH ⇒
trace dτ(i) = mtrace dH
Apì tic duo teleutaÐec sqèseic èqoume
τ2(i) = mtrace dH+mτ(i) = mtrace dH+m.mH = m trace dH+mH = 0
'An xi m
i=1
eÐnai èna sÔsthma orjog¸niwn suntetagmènwn sth perioq tou
tuqaÐou shmeÐou p ∈ M kai ei = ∂
∂xi
m
i=1
èna orjog¸nio sÔsthma suntetag-
mènwn ston efaptìmeno q¸ro TpM thc M tìte
trace dH =
m
i=1
Sn
ei
Sn
ei
H

51
Apì ton tÔpo tou Weingarten èqoume ìti
trace dH =
i=1
m Sn
ei
−AH(ei) + ⊥
ei
H
ìpou
⊥
ei
: TM⊥
→ TM⊥
H → ⊥
ei
H ∈ TM⊥
kai
−AH(ei) ∈ TM
Apì to tÔpo tou Gauss
Sn
ei
AH(ei) = ei
AH(ei) + B(ei, AH(ei))
ìpou
Sn
ei
AH(ei) ∈ TSn
ei
AH(ei) ∈ TM
B(ei, AH(ei)) ∈ TM⊥
kai apì ton tÔpo tou Weingarten
Sn
ei
( ⊥
ei
H) = −A ⊥
ei
H(ei) + ⊥
ei
( ⊥
ei
H)
ìpou
Sn
ei
( ⊥
ei
H) ∈ TSn
−A ⊥
ei
H(ei) ∈ TM
⊥
ei
( ⊥
ei
H) ∈ TM⊥
Epomènwc,
trace dH =
m
i=1
Sn
ei
−AH(ei) + ⊥
ei
H =

m
i=1
⊥
ei
( ⊥
ei
H) − A ⊥
ei
H(ei) − ei
AH(ei) − B(ei, AH(ei))
'Omwc
⊥
H = −
m
i=1
⊥
ei
( ⊥
ei
H) − ⊥
⊥
ei
ei
H = −
m
i=1
⊥
ei
⊥
ei
H
'Ara,
trace dH = − ⊥
H − traceB(−, AH−) −
m
i=1
A ⊥
ei
H(ei) + ei
AH(ei)
Sth sunèqeia ja apodeÐxoume mia prìtash pou mac eÐnai qr simh sthn olokl -
rwsh thc apìdeixhc tou jewr matoc.
JewroÔme ton mousikì isomorfismì : TM → T∗
M mèsw tou opoÐou
tautÐzontai ta dianusmatikˆ pedÐa me tic 1-morfèc. H apeikìnish aut orÐze-
tai wc ex c: 'Estw V ∈ TM kai V ∗
oi 1-morfèc sto q¸ro T∗
M ètsi ¸ste
V ∗
(X) = V, X gia kˆje X ∈ TM.
Prìtash
'Estw V ∈ TM kai V ∗
oi 1-morfèc sto q¸ro T∗
M ètsi ¸ste V ∗
(X) = V, X
gia kˆje X ∈ TM.
Tìte h apeikìnish : TM → T∗
M eÐnai ènac isomorfismìc.
Apìdeixh
Gia na deÐxoume ìti h apeikìnish eÐnai isomorfismìc prèpei na deÐxoume ìti aut
eÐnai 1-1 kai epÐ.
Gia na deÐxoume ìti eÐnai 1-1 arkeÐ na deÐxoume ìti an V ∗
(X) = W∗
(X) gia
kˆje X ∈ TM tìte V = W. 'H isodÔnama an V, X = W, X gia kˆje
X ∈ TM tìte V = W. Prˆgmati, èstw U = V − W. ArkeÐ na deÐxw ìti eˆn
Up, Xp = 0 gia kˆje p ∈ M kai X ∈ TM tìte U = 0. Autì ìmwc isqÔei
apì ton orismì tou metrikoÔ tanust Riemann.

53
Gia na deÐxoume ìti eÐnai epÐ, prèpei na deÐxoume ìti dojeÐshc miac 1-morf c
θ ∈ T∗
M upˆrqei monadikì dianusmatikì pedÐo V ∈ TM tètoio ¸ste θ(X) =
V, X gia kˆje X ∈ TM.
Prˆgmati, jewroÔme èna topikì sÔsthma suntetagmènwn {xi}m
i=1 kai mia to-
pik orjokanonik bˆsh {∂i}m
i=1 tou q¸rou TM, kai {dxi}m
i=1 thn antÐstoiqh
orjokanonik bˆsh tou duikoÔ q¸rou T∗
M.
Tìte, h 1-morf θ kai to dianusmatikì pedÐo V grˆfontai wc ex c :
θ = i θidxi kai V = i,j gij
θi∂j.
Tìte, èqoume
V, ∂k M = i,j gij
θi∂j, ∂k
M
= i,j gij
θi ∂j, ∂k M = i,j θigij
gjk = i θiδik =
θk = θ(∂k).
Epomènwc, gia kˆje X = i Xi
∂i ìpou X ∈ TM èqoume
V, X M = V, i Xi
∂i M = i Xi
V, ∂i M = i Xi
θ(∂i) = θ i Xi
∂i =
θ(X).
Sth sunèqeia ja deÐxoume ìti to dianusmatikì pedÐo V ∈ TM tètoio ¸ste
θ(X) = V, X gia kˆje X ∈ TM eÐnai monadikì.
Prˆgmati, jewroÔme èna ˆllo dianusmatikì pedÐo W ∈ TM tètoio ¸ste
θ(X) = W, X gia kˆje X ∈ TM. Tìte, èqoume V, X = W, X gia
kˆje X ∈ TM. Autì shmaÐnei ìti V = W.
Epistrèfoume sthn apìdeixh tou jewr matoc 4.3.1. kai èqoume
trace dH = − ⊥
H − traceB(−, AH−) −
m
i=1
A ⊥
ei
H(ei) + ei
AH(ei)
'Omwc
m
i=1
A ⊥
ei
H(ei) + ei
AH(ei) = 2
m
i=1
A ⊥
ei
H(ei) +
m
2
(d|H|2
) =
2traceA ⊥
(−)
H(−) +
m
2
grad(|H|)2
)

ìpou
d(|H|)2
→ (d(|H|)) ≡ grad(|H|2
) ∈ TM
Anakefalai¸noume lègontac ìti h apeikìnish i eÐnai diarmonik an kai mìno an
τ2(i) = 0. Jètoume th tim tou trace dH sth sqèsh τ2(i) = m trace dH +
mH = 0 kai èqoume
− ⊥
H − traceB(−, AH−) + mH = 2traceA ⊥
(−)
H(−) +
m
2
grad(|H|2
)
Efìson to aristerì mèloc thc sqèshc an kei sto kˆjeto q¸ro thc M kai to
dexiì mèloc thc sqèshc ston efaptìmeno q¸ro thc M, èqoume
− ⊥
2traceA ⊥
(−)
H(−) +
m
2
grad(|H|2
) = 0
kai to je¸rhma èqei apodeiqjeÐ.
Pìrisma 4.3.1. 'Estw M mia upopollaplìthta thc Sn
me ⊥
H = 0.
Tìte h apeikìnish ègklishc i : M → Sn
eÐnai diarmonik an kai mìno an
mH = traceB(−, AH−).
Apìdeixh
Apì thn upìjesh gnwrÐzw ìti ⊥
H = 0, dhlad h sunˆrthsh ègklishc
i : M → Sn
èqei parˆllhlo dianusmatikì pedÐo mèshc kampulìthtac kai katˆ
sunèpeia to |H| eÐnai stajerì katˆ m koc thc M. Sto prohgoÔmeno je¸rhma
apodeÐxame pwc h i eÐnai diarmonik an kai mìno an isqÔoun ta ex c:
(1) − ⊥
(2) 2traceA ⊥
(−)
H(−) + m
2
grad(|H|2
) = 0
Epeid ⊥
H = 0 h pr¸th sqèsh gÐnetai
traceB(−, AH−) = mH

55
kai to pìrisma apedeÐqjhke.
Prìtash 4.3.1. 'Estw M mia uperepifˆneia thc Sn
. Tìte h apeikìnish
ègklishc i : M → Sn
eÐnai diarmonik an kai mìno an
(1) ⊥
H = (m − |B|2
)H
(2) 2traceA ⊥
(−)
H(−) + m
2
grad(|H|2
) = 0
Apìdeixh
'Eqoume
traceB(−, AH−) =
1
m
(traceA)η|B|2
= |B|2
H
ìpou
H =
1
m
(traceA)ηa =
1
m
(traceA)η
kai h ηa
m
a=1
eÐnai mia orjokanonik bˆsh tou TM⊥
.
Sto Je¸rhma 4.3.1. apodeÐxame ìti h i eÐnai diarmonik an kai mìno an
(1) − ⊥
(2) 2traceA (−)
H(−) + m
2
grad(|H|2
) = 0
H pr¸th sqèsh gÐnetai
− ⊥
H − |B|2
H + mH = 0 ⇔
⊥
H = (m − |B|2
)H
Epomènwc, h i eÐnai diarmonik an kai mìno an
(1) ⊥
H = (m − |B|2
)H
(2) 2traceA (−)H(−) + m
2
grad(|H|2
) = 0

Prìtash 4.3.2. 'Estw M = Sm
(a) × b =
p = (x1, ..., xm+1, b); x2
1 + ... + x2
m+1 = a2
, a2
+ b2
= 1, 0 a 1 mia
parˆllhlh upersfaÐra thc Sm+1
.
H M eÐnai diarmonik upopollaplìthta thc Sm+1
an kai mìno an a = 1√
2
kai
b = + 1√
2
b = − 1√
2
Apìdeixh
JewroÔme to sÔnolo Γ(TM) = X = (X1
, ...Xm
, 0) ∈ Rm+2
; x1
X1
+ ... +
xm+1
Xm+1
= 0 twn tom¸n (section) thc efaptìmenhc dèsmhc thc M kai
ξ = (x1
, ..., xm+1
, −a2
b
) èna dianusmatikì pedÐo thc M.
'Eqoume
ξ, X = x1
X1
+ ... + xm+1
Xm+1
+ (−
a2
b
)0 = 0
kai
ξ, p = (x1
)2
+ ... + (xm+1
)2
−
a2
b
b = a2
− a2
= 0
ξ, ξ = (x1
)2
+ ... + (xm+1
)2
+ (−
a2
b
)2
= a2
+
a4
b2
= c2
ìpou c 0. Apì tic duo pr¸tec sqèseic sumperaÐnoume ìti to ξ eÐnai tom
(section) thc orjog¸niac dèsmhc thc M, dhlad ξ ∈ Γ(TM⊥
).
Prìkeitai dhlad gia mia C∞
−apeikìnish
ξ : M → TM⊥
p → ξ(p) tètoia ¸ste π ◦ ξ = id, ìpou π ◦ ξ : M → M me tim (π ◦ ξ)(p) = p
gia kˆje p ∈ M kai π : TM⊥
→ M h dianusmatik dèsmh pˆnw sth M.
Jètoume η = 1
c
ξ kai sumbolÐzoume me −AηX to efaptìmeno dianusmatikì pedÐo
thc Sm+1
, dhlad
−AηX = ( Sm+1
X η)

57
ìpou h apeikìnish
Aη : C(TM) → C(TM)
X → AηX
eÐnai digrammik , autosuzug c kai kaleÐtai telest c sq matoc deÔterh jemeli¸dhc
morf sth kˆjeth dieÔjunsh ξ.
Apì to tÔpo tou Weingarten èqoume ìti
Sm+1
X η = ⊥
Xη − AηX
ìpou to dianusmatikì pedÐo ⊥
Xη orÐzei mia sunoq pou eÐnai sumbat sto
sÔnolo twn tom¸n thc orjog¸niac dèsmhc TM⊥
.
Jètw η = 1
c
ξ kai h sqèsh grˆfetai
Sm+1
X
1
c
ξ = ⊥
X
1
c
ξ − AηX ⇔
1
c
Sm+1
X ξ =
1
c
( ⊥
Xξ − AξX) =
1
c
( Rm+1
X ξ − AξX)
=
1
c
( Rm+1
X ξ + ξ, X p) =
1
c
(X1,...,Xm+1,0)(x1
, ..., xm+1
, −
a2
b
) =
1
c
X
Epomènwc,
⊥
Xη − AηX =
1
c
X ⇔ ⊥
(−)η − Aη(−) =
1
c
(−)
Apì th teleutaÐa sqèsh, èqoume ⊥
η = 0 kai Aη = 1
c
I kai to diˆnusma mèshc
kampulìthtac gÐnetai
H =
1
m
(traceA)η = −
1
c
η
AH = A−1
c
η = −
1
c
Aη = −
1
c
(−
1
c
)I =
1
c2
I
ApodeÐxame sto pìrisma 4.3.1 ìti h apeikìnish ègklishc miac upopollaplìth-
tac M thc Sn
me ⊥
H = 0 eÐnai diarmonik an kai mìno an
mH = traceB(−, AH−) = |B|2
H

Me efarmog tou porÐsmatoc autoÔ, h teleutaÐa sqèsh mac dÐnei
c2
= 1 ⇔ a2
+
a4
b2
= 1
ìpou a2
+b2
= 1 kai 0 a 1 . Oi lÔseic tou sust matoc twn duo exis¸sewn
eÐnai (a = 1√
2
, b = + 1√
2
) kai (a = 1√
2
, b = − 1√
2
).
'Ara h upopollaplìthta M eÐnai diarmonik thc Sm+1
an kai mìno an
a = 1√
2
kai b = 1√
2
b = − 1√
2
.
EÐdame ìti oi mh armonikèc diarmonikèc kampÔlec thc S3
èqoun stajer gew-
daisiak kampulìthta. Oi B.Y. Chen kai S. Ishikawa sthn ergasÐa touc [5],
apèdeixan ìti to mètro tou dianÔsmatoc mèshc kampulìthtac twn mh armonik¸n
diarmonik¸n epifanei¸n thc S3
eÐnai stajerì.
DiatÔpwsan kai apèdeixan to parakˆtw je¸rhma :
Je¸rhma 4.3.2. 'Estw M mia epifˆneia thc S3
. H M eÐnai mh armonik di-
armonik upopollaplìthta an kai mìno an to |H| eÐnai stajerì kai to |B|2
= 2.
Prokeimènou na taxinom soume tic diarmonikèc epifˆneiec thc S3
parajètoume
to apotèlesma thc ergasÐac [13] tou Z.H. Hou.
Je¸rhma 4.3.3. 'Estw M mia uperepifˆneia thc S3
me stajer mèsh kam-
pulìthta.
(1) An |B|2
= 2, tìte h M eÐnai eÐte topikˆ isometrik me èna tm ma thc uper-
sfaÐrac S2
( 1√
2
) sthn S3
eÐte eÐnai topikˆ isometrik me èna tm ma tou tìrou
S1
( 1√
2
) × S1
( 1√
2
)
(2) An h M eÐnai sumpag c kai prosanatolismènh kai |B|2
= 2, tìte h M
eÐnai eÐte isometrik mia mikr upersfaÐra S2
( 1√
2
) eÐte isometrik me ton tìro
S1
( 1√
2
) × S1
( 1√
2
).

59
An lˆboume upìyh mac ìti o tìroc tou Cliﬀord S1
( 1√
2
) × S1
( 1√
2
) eÐnai ar-
monik epifˆneia thc S3
tìte sundiˆzontac to je¸rhma 4.3.2. kai to je¸rhma
4.3.3., èqoume :
Je¸rhma 4.3.4. 'Estw M mia mh armonik diarmonik epifˆneia thc S3
.
(1) An h M eÐnai mh sumpag c, tìte aut eÐnai topikˆ isometrik me èna tm ma
thc sfaÐrac S2
( 1√
2
) sthn S3
.
(2) An h M eÐnai sumpag c kai prosanatolismènh, tìte eÐnai isometrik me th
sfaÐra S2
( 1√
2
) aktÐnac 1√
2
.
Anakefalai¸nontac, ta apotelèsmata pou katal goume eÐnai ta ex c :
'Estw Mm
mia diarmonik upopollaplìthta thc tridiˆstathc sfaÐrac S3
.
Tìte,
(1) An m = 1, dhlad h M eÐnai mia kampÔlh thc S3
, tìte h M eÐnai isometrik
eÐte
(i) me ènan kÔklo aktÐnac 1√
2
, ìtan h gewdaisiak kampulìthta eÐnai Ðsh me th
monˆda, dhlad kg = 1, eÐte
(ii) me mia gewdaisiak kampÔlh tou tìrou tou Cliﬀord
S1
( 1√
2
) × S1
( 1√
2
), ìtan h gewdaisiak kampulìthta ikanopoieÐ th sqèsh
0 kg 1.
(2) An m = 2, dhlad h M eÐnai mia uperepifˆneia tìte:
(i) an h M eÐnai mh sumpag c tìte aut eÐnai topikˆ isometrik me èna tm ma
thc sfaÐrac S2
( 1√
2
) sthn S3
, kai
(ii) an h M eÐnai sumpag c kai prosanatolismènh tìte eÐnai isometrik me th
sfaÐra S2
( 1√
2
) aktÐnac 1√
2
.

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