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# Reflect tsukuba524

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### Reflect tsukuba524

1. 1. Lorentz surfaces in pseudo-Riemannian space forms of neutral signature with horizontal reﬂector lifts Kazuyuki HASEGAWA (Kanazawa University) Joint work with Dr. K. Miura (Ochanomizu Univ.)0. Introduction. f e(M , g ) : oriented n-dimensional pseudo-Riemannian manifold(M, g) : oriented pseudo-Riemannian submanifold ff : M → M : isometric immersionH : mean curvature vector ﬁeld (1)
2. 2. Problem. f : Classify f : M → M with H = 0• f Riemannian case : When M = S n(c), M = S 2(K) of constantcurvatures c > 0 and K > 0, a full immersion f with H = 0 is congru-ent to the standard immersion (E. Calabi, J. Diﬀ. Geom., 1967). The ftwistor space of M = S n(c) and the twistor lift play an importantrole. f• In this talk, we consider the problem above in the case of M = Qn(c) = spseudo-Riemannian space form of constant curvature c and index s, and M =Lorentz surfaces , in particular, n = 2m and s = m. (2)
3. 3. • This talk is consists of1. Examples of Lorentz surfaces in pseudo-Riemannian space forms with H = 0.2. Reﬂector spaces and reﬂector lifts.3. A Rigidity theorem.4. Applications. (3)
4. 4. 1. Examples of Lorentz surfaces with H = 0.• In “K. Miura, Tsukuba J. math., 2007”, pseudo-Riemannian subman-ifolds of constant curvature in pseudo-Riemannian space forms withH = 0 are constructed from the Riemannian standard immersion us-ing “ Wick rotations ”.K[x] := K[x1, . . . , xn+1] : polynomial algebra in n + 1 variables x1, . . . , xn+1 over K (K = R or C).Kd[x] ⊂ K[x] : space of d-homogeneous polynomialsRn+1 : pseudo-Euclidean space of dim=n + 1 and index=t t (4)
5. 5. ⎧ n+1 X ⎪ ⎨ −1 ∂ (1 ≤ i ≤ t)4Rn+1 := − εi , where εi = t ∂x2 i ⎪ ⎩ 1 (t + 1 ≤ i ≤ n + 1) i=1H(Rn+1) := Ker4Rn+1 t tHd(Rn+1) := H(Rn+1) ∩ Kd[x] t t (5)
6. 6. We deﬁne ρt : C[x] → C[x] by ⎧ ⎪√ ⎨ −1xi (1 ≤ i ≤ t) ρt(xi) = ⎪ ⎩ xi (t + 1 ≤ i ≤ n + 1)• ρt is called “Wick rotation”.Set σt := ρt ◦ ρt and Pt± := Ker(σt|R[x] ∓ idR[x]).We deﬁne H± (Rn+1) := Pt± ∩ Hd(Rn+1). d,t 0 0 (6)
7. 7. • Hd(Rn+1) = H+ (Rn+1) ⊕ H− (Rn+1) 0 d,t 0 d,t 0{ui}i :orthonormal basis Hd(Rn+1) s.t. 0 u1, . . . , ul ∈ H− (Rn+1), d,t 0 ul+1, . . . um+1 ∈ H+ (Rn+1), d,t 0 m+1 X (ui)2 = (x2 + · · · + x2 )2, 1 n+1 i=1where l = l(d, t) = dim H− (Rn+1) and m+1 = m(d, t)+1 = dim Hd(Rn+1). d,t 0 0 (7)
8. 8. Hereafter, we assume n = 2 and t = 1 (for simplicity). Then we seethat m = 2d + 1 and l = d. We deﬁne Ui := Imρt(ui) (i = 1, . . . , d)and Ui := Reρt(ui) (i = d + 1, . . . , 2d + 1) d(d+1)Using these function, we deﬁne an immersion from φ : Q2(1) → Q2d( 1 d 2 )by φ = (U1, . . . , U2d+1). Note that φ satisﬁes H = 0. (8)
9. 9. For example, when d = 2 (n = 1, t = 1), φ : Q2(1) → Q4(3) is given by 1 2 √ 3 1 2 U1 = ( +2x + y + z ), U2 = (y − z 2) 2 2 2 6 2 U3 = xy, U4 = zx, U5 = yz, d(d+1)• Composing homotheties and anti-isometries of Q2(1) and Q2d( 1 d 2 ),we can obtain immersions from Q2(2c/d(d + 1)) to Q2d(c) with H = 0 1 d 6(c = 0). We denote this immersion by φd,c . (9)
10. 10. 2. Reﬂector spaces and reﬂector lifts.(See “G. Jensen and M. Rigoli, Matematiche (Catania) 45 (1990), 407-443. ”) f e(M , g ) : oriented 2m-dim. pseudo-Riemannian manifold of neutral signature (index= 1 dim M ) f 2 fJx ∈ End(TxM ) s.t. 2 ◦ Jx = I, ◦ (Jx)∗gx = −ex, e g ◦ dim Ker(Jx − I) = dim Ker(Jx + I) = m, ◦ Jx preserves the orientation. (10)
11. 11. fZx : the set of such Jx ∈ End(TxM ) [ fZ(M ) := Zx f x∈M(the bundle whose ﬁbers are consist of para-complex structures) f• Z(M ) is called the f reﬂector space of M , which is one of corre-sponding objects to the twistor space in Riemannian geometry.(M, g) : oriented Lorentz surface with para-complex structure J ∈ Z(M ). ff : M → M : isometric immersion (11)
12. 12. e f e Def. : The map J : M → Z(M ) satisfying J (x)|TxM = f∗ ◦ Jx is reﬂector lift of f . e e e Def. : The reﬂector lift is called horizontal if ∇J = 0, where ∇ is f the Levi-Civita connection of M . e e• ∇J = 0 ⇒ H = 0• Surfaces with horizontal reﬂector lifts are corresponding to superminimalsurfaces in Riemannian cases. (12)
13. 13. Prop. The immersion φm,c : Q2(2c/m(m + 1)) → Q2m(c) in §1 1 madmits horizontal reﬂector lift. (13)
14. 14. 3. A Rigidity theorem.At ﬁrst, we recall the k-th fundamental forms αk . f e(M , g ) : oriented n-dim pseudo-Riemannian manifold(M, g) : oriented pseudo-Riemannian submanifold ff : M → M : isometric immersione f∇ : Levi-Civita connection of M . e e eWe deﬁne ∇X1 := X1, ∇(X1, X2) = ∇X1 X2 and inductively for k ≥ 3 e e e ∇(X1, . . . , Xk ) = ∇X1 ∇(X2, . . . , Xk ),where Xi ∈ Γ(T M ). (14)
15. 15. We deﬁne e Osck (f ) := {(∇(X1, . . . , Xk ))x | Xi ∈ Γ(T M ), 1 ≤ i ≤ k} xand [ k Osc (f ) := Osck (f ). x x∈M f• T M = Osc1(f ) ⊂ Osc2(f ) ⊂ · · · ⊂ Osck (f ) ⊂ · · · ⊂ T M .• We deﬁne deg(f ) =the minimum integer d such that Oscd−1(f ) (Oscd(f ) and Oscd(f ) = Oscd+1(f ). (15)
16. 16. • There exists the maximum number n (1 ≤ n ≤ deg(f )) such that fOsc1(f ), . . . , Oscn(f ) are subbundles on T M . Then we call f to benicely curved up to order n.• When f is nicely curved up to order n, there exists the maximumnumber m (1 ≤ m ≤ n) such that all induced metrics of subbundlesOsc1(f ), . . . , Oscm(f ) are nondegenerate. Then f is called nondegenerately nicely curvedup to m. (16)
17. 17. • 1 ≤ m ≤ n ≤ deg(f ).• When f is called nondegenerately nicely curved up to m, we candeﬁne the (k −1)-th normal space N k−1 which is deﬁned the orthogonalcomplement subspace of Osck−1(f ) in Osck (f ). Def. : When f is nondegenerately nicely curved up to m, we deﬁne (k + 1)-th fundamental form αk+1 as follows : k e αk+1(X1, . . . , Xk+1) := (∇(X1, . . . , Xk+1)N , where Xi ∈ Γ(T M ) (1 ≤ i ≤ k + 1) and 1 ≤ k ≤ m − 1 f• If M = Qn(c), αk+1 is symmetric. s (17)
18. 18. Thm. ¯ : Let f , f : M → Q2m(c) be an isometric immersions from ma connected Lorentz surface M with horizontal reﬂector lifts. If ¯both immersions f and f are nondegenerately nicely curved up to ¯m, then there exist an isometry Φ of Q2m(c) such that f = Φ ◦ f . mRemark. The immersion φm,c : Q2(2c/m(m + 1)) → Q2m(c) in §1 1 mis nondegenerately nicely curved up to m. (18)
19. 19. Cor. : Let f : M → Q2m(c) be an isometric immersions from a mconnected Lorentz surface M with horizontal reﬂector lifts. If f isnondegenerately nicely curved up to m and the Gaussian curvatureK is constant, then K = 2c/m(m + 1) and f is locally congruentto φm,c. (19)
20. 20. 4. Applications.V , W : vector spaces with inner products h , iα : V × V · · · × V → W : symmetric k-multilinear map to W .For the sake of simpliﬁcation, we set α(X k ) := α(X, .{z , X ) | .. } kfor X ∈ V . (20)
21. 21. Def. : We say that α is spacelike (resp. timelike) isotropic if hα(uk ), α(uk )i is independent of the choice of all spacelike (resp. timelike) unit vectors u. The number hα(uk ), α(uk )i is called space- like (resp. timelike) isotropic constant of α.• α is spacelike isotropic with spacelike isotropic constant λ ⇐⇒ α istimelike isotropic with timelike isotropic constant (−1)k λ. (21)
22. 22. Def. : We say that the k-th fundamental form αk is spacelike(resp. timelike) isotropic if αk is spacelike (reps. p timelike)isotropic at each point p ∈ M . The function λk : M → R de-ﬁned by λk (p) := spacelike isotropic constant of αk p is called thespacelike (resp. timelike) isotoropic function. If the spacelike (resp.timelike) isotoropic function λk is constant, then k-th fundamentalform αk is called constant spacelike (resp. timelike) isotropic . (22)
23. 23. Prop. : Let f : M → Qn(c) be an isometric immersion with H = s0. If the higher fundamental forms αk are spacelike isotropic for2 ≤ k ≤ m and their spacelike isotropic functions are everywherenonzero on M . Then we have n ≥ 2m and s ≥ m. In particular, ifn = 2m, then s = m and f admits horizontal reﬂector lift. (23)
24. 24. Cor. : If f : M → Q2m(c) be an isometric immersion with H = 0 ssuch that αk are spacelike isotropic for 2 ≤ k ≤ m whose isotropicfunctions are everywhere nonzero and M has a constant Gaussiancurvature K, then we have s = m, K = 2c/m(m + 1) and f islocally congruent to the immersion φm,c (24)
25. 25. Consider the following condition : (†) An isometric immersion f maps each null geodesic of M into f a totally isotropic and totally geodesic submanifold L of M . e• L is totally isotropic : ⇐⇒ g |L = 0. e• L is totally geodesic : ⇐⇒ ∇X Y ∈ Γ(T L) for all X, Y ∈ Γ(T L). Remark. The immersion φm,c satisﬁes the condition (†). (25)
26. 26. Cor. : Let f : M → Q2m(c) be an isometric immerison with sH = 0. If K = 2c/m(m + 1) and the condition (†) holds, thens = m and f is locally congruent to the immersion φm,c. (26)