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  • 1. Lorentz surfaces in pseudo-Riemannian space forms of neutral signature with horizontal reector 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 eld (1)
  • 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. Di. 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. This talk is consists of1. Examples of Lorentz surfaces in pseudo-Riemannian space forms with H = 0.2. Reector spaces and reector lifts.3. A Rigidity theorem.4. Applications. (3)
  • 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. n+1 X 1 (1 i t)4Rn+1 := i , where i = 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. We dene t : C[x] C[x] by 1xi (1 i t) t(xi) = xi (t + 1 i n + 1) t is called Wick rotation.Set t := t t and Pt := Ker(t|R[x] idR[x]).We dene H (Rn+1) := Pt Hd(Rn+1). d,t 0 0 (6)
  • 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. Hereafter, we assume n = 2 and t = 1 (for simplicity). Then we seethat m = 2d + 1 and l = d. We dene Ui := Imt(ui) (i = 1, . . . , d)and Ui := Ret(ui) (i = d + 1, . . . , 2d + 1) d(d+1)Using these function, we dene an immersion from : Q2(1) Q2d( 1 d 2 )by = (U1, . . . , U2d+1). Note that satises H = 0. (8)
  • 9. For example, when d = 2 (n = 1, t = 1), : 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 d,c . (9)
  • 10. 2. Reector spaces and reector 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. fZx : the set of such Jx End(TxM ) [ fZ(M ) := Zx f xM(the bundle whose bers are consist of para-complex structures) f Z(M ) is called the f reector 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. e f e Def. : The map J : M Z(M ) satisfying J (x)|TxM = f Jx is reector lift of f . e e e Def. : The reector 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 reector lifts are corresponding to superminimalsurfaces in Riemannian cases. (12)
  • 13. Prop. The immersion m,c : Q2(2c/m(m + 1)) Q2m(c) in 1 1 madmits horizontal reector lift. (13)
  • 14. 3. A Rigidity theorem.At rst, we recall the k-th fundamental forms k . 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 dene X1 := X1, (X1, X2) = X1 X2 and inductively for k 3 e e e (X1, . . . , Xk ) = X1 (X2, . . . , Xk ),where Xi (T M ). (14)
  • 15. We dene e Osck (f ) := {((X1, . . . , Xk ))x | Xi (T M ), 1 i k} xand [ k Osc (f ) := Osck (f ). x xM f T M = Osc1(f ) Osc2(f ) Osck (f ) T M . We dene deg(f ) =the minimum integer d such that Oscd1(f ) (Oscd(f ) and Oscd(f ) = Oscd+1(f ). (15)