To Reiko MIYAOKA and Keizo YAMAGUCHI. Shyuichi IZUMIYA. Department of Mathematics, Hokkaido University (SAPPORO, JAPAN) February 6, 2011

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1 The lightlike geometry of spacelike submanifolds in Minkowski space joint work with Maria del Carmen ROMERO FUSTER (Valencia, Spain) and Maria Aparecida SOARES RUAS (São Carlos, Brazil) To Reiko MIYAOKA and Keizo YAMAGUCHI Shyuichi IZUMIYA Department of Mathematics, Hokkaido University (SAPPORO, JAPAN) February 6, 20 Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 / 6

2 Gaussian Differential Geometry; brief review 2 Lorentz-Minkowski space:r n+ 3 Spacelike submanifolds with codimension two 4 Spacelike submanifolds with general codimension 5 Spacelike submanifolds with codimension two, revisited Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 2 / 6

3 Gaussian Differential Geometry; brief review X : U R n : an embedding (hypersurface) (U R n : open, X(U) = M.) n(u) :Unit normal vector at p = X(u). The Gauss map G M : U S n : G M (u) = n(u) S p = dg M (u) : T p M T p M : the shape operator. The Gauss-Kronecker curvature : K(p) = det S p. The mean curvature : H(p) = n Trace S p. Related results on hypersurfaces: The Gauss-Bonnet theorem, The Weierstrass representation formula for a minimal surface, etc. For a general submanifolds M s R n, consider the unit normal bundle N (M). The (generalized) Gauss map G N (M) : N (M) S n : G N (M)(p, ξ) = ξ The Lipschit z-killing curvature at (p, ξ) : K(p, ξ) (can be defined). The total (absolute) curvature of M at p : K (p) = K(p, ξ) dσ n s. ξ N (M) p The related reults: The Chern-Lashof theorem, Convexity, Tightness etc. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 3 / 6

4 2 Lorentz-Minkowski sapce:r n+ R n+ = (R n+,, ): Lorentz-Minkowski n + -space x, y = x 0 y 0 + n x i y i, where x = (x 0, x,..., x n ), y = (y 0, y..., y n ) x R n+ \ {0} is i= spacelike if x, x > 0 light like if x, x = 0 timelike if x, x < 0, For any lightlike vector x, define ( x =, x,..., x ) n S n = {x = (x x 0 x + 0, x,..., x n ) x, x = 0, x 0 = }. 0 Hyperplane with pseudo normal n : HP(n, c) = {x R n+ x, n = c} for any n R n+ \ {0} and c R. HP(n, c) is spacelike if n is timelike light like if n is ligtlike timelike if n is spacelike, Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 4 / 6

5 2 Lorentz-Minkowski sapce:r n+ Pseudo-spheres in R n+ : H n ( ) = {x R n+ x, x = } : Hyperbolic n-space S n = {x Rn+ x, x = } : de Sit ter n-space LC = {x 0 R n+ x, x = 0 } : (open) lightcone For any x, x 2,..., x n R n+, e 0 e e n x x x 0 n x x 2 x n = x 2 x 2 x 2 0 n... x n n x n 0 x n, x i = (x i, 0 xi,..., xi n ). x x 2 x n : pseudo orthogonal to any x i (i =,..., n). We choose e 0 = (, 0,..., 0) as the future timelike vector field. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 5 / 6

6 3 Spacelike submanifolds with codimension two X : U R n+ : a spacelike embedding (U R n : open, M = X(U). ) T p M : a spacelike subspace at any point p = X(u) M. N p M : the pseudo-normal space, a timelike plane (i.e., Lorentz plane). N(M) : the pseudo-normal bundle over M. n T (u) N p (M): arbitrarily future directed unit timelike normal vector f ield. n S (u) N p (M) : the spacelike unit normal vector f ield defined by n S (u) = nt (u) X u (u) X un (u) n T (u) X u (u) X un (u) {n T, n S } : a pseudo-orthonormal f rame f ield along M. d(n T ± n S ) u : T p M T p R n+ = T p M N p (M) : linear mapping. Consider the psudo-orthogonal projection : π t : T p M N p (M) T p M S p (n T, ±n S ) = π t d(n T ± n S ) t u : T pm T p M: (n T, ±n S )-shape operator Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 6 / 6

7 3 Spacelike submanifolds with codimension two The lightcone Gauss-Kronecker curvature with respect to (n T, n S ): K ± l (nt, n S )(u) = det S p (n T, ±n S ). The lightcone mean curvature with respect to (n T, n S ): H ± l (nt, n S )(u) = n Trace S p(n T, ±n S ). n T (u) : another future pointed nomal vector field; (nt ± n S ) = (n T ± n S ) S n +. The lightcone Gauss map : L ± : U S n : L ± (u) = (nt ± n + S )(u) S ± p = πt d L ± p : T pm T p M : the normalized lightcone shape operator. The normalized lightcone Gauss-kronecker curvature : K ± l (u) = det S ± p. The normalized lightcone mean curvature : H ± l (u) = n Trace S ± p. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 7 / 6

8 3 Spacelike submanifolds with codimension two Proposition (The relation between curvatures) n K ± (u) = l l ± (u) K ± l (nt, n S )(u), H ± (u) = l l ± (u) H± l (nt, n S )(u), 0 0 where (n T ± n S )(u) = (l ± (u), 0 l± (u),..., l± n (u)). Corollary () K ± l (u) = 0 if and only if K± l (nt, n S )(u) = 0. (2) H ± l (u) = 0 if and only if H± l (nt, n S )(u) = 0. The flatness is independent of the choice of n T (u). Proposition Suppose n = 3. Let H be the mean curvature vector along M. Then H ± l 0 H : lightlike M : a marginary trapped sur f ace. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 8 / 6

9 Example 3 Spacelike submanifolds with codimension two () n T v:constant M H(v : c): a hypersurface in a spacelike hyperplane. Special case :n T (u) = e 0 H(e 0, 0) = R n : Eudclidean space 0 n S (u) : the ordinary unit normal in the Euclidean sence. K ± (u) = K(u): The Gauss-Kronecker curvature, l H ± (u) = ±H(u): The mean curvature. H 0: Minimal surfaces l (2) n T (u) = X(u) M = X(U) H n ( ): a hypersurface in Hyperbolic space L ± (u) = X(u) ± ns (u) : the hyperbolic Gauss map K ± (u) = K ± (u): The horospherical Gauss-Kronecker curvature, l h H ± (u) = H ± (u): The horospherical mean curvature. l h H ± 0: CMC ± surfaces h (3) n S v:constant M:a spacelike hypersurface in a timelike hyperplane H ± 0: Maximal surfaces l (4) n S (u) = X(u) M = X(U) S n : a spacelike hypersurface in de Sitter space L ± (u) = nt (u) ± X(u) : the de Sitter horospherical Gauss map Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 9 / 6

10 3 Spacelike submanifolds with codimension two M : closed orientable (n )-manifold, f : M R n+ : spacelike embedding. R n+ : time-oriented globally exists n T : M H n ( ) :future directed timelike unit normal vector field along f(m) The global ligthtcone Gauss map : L ± : M S n : L ± (p) = n + T (p) ± n S (p). The global normalized light cone Gauss-Kronecker curvature f unction: K ± l (p) = det( πt d L ± p ). Theorem (The Gauss-Bonnet type theorem) M : a closed orientable, spacelike submanifold of codimension two in R n+. Suppose that n is odd. Then K l dv M = 2 γ n χ(m), M χ(m) : the Euler characteristic of M, dv M : the volume form of M, γ n : the volume of the unit (n )-sphere S n. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 0 / 6

11 4 Spacelike submanifolds with general codimension X : U R n+ : a spacelike embedding of codimension k (U R s, M = X(U)) N p (M) : the pseudo-normal space at p = X(u), a k-dim Lorentz vector space. Two kinds of pseudo spheres: N p (M; ) = {v N p (M) v, v = } N p (M; ) = {v N p (M) v, v = } Two unit normal sperical normal bundles over M: N(M; ) = p M N p (M; ) and N(M; ) = p M N p (M; ). Remark that N p (M; ±) are non-compact we cannot integrate on the fiber. future directed unit timelike normal vector field n T (p) N p (M; ) (fix!!) N (M) p [n T ] = {ξ N p (M; ) ξ, n T (p) = 0 }:k -spacelike normal sphere. N (M)[n T ] = p M N (M) p [n T ] : spacelike unit normal bundle w.r.t n T Remark that N (M) p [n T ] is compact we can integrate on the fiber. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 / 6

12 4 Spacelike submanifolds with general codimension The lightcone Gauss map of N (M)[n T ] : Π t : LG(n T ) TR n+ S(n T ) (p,ξ) = Π t LG(n T ) : N (M)[n T ] S n + : LG(n T )(p, ξ) = nt (p) + ξ LG(n T )(p,ξ) = TN (M)[n T ] R k+ TN (M)[n T ] : the projection. d (p,ξ) LG(n T ) : T (p,ξ) N (M)[n T ] T (p,ξ) N (M)[n T ] :the light cone shape operat or. K l (n T )(p, ξ) = det S(n T ) (p,ξ) : the lightcone Lipschit z-killing curvature of N (M)[n T ] at (p, ξ) Remark: We can apply the theory of Lagarangian/Legendrian singularities to investigate local properties o the lightcone Lipschitz-Killing curvature. However, we do not mention these results here. Theorem ( LG(n T ) dv S n ) (p,ξ) = K l (n T )(p, ξ) dv N (M)[n T ](p,ξ), + where dv N (M)[n T ] : the canonical volume form of N (M)[n T ], dv S n + volume form of S n +. : the canonical Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 2 / 6

13 4 Spacelike submanifolds with general codimension a differential form dσ k 2 (n T ) of degree k 2 on N (M)[n T ] s.t its restriction to a fiber is the volume element of the unit k 2-sphere and dv N (M)[n T ] = dv M dσ k 2 (n T ). Proposition (Uniqueness) Let n T, n T be future directed unit timelike normal vector fields along M. Then we have K l (n T )(p, ξ) dσ k 2 (n T ) = K l (n T )(p, ξ) dσ k 2 (n T ). N (M) p [n T ] N (M) p [n T ] The total absolute lightcone curvature of M at p (well-defined): K (p) = K l l (n T )(p, ξ) dσ k 2 (n T ). N (M) p [n T ] Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 3 / 6

14 4 Spacelike submanifolds with general codimension f : M R n+ (M: s-dim closed orientable manifold): a spacelike immersion The total absolute lightcone curvature of M : τ l (M, f) = γ n K (p)dv l M = M γ n N (M)[n T ] where γ n is the volume of the unit n -sphere S n. Theorem (The Chern-Lashof type theorem) () τ l (M, f) γ(m) 2, (2) if τ l (M, f) < 3, then M is homeomorphic to the sphere S s, where γ(m) is the Morse number of M. K l (n T )(p, ξ) dv N (M)[n T ], Problem: Suppose τ l (M, f) = 2. What kind of immersed spheres in R n+ we have? This problem leads the notion of lightlike convexity and lightlike tightness. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 4 / 6

15 4 Spacelike submanifolds with codimension two, revisited If s = n, then N (M)[n T ] is a double covering of M. σ(p) = (p, n S (p)): global section of N (M)[n T ]. K (p) = K + (p) + K (p). l l l The positive/or negative total absolute curvature of M: τ ± (M, f) = l γ n K ± dv l M. M Theorem (The strong Chern-Lashof type theorem) τ ± (M, f). l Remark that M such that τ + (M, f) l τ (M, f). l Independent lightlike vectors v i (i =, 2) lightlike hyperplanes HP(v i : c i ). If HP(v : c ) HP(v 2 : c 2 ), then HP(v : c ) HP(v 2 : c 2 ) dvides R n+. into 4 regions. ; Two timelike regions and two spacelike regions. f(m) is lightlike convex if f(m) lies entirely in one of the closed half-spacelike regions determined by the tangent lightlike hypersurfaces of f(m) at any p M. Theorem τ ± l (M, f) = M is homeomorphic to Sn and f(m) is lightlike convex. Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 5 / 6

16 Thank you very much for your attention! And Happy birthday Reiko and Keizo! Shyuichi IZUMIYA (Hokkaido University) Spacelike submanifolds in Minkowski space February 6, 20 6 / 6