A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space

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1 A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), (Partially supported by DGICYT grant PB Fundación Séneca PB/5/FS/97) Abstract We first show that the geometries of a hypersurface seen either as a lightlike hypersurface or as a Riemannian one are closely related Furthermore, the Bernstein theorem for lightlike hypersurfaces is proved 1 Introduction As it is well known, the main difference between the geometry of submanifolds in Riemannian manifolds in semi-riemannian manifolds is that in the latter case the induced metric tensor field by the semi-riemannian metric on the ambient space is not necessarily non-degenerate If the induced metric tensor field is degenerate the classical theory of Riemannian submanifolds fails since the normal bundle the tangent bundle of the submanifold have a non-zero intersection The main purpose of the present paper is to show that the geometry of a lightlike (degenerate, null) hypersurface M in a semi-euclidean space R m+ ν can be investigated by using the geometry of M as a Riemannian hypersurface in a Euclidean space R m+ 0 In the first section we present the main tools in studying the geometry of a lightlike hypersurface: the screen distribution the lightlike transversal vector bundle (see [] for details) Then, in Section, we show that the canonical lightlike transversal vector bundle of a lightlike immersion is just the normal bundle of a Riemannian immersion This enables us, in Section 3, to prove the Bernstein theorem for lightlike hypersurfaces of Lorentz spaces Much of this work was done while the first author was visiting the Department of Mathematics at Universidad de Murcia He is very grateful to the differential geometry group for their kind invitation hospitality Preliminaries Let ( M, g) be a semi-riemannian manifold (see [7]), where g is a semi-riemannian metric of index 0 < ν < dim( M) Consider a hypersurface M of M denote by g the induced tensor field by g on M Then we say that M is a lightlike (degenerate, null) hypersurface if rank g = dim(m) 1 In order to get another characterization for a lightlike hypersurface we consider T x M = {V x T x M : g(vx, Y x ) = 0, Y x T x M} 1

2 Saitama Math J 16 (1998), T M = x M T x M Then it is easy to see that M is a lightlike hypersurface if only if T M is a distribution on M Next we consider a screen distribution ST M on M, which is a complementary non-degenerate vector bundle to T M in T M As T M lies in the tangent bundle, the following theorem has an important role in studying the geometry of a lightlike hypersurface Theorem 1 (Bejancu []) Let (M, ST M) be a lightlike hypersurface of ( M, g) Then there exists a unique vector bundle tr(t M) of rank 1 over M, such that for any non-zero section ξ of T M on a coordinate neighborhhod U M, there exists a unique section N of tr(t M) on U satisfying g(n, ξ) = 1 g(n, N) = g(n, W ) = 0, W Γ(ST M U ) By this theorem we may write down the decomposition T M M = T M tr(t M), which enables us to call tr(t M) the lightlike transversal vector bundle of M with respect to the screen distribution ST M Here in the sequel we denote by Γ(E) the F(M)-module of smooth sections of a vector bundle E over M, F(M) being the algebra of smooth functions on M Also by we denote the orthogonal non-orthogonal direct sum of two vector bundles In order to get an expression for N, we consider the orthogonal decomposition T M M = ST M ST M, where ST M is the orthogonal complementary vector bundle to ST M in T M M As T M is a vector subbundle of ST M we take a complementary vector bundle V of T M in ST M a non-zero V Γ(V U ) Then g(v, ξ) 0 on U, otherwise ST M would be degenerate at a point of U Finally, the vector field N from Theorem 1 is given by N = 1 g(v, ξ) { V g(v, V ) g(v, ξ) ξ The whole study of geometry of the lightlike hypersurface M is based on both the screen distribution the lightlike transversal vector bundle In case the ambient space is a semi-euclidean space it is constructed in [1] a canonical screen distribution that induces a canonical lightlike transversal vector bundle These vector bundles will play an important role in the rest of the paper 3 The geometry of a lightlike hypersurface in R m+ ν in R m+ 0 } via its geometry It is the purpose of this section to show that the canonical lightlike transversal vector bundle of a lightlike hypersurface M in R m+ ν is just the usual normal bundle of M as Riemannian hypersurface of the Euclidean space R m+ 0 As a consequence, we derive the interrelations between the geometric objects induced by the lightlike immersion the Riemannian immersion

3 Aurel Bejancu, Angel Ferrández Pascual Lucas, A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Let R m+ ν = (R m+, g SE ) R m+ 0 = (R m+, ) be the (m + )-dimensional semi- Euclidean space Euclidean space, where g SE st for the semi-euclidean metric of index 0 < ν < m + the Euclidean metric given by ν 1 g SE (x, y) = x i y i + x a y a, (x, y) = A=0 a=ν x A y A Consider a hypersurface M of R m+ given by the quation F (x o,, x ) = 0, where F is smooth on an open set Ω R m+ rank[f x o,, F x ] = 1 on M Then the normal bundle of M with respect to g SE is spanned by ξ = ν 1 SE F = F x i x i + a=ν F x a x a, where, as usual, SE F sts for the gradient of F with respect to g SE Thus M is a lightlike hypersurface if only if ξ is a null vector field with respect to g SE Hence we may state the following Theorem 31 M is a lightlike hypersurface if only if F is a solution of the partial differential equation ν 1 (F x ) = (F i x a) In order to obtain both the canonical transversal vector bundle the canonical screen distribution, we consider along M the vector field ν 1 V = note that V is nowhere tangent to M since a=ν F x i x i, ν 1 g SE (V, ξ) = (F x ) 0 i on M Then applying (1) we obtain that the canonical lightlike transversal vector bundle tr(t M) is spanned by N = 1 ( ν 1 ) 1 (F x ) i 3 A=0 F x A x A

4 Saitama Math J 16 (1998), Next we denote by N 0 the unit normal vector field on M with respect to the Euclidean metric Then due to (4) (5) we deduce This enables us to state ( N = ε(x)n 0, ε(x) = 1 ν 1 ) 1 (F x ) i Proposition 3 The canonical lightlike transversal vector bundle of the lightlike hypersurface M in R m+ ν is just the normal vector bundle of M as Riemannian hypersurface of R m+ 0 Taking into account that the canonical screen distribution ST M is complementary orthogonal to span{n, ξ}, (see [1], []) we get that in case ν > 1, ST M is locally spanned in a coordinate neighborhood U M by W i = F x 0 x i F x, i {1,, ν 1}, i x0 W a = F x x a F x a x, a {ν,, m}, provided that F x 0 F 0 x 0 on U In general, ST M is not integrable Indeed, the canonical screen distribution on the lightlike hypersurface M in R 4 defined by x (x1 + x ) x 3 = 0 is not integrable (see []) However, we shall prove that ST M is the orthogonal sum of two integrable distributions To this end we consider another coordinate neighborhood U M such that U U Then ST M is spanned on U by vector fields {W i, W a }, constructed as in (7) (8) provided one of the partial derivatives from each group {F x i } {F x a} is non-zero on U By direct computations we obtain span{w i } = span{w i} span{w a } = span{w a } on U U Hence we obtain two complementary distributions ST M ST M + locally spanned on U by {W i } {W a }, respectively Also we note that ST M ST M + are timelike spacelike orthogonal vector subbundles of ST M with respect to the semi-euclidean metric g SE, respectively Moreover, we prove Proposition 33 Both distributions ST M ST M + on a lightlike hypersurface M in R m+ ν are integrable Proof By using (7) we obtain F x j W i F x i W j = F x 0 ( F x j x i F x i x j Then by direct calculations using again (7) (9) we deduce that [W i, W j ] = ( F x 0 x 0 F x j (F x 0 ) 1 F x 0 x j ) Wi + ( F x 0 x i F x 0 x 0 F x i (F x 0 ) 1) W j, which proves that ST M is integrable In a similar way it follows that ST M + is integrable too In particular, if the ambient space is the Lorentz space R m+ 1 we have ST M = ST M + from Proposition 33 we deduce that 4 )

5 Aurel Bejancu, Angel Ferrández Pascual Lucas, A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Corollary 34 ([1]) Let M be a lightlike hypersurface of a Lorentz space R m+ 1 Then the canonical screen distribution ST M is integrable Due to Proposition 3 we shall write down some useful relations between the geometric objects induced by both the lightlike Riemannian immersion of M in R m+ Let be the Levi-Civita connection on R m+ with respect to both metrics g SE Then taking into account (6) we obtain that the Gauss Weingarten equations for the lightlike immersion of M in R m+ ν are given by (see []) X Y = X Y + B(X, Y )N = X Y + εb(x, Y )N 0, X, Y Γ(T M) X N = A N X + τ(x)n = A N X + ετ(x)n 0, X Γ(T M), respectively Here, as usual, we denote by, B A N the induced linear connection, the second fundamental form the shape operator of the lightlike immersion The lightlike transversal 1-form τ is specific to the lightlike immersion By using (10), (11) (6) we get B 0 (X, Y ) = εb(x, Y ), X, Y Γ(T M) A N X = εa N0 X, τ(x) = X(logε), X Γ(T M), where B 0 A N0 are the second fundamental form the shape operator of the Riemannian immersion of M in R m+ 0 Finally, we note that both immersions of M in R m+ ν in R m+ 0 induce the same linear connection on M, namely, the Levi-Civita connection with respect to the Riemannian metric induced by on M 4 The Bernstein theorem for lightlike hypersurfaces of a Lorentz space Bernstein s Theorem, which states that the only entire minimal surfaces in R 3 are planes, is one of the most striking theorems in global geometry In a famous paper [4], Cheng Yau proved that the Bernstein Theorem holds good for maximal spacelike hypersurfaces of a Lorentz space It is well known that such a result does not hold for entire timelike surfaces in the Minkowski 3-space R 3 1 A few years later TK Milnor [6] presented a generalization of the Hilbert-Holmgren theorem used it to discuss the indefinite Bernstein problem prove a conformal analog of Bernstein s Theorem for timelike surfaces in R 3 1 In [5] Magid gave one version of a solution to the indefinite Bernstein problem Actually he looked at entire timelike surfaces in R 3 1 which have zero mean curvature showed that such a graph over a timelike or spacelike plane is a global translation surface It is the purpose of the present section to prove the Bernstein Theorem for lightlike hypersurfaces with vanishing lightlike mean curvature in Lorentz spaces Let M be a hypersurface of the Lorentz space R m+ 1 given by equation () Then according to Theorem 31 M is a lightlike hypersurface if only if (F x ) = (F 0 x a) 5

6 Saitama Math J 16 (1998), As rank[f x 0,, F x ] = 1 on M, we deduce that F x 0 is nowhere zero on M Hence we may state the following Proposition 41 A connected lightlike hypersurface M of R 1 given by () is globally expressed by a Monge equation x 0 = f(x 1,, x ), where f is a smooth function on a domain Ω R such that (f x a) = 1 Since g SE becomes now a Lorentz metric, ie, it has index ν = 1, we shall denote it by g L In the next proposition we find the iterrelation between g L on M Proposition 4 Let (M, ST M) be a lightlike hypersurface of the Lorentz space R m+ 1, where ST M is the canonical screen distribution on M Suppose {V 1,, V m } is an orthonormal basis of Γ(ST M) with respect to g L Then {V 0, V 1,, V m } is an orthonormal field of frames on M with respect to, where we set Proof By using (3), (8) (13) we get V 0 = ( (ξ, ξ)) 1 ξ ξ = x 0 + W a = f x x a f x a x, Then by direct calculations we obtain f x a x a, a {1,, m} (W a, ξ) = g L (W a, ξ) = 0, a {1,, m} (W a, W b ) = g L (W a, W b ), a, b {1,, m}, which prove the assertion of the proposition As B(ξ, ξ) = 0 (see []), we define the lightlike mean curvature of M by α L = m B(V a, V a ), where {V 1,, V m } is an orthonormal basis of Γ(ST M) It is easy to show that α L does not depend on both the screen distribution the orthonormal basis By α E we shall denote the mean curvature of the Riemannian immersion of M in R m+ 0 Then by using (15), (1) Proposition 4 we deduce α E = B 0 (V 0, V 0 ) + m B 0 (V a, V a ) = 1 m B(V a, V a ) = 1 α L, since from (6) (13) we have ε = 1 Due to (16) we may state the following important result 6

7 Aurel Bejancu, Angel Ferrández Pascual Lucas, A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Theorem 43 Let M be a lightlike hypersurface of the Lorentz space R m+ 1 Then M has zero lightlike mean curvature if only if M is a minimal hypersurface of the Euclidean space R m+ We now examine the partial differential equation f x a x a 1 + = 0, E f 0 whose solutions give all minimal hypersurfaces (13) of the Euclidean space R m+ 0 In 1914 Bernstein proved a theorem that states that for m = the only entire solution of (17) is linear For the higher dimensional version of Bernstein s Theorem we need the following result Theorem 44 (Bompieri-Giusti [3]) Let f be a C solution of the minimal hypersurface equation (17) in R such that m partial derivatives f x a are uniformly bounded in R Then f must be linear For spacelike hypersurfaces of R m+ 1 the Bernstein Theorem was proved by Cheng Yau [4] By combining Theorems 43 44, using (14), we obtain the following result on Bernstein s Theorem for lightlike hypersurfaces Theorem 45 Let M be a lightlike hypersurface of R m+ 1 given by (13), where f is defined on R If M has zero lightlike mean curvature then f must be linear, thus M is a lightlike hyperplane of R m+ 1 References [1] A Bejancu, A canonical screen distribution on a degenerate hypersurface, Scientific Bulletin, Series A, Applied Math Physics, 55(1993), [] A Bejancu, Null hypersurfaces of semi-euclidean spaces Saitama Math J, 14(1996), 5-40 [3] E Bompieri E Giusti, Harnack s inequality for elliptic differential equations on minimal surfaces, Inventiones Math, 15(197), 4-47 [4] S Cheng ST Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski space, Ann of Math, 104(1976), [5] M Magid, The Bernstein problem for timelike surfaces, Yokohama Math J, 37(1989), [6] T K Milnor, A conformal analog of Bernstein s theorem for timelike surfaces in Minkowski 3-space, Contemporary Math, 64(1987), [7] B O Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York,