SPACELIKE SUBMANIFOLDS IN DE SITTER SPACE

We investigate the differential geometry of spacelike submanifolds of codimension at least two in de Sitter space as an application of the theory of Legendrian singularities. We also discuss related geometric property of spacelike hypersurfaces in de Sitter space. Mathematics Subject Classification (2000): 53A35, 53B30, 58C25.


Introduction
It is known that de Sitter space is a Lorentzian space form with a positive curvature. Recently, Izumiya, Pei and Sano [3] investigated the extrinsic differential geometry of hypersurfaces in the hyperbolic space by applying the theory of Legendrian singularities. The main tool is a lightcone Gauss indicatrix, which is defined by a lightlike normal of hypersurface, and their singularity sets correspond to lightcone parabolic sets of hypersurfaces. For higher codimension case, the normal vector is not uniquely determined, however it is possible to construct hypersurfaces from the normal unit vector fields of the subspace. Izumiya, Pei, Romero Fuster and Takahashi [6] introduced the notion of canal hypersurfaces and horospherical hypersurfaces from the normal frames of submanifolds in the hyperbolic space, and investigated submanifolds of higher codimension in the hyperbolic space from the viewpoint of singularity theory. On the other hand, the differential geometry of de Sitter space is also studied. In [7] we introduced the notion of lightcone Gauss image which is an analogous tool introduced in [3], and investigate the case of spacelike hypersurface in de Sitter space. For codimension two case, Fusho and Izumiya [2] firstly introduced the notion of lightlike surface of a spacelike curve in the de Sitter three-space. In [8] we investigated singularities of lightlike hypersurface of spacelike submanifold of codimension two in de Sitter n-space for n ≥ 3, which is an analogous study in the Minkowski space [4,5].
In this paper, we argue an analogous study of the submanifolds of higher codimension in hyperbolic space [6] and introduce the notions of horospherical hypersurfaces and spacelike canal hypersurfaces by using timelike unit normal vector fields. The singular point of horospherical surface corresponds to the parabolic point of spacelike canal hypersurface, which we call a horospherical point, and the spacelike submanifold is tangent to a de Sitter hyperhorosphere at the horospherical point. If we assume a hypothesis of Theorem 6.5, then a contact type of a de Sitter hyperhorosphere and a spacelike submanifold corresponds to a singular type of horospherical hypersurface, and also corresponds to a singular type of lightcone Gauss image of spacelike canal surface. In §2 we review briefly the basic notions of differential geometry of spacelike hypersurfaces [7]. In §3,4 we define a timelike normal vector field of spacelike submanifolds in de Sitter space. and introduce a notion of horospherical height function and horospherical hypersurface. We also define a spacelike canal hypersurface, whose lightcone Gauss image is diffeomorphic to a horospherical hypersurface. In §5 we naturally interpret a horospherical hypersurfaces of a spacelike submanifold as a wave front set of horospherical height functions in the theory of Legendrian singularities. In §6 we use the theory of contacts between the submanifolds due to Montaldi [10], and we discuss geometric properties of singularities of horospherical hypersurfaces. We consider generic properties of spacelike submanifolds in §7.

Spacelike hypersurfaces in de Sitter space
In this section we review the extrinsic differential geometry of spacelike hypersurfaces in de Sitter space [7], which is an analogous study of [3]. Let R n+1 = {x = (x 0 , . . . , x n ) | x i ∈ R (i = 0, . . . , n)} be an (n + 1)-dimensional vector space. For any vectors x = (x 0 , . . . , x n ), y = (y 0 , . . . , y n ) in R n+1 , the pseudo scalar product of x and y is defined by and a real number c, we define a hyperplane with pseudo normal v in the Minkowski space by HP(v, c) = {x ∈ R n+1 1 | x, v = c}. We say that a hyperplane HP (v, c) is spacelike, timelike or lightlike if the vector v is timelike, spacelike or lightlike.
We now respectively define hyperbolic n-space and de Sitter n-space by , we can define a vector x 1 ∧ x 2 ∧ . . . ∧ x n with the property x, x 1 ∧ . . . ∧ x n = det(x, x 1 , . . . , x n ), so that x 1 ∧ . . . ∧ x n is pseudo-orthogonal to any x i for i = 1, . . . , n. We also define future (resp. past) lightcone at the origin by We say X is a spacelike hypersurface in S n 1 if every non zero vector generated by {X u i (u)} n−1 i=1 is always spacelike, where u = (u 1 , . . . , u n−1 ) is an element of U and X u i is a partial derivative of X with respect to u i . We denote M = X(U ) and identify M with U through the embedding X. Since X, X ≡ 1, we have X u i , X ≡ 0 for i = 1, . . . , n − 1. It follows that a hyperplane spanned by {X, X u 1 , . . . , X u n−1 } is spacelike. We define a vector Then e is pseudo orthogonal to X and X u i for i = 1, . . . , n − 1. We define a map L ± : U −→ LC * ± by L ± (u) = X(u) ± e(u), which is called a positive (resp. negative) lightcone Gauss image of X.
We now consider a hypersurface defined by HP (v, c) ∩ S n 1 . We say that HP (v, c) ∩ S n 1 is an elliptic hyperquadric or a hyperbolic hyperquadric if HP (v, c) is spacelike or timelike respectively. We say that HP (v, 1) ∩ S n 1 is a de Sitter hyperhorosphere if HP (v, 1) is lightlike. We have the following proposition analogous to ( [3], Proposition 2.2). Since X u i (for i = 1, . . . , n − 1) are spacelike vectors, we have the Riemannian metric (first fundamental form) We define a lightcone second fundamental invariants byh ± ij (u) = −L ± u i (u), X u j (u) for any u ∈ U . In [7] we obtained explicit expression for the lightcone Gauss-Kronecker curvature: We say that p = X(u 0 ) is a lightcone parabolic point of X if K ± (u 0 ) = 0.
We We also naturally interpreted the lightcone Gauss image of a spacelike hypersurface as a wave front set in the frame work of contact geometry in [7]. This is the analogous way to the differential geometry of hypersurfaces in hyperbolic space [3].
It is easy to show that is non singular. In this case, we have a smooth (k − 1)-dimensional smooth submanifold, is a Legendrian immersion germ. Then we have the following fundamental theorem of Arnol'd and Zakalyukin [1,12]. Proposition 2.3. All Legendrian submanifold germs in P T * R k are constructed by the above method.
We call F a generating family of L F (Σ * (F )). Therefore the wave front is We call it the discriminant set of F . In [9] we showed that the lightcone height function H is a Morse family of hypersurface and its discriminant set is the image of Gauss images L ± (U ). Therefore we have a immersion germ L ± : where v ± = L ± (u) and Σ ± * (H) is a singular set of H.

Spacelike submanifolds in de Sitter space
In this section, we consider the differential geometry of spacelike submanifolds in de Sitter space, which is analogous to [6]. Let r be an integer at least two and X : U −→ S n 1 be an embedding from an open set U ⊂ R n−r . We say that X is spacelike in S n 1 if every non zero vector generated by is spacelike, where u ∈ U and X u i = ∂X/∂u i . We identify M = X(U ) with U through the embedding X and call M a spacelike submanifold of codimension r in de Sitter space. Since be a timelike unit normal of M . Since n, n ≡ −1 and X, n ≡ 0, n u i is pseudo orthogonal to both of X and n for i = 1, . . . , n − r. Therefore we have Let d u n be the derivative of n at u, under the identification of M and U through X, we have the linear transformations on We respectively call the linear transformation A p (n) = −d p n T and S p (n) = −(id TpM + d p n T ) an n-shape operator and a horospherical n-shape operator of M at p = X(u). We also call the linear map d u n N a normal connection with respect to the timelike normal n of M . We denote eigenvalues of A p (n) and S p (n) by κ p (n) andκ p (n), which we respectively call an n-principal curvature and a horospherical n-principal curvature. The horospherical Gauss-Kronecker curvature with respect to n at p = X(u) is defined to be We say that a point p 0 = X(u 0 ) is an n-umbilic point if det S p 0 (n) =κ p 0 (n)id Tp 0 M . Since the eigenvectors of det S p 0 (n) and det A p 0 (n) are the same, the above condition is equivalent to det A p 0 (n) = κ p 0 (n)id Tp 0 M . We say that the spacelike submanifold M is totally n-umbilic if every point on M is n-umbilic. We also say that the timelike unit normal vector field n is parallel at p if d p n N = 0 TpM . The timelike unit normal field n is parallel if n is parallel at any points on M . Then we have the following result which is analogous to ([6], Proposition 3.1).
is totally n-umbilic and timelike unit normal vector field n is parallel. Then κ p (n),κ p (n) are constant κ(n),κ(n) and there exists a vector v ∈ R n+1 1 such that M is a part of hyperplanes HP (v, c) ∩ S n 1 in de Sitter space for some vector v and real number c. Under this condition we have following cases: (1) If 1 < |κ(n) + 1| = |κ(n)| then M is a part of a hyperbolic hyperquadric HP (v, +1).
Proof. By the assumption, we have dn . Since X i (u) and X j (u) are linearly independent, Therefore κ u i ,p = κ u j ,p = 0, this means that κ p andκ p are constant κ,κ.
On the other hand, ifκ + 1 = κ = 0 then there exists a constant timelike vector v such that n(u) = v for any u ∈ U . So that X(u), v = X(u), n(u) = 0 for any u ∈ U . This means that M ⊂ HP (v, 0) Therefore (3) holds. This completes the proof.
We now consider the following Weingarten type formula. Since {X u i } n−r i=1 spans a spacelike vector subspace, we induce a Riemannian metric (the horospherical first fundamental form) We respectively define the second fundamental invariant and horospherical second fundamental invariant with respect to the timelike unit normal vector field n by We have the relationh ij (n) = −g ij + h ij (n) (for i, j = 1, . . . , n − r).
Under the above notations, we have the following Weingarten type formula with respect to the timelike unit normal vector field n, which is analogous to ([6], Proposition 3.2).
where (h j i (n)) ij = (h ik (n)) ik (g kj ) kj and (g kj ) = (g kj ) −1 . Therefore, the Gauss-Kronecker curvature with respect to n is given by Since X + n, X u j ≡ 0, the coefficient of the second fundamental form with respect to the timelike parallel unit normal vector field n is expressed bȳ Therefore the horospherical second fundamental invariant at a point p 0 = X(u 0 ) depends only on the timelike vector n 0 = n(u 0 ). It is independent of the choice of timelike parallel unit normal vector field n with n 0 = n(u 0 ). Let n 0 be a timelike unit normal vector, we say that a point p 0 = X(u 0 ) is an n 0 -parabolic point (resp. n 0 -umbilic point) of M if K(n)(u 0 ) = 0 (S p 0 (n) =κ p 0 (n)id Tp 0 M ) for some timelike parallel unit normal vector field n with n(u 0 ) = n 0 . We also say that p 0 is an n 0 -horospherical point if it is an n 0 -parabolic point and an n 0 -umbilic point.
We remark that for any spacelike submanifold X and point (u 0 , µ 0 ) ∈ U × H r−1 (−1), there are a real number θ = 0 and an open neighborhood V of (u 0 , µ 0 ) such thatX θ is spacelike embedding on V . We assume that for any (u, µ) ∈ V then (u, −µ) ∈ V . We write CM as an imageX θ (V ) and call it a spacelike canal hypersurface of M = X(U ). Izumiya, Pei, Romero Fuster, Takahashi [6] has introduced the notion of canal surfaces of submanifolds in the hyperbolic space. We now consider the horospherical height function on a spacelike submanifold. For a spacelike submanifolds X of codimension r, we define a function  The proof for above proposition are parallel to those of Proposition 3.4 in [6], so that we omit it. Therefore the discriminant set of the horospherical height function H is We .
We also define e (u, µ) = ∑ r−1 i=0 µ i n i (u). It follows from the above definition that e(u, µ) = e • Φ(u, µ) Therefore we have where HS X = X(u) + e (u, µ). This means that HS X is independent to the choice of orthonormal frames of N (M ) up to the diffeomorphic parametrization. We have a following proposition which is analogous to ([6], Proposition 3.5).  (u, µ(u))-umbilic for some parallel normal vector field e(u, µ(u)) and K(e(u, µ(u)))(u) = 0, then the above assertion holds.
Proof. Suppose that v 0 = X(u)+e(u, µ) is a constant vector. Since e(u, µ) is pseudo orthogonal to X(u), then we have X(u), v 0 = +1 for any u ∈ U . This means that M is a part of a de Sitter hyperhorosphere HP (v 0 , 1) ∩ S n 1 . On the other hand, if M ⊂ HP (v 0 , 1) ∩ S n 1 for some lightlike vector, then v 0 − X(u), X(u) = 0 for any u ∈ U . Since X(u) is pseudo orthogonal to X u i (u), it follows that v 0 − X(u), X u i (u) = 0. This means that X(u) − v 0 is a normal vector of M at p = X(u). We define a function µ(u) by e(u, µ). This completes the proof.
Since the image of HS X is the discriminant set of the horospherical height function H on M , the singular set of HS X corresponds to the null set of the Hessian matrix of the horospherical height function with the fixed parameter v at each point. Therefore we have the following proposition which is analogous to ([6], Proposition 3.6).

Proposition 4.3.
The singular set of HS X is given by ) is the horospherical second fundamental invariant with respect to the timelike direction e(u 0 , µ 0 ). The horospherical Gauss-Kronecker curvature is K(e(u, µ))(u) = det( X u i u j (u), X(u) + e(u) )/ det(g ij (u)) = det Hess h v (u)/ det(g ij (u)), where (g ij (u)) is the first fundamental invariant of M . Therefore Hess h v (u) = 0 if and only if K(e(u, µ))(u) = 0. This completes the proof.

By Proposition 2.2,
We now define a diffeomorphism By the above lemma, the horospherical hypersurface HS X is locally diffeomorphic to the lightcone Gauss image of the spacelike canal hypersurfaceX θ .

Horospherical hypersurfaces as wave fronts
In this section we naturally interpret the horospherical hypersurfaces of M as a wave front set of the horospherical height functions in the theory of Legendrian singularities.
By proceeding arguments in §2, the horospherical hypersurface HS X is the discriminant set of the horospherical height function H, and the singular point set of the horospherical hypersurface is the horospherical point set. We have the following proposition which is analogous to ([6], Proposition 4.1). Proof. We denote X(u) = (X 0 (u), . . . , X n (u)) and X u i (u) = (X 0,u i (u) , . . . , X n,u i (u)). For any v = (v 0 , . . . , v n ) ∈ LC * , we have v 0 = 0. Without loss of generality, we assume that v 0 = √ v 2 1 + · · · + v 2 n > 0, so that we have We now prove a map is non singular at any (u, v) ∈ Σ * (H). The Jacobian matrix of ∆ * H is We denote an (n − r + 1) × n matrix B by J∆ * H = ( * | B). It is sufficient to show that rank B = n − r + 1 at (u, v) ∈ Σ * (H). We also denote an (n − r + 3) × (n + 1) matrix C by We now show that the rank of the matrix C is equal to n − r + 3. Since v, X(u) and X u i (u) are linearly independent for all (u, v) ∈ Σ * (H), it is sufficient to show that timelike unit vector e = (1, 0, . . . , 0) can not be written by a linear combination of v, X(u) and X u i (u). If that is not so, there exists some real numbers η, µ, ξ i such that e = ηv +µX(u)+w and w = ∑ n−r i=1 ξ i X u i (u). Then we have e, e = µ 2 + w, w . However, w is a spacelike vector, so that e, e would not be negative, which contradicts our assumption. This means that e, v, X(u) and X u i (u) are linearly independent, therefore we have rank C = n − r + 3.
We now show rank B = rank C − 2. We subtract the second row multiplied by X 0 /v 0 from the third row of the matrix C, and add the second row multiplied by X 0,u k (u)/v 0 from the (3 + k)-th row for k = 1, . . . , n − r. Then we have a matrix Therefore we have rank B = rank C − 2 = n − r + 1. This completes the proof.
Since H is a Morse family of hypersurfaces, we have the Legendrian immersion germ L H : We remark that the wave front set of the Legendrian immersion germ L H is the horospherical hypersurfaces HS X of M . On the other hand, we define a contact diffeomorphismM c : where c is a fixed real parameter with c = 0. By definition, we have the following theorem.

Contact with de Sitter hyperhorospheres
In this section we use the theory of contacts between the spacelike submanifolds and the de Sitter hyperhorospheres due to Montaldi [10].
We now consider the function H : 1 be a spacelike submanifold of codimension r ≥ 2. For any u 0 ∈ U and µ 0 ∈ H r−1 (−1), we take a point v 0 = X(u 0 ) + e(u 0 , µ 0 ). By Proposition 4.1, we have It follows that the de Sitter hyperhorosphere h −1 v 0 (0) = HP (v 0 , +1) ∩ S n 1 is tangent to M at p 0 = X(u 0 ). In this case we call HP (v 0 , +1) ∩ S n 1 a tangent de Sitter hyperhorosphere (briefly, tangent hyperhorosphere) with respect to X(u 0 ) + e(u 0 , µ 0 ). We may also consider the contacts of the spacelike canal surface CM =X(V ) and the de Sitter horospheres. (see [7]) We now review some notions of Legendrian singularity theory to study the contact between hypersurfaces and de Sitter hyperhorospheres. We say that Legendrian immersion germs ι i : (U i , u i ) −→ (P T * R n , p i ) (i = 1, 2) are Legendrian equivalent if there are a contact diffeomorphism germ H : (P T * R n , p 1 ) −→ (P T * R n , p 2 ) and a diffeomorphic germ τ : (U 1 , u 1 ) −→ (U 2 , u 2 ) such that H preserves fibers of π and H •ι 1 = ι 2 •τ . A Legendrian immersion germ at a point is said to be Legendrian stable if for every map with the given germ there are a neighborhood in the space of Legendrian immersions (in the Whitney C ∞ -topology) and a neighborhood of the original point such that each Legendrian map belonging to the first neighborhood has in the second neighborhood a point at which its germ is Legendrian equivalent to the original germ. (see [1]) Proposition 6.2. ( [13]) Let i 1 , i 2 be Legendrian immersion germs such that regular sets of π • i 1 and π • i 2 are respectively dense. Then i 1 , i 2 are Legendrian equivalent if and only if corresponding wave front sets W (i 1 ) and W (i 2 ) are diffeomorphic as set germs.

Proof.
Since L H 1 and L H 2 are Legendrian stable, regular sets of HS X 1 and HS X 2 are respectively dense, by applying Proposition 6.2, the conditions (1) and (2) are equivalent. And we apply Theorem 6.3, the conditions (2) and (3) are equivalent. By the previous arguments from Theorem 6.1, the conditions (4) and (5) are equivalent. If we assume the condition (3), then the P-K-equivalence of H i (i = 1, 2) preserves the K-equivalence of h i,v i , so that the condition (4) holds. Since the local rings Q(X i , u i ) are K-invariant, this means that the condition (6) holds. By Proposition 6.4, the condition (6) implies the condition (2). Therefore the conditions from (1) to (6) are equivalent.
By theorem 5.2, (2) and (8) are equivalent. Since L H i are Legendrian stable, LH i are also Legendrian stable. So that we may similarly show the equivalence of the conditions from (7) to (12). On the other hand, h −1 i,v i (0) = (X i −1 (HP (v i , 1) ∩ S n 1 ), u i ) and K-equivalence preserves the zero level sets, so that (X i −1 (HP (v i , 1) ∩ S n 1 ), u i ) (i = 1, 2) are diffeomorphic as set germs. This completes the proof.

Generic properties
In this section we consider generic properties of spacelike submanifolds of codimension r ≥ 2 in S n 1 . Let U be an open subset of R n−r . We consider the space of spacelike embeddings Sp-Emb(U, S n 1 ) with Whitney C ∞ -topology. We define a function H : S n 1 × LC * −→ R by H(x, v) = x, v , and denote h v (x) = H(x, v). Then h v is a submersion for any v ∈ LC * . For spacelike submanifolds X ∈ Sp-Emb(U, S n 1 ), we have H = H • (X × id LC * ). We also have the -jet extension j 1 H : U × S n 1 −→ J (U, R) defined by j 1 H(x, v) = j h v (u). We consider the trivialization J (U, R) ≡ U × R × J (n − r, 1). For any submanifold Q ⊂ J (n − r, 1), we denote Q = U × {0} × Q. Then we have the following proposition as a corollary of Lemma 6 of Wassermann [11]. Proposition 7.1. ( [11]) Let Q be a submanifold of J (n − 1, 1). Then the set T Q = {x ∈ Sp-Emb(U, S n 1 ) | j 1 H is transversal to Q} is a residual subset of Sp-Emb(U, S n 1 ). If Q is a closed subset, then T Q is open. We remark that if the corresponding horospherical height function h v 0 is -determined relative to K, then H is a K-versal deformation if and only if j 1 H is transversal to K h,v 0 , where K h,v 0 is the K-orbit through j h v 0 (0) ∈ J (n − r, 1). Applying Theorem 6.3, this condition is equivalent to the condition that the corresponding Legendrian immersion germ is Legendrian stable. From the previous arguments and §5 in [6], we have the following proposition. (See also [1].) Theorem 7.2. If n ≤ 6, there exists an open subset O ⊂ Sp-Emb(U, S n 1 ) such that for any X ∈ O, the corresponding Legendrian immersion germ L is Legendrian stable.