PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 63 (77), 1998, 81–101

SINGULAR POINTS OF LIGHTLIKE HYPERSURFACES OF THE DE SITTER SPACE

M.A. Akivis and V.V. Goldberg

Communicated by Mileva Prvanović

Abstract. We study singular points of lightlike hypersurfaces of the de Sitter space S_{1}^{n+1} and the geometry of such hypersurfaces and use them for construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces.

Introduction. It is well-known that the pseudo-Riemannian manifolds (M, g) of Lorentzian signature play a special role in geometry and physics, and that they are models of spacetime of general relativity. At the tangent space T_{x} of an arbitrary point x of such a manifold, one can invariantly define a real isotropic cone C_{x}. From the point of view of physics, this cone is the light cone: trajectories of light impulses emanating from the point x are tangent to this cone.

Hypersurfaces of a Lorentzian manifold (M, g) can be of three types: spacelike, timelike, and lightlike (see, for example, [14] or [4]). For definiteness, we will assume that dim M = n + 1 and sign g = (n, 1).

The tangent hyperplane to a spacelike hypersurface U^{n} at any point does not have real common points with the light cone C_{x}. This implies that on U^{n} a proper Riemannian metric is induced. The tangent hyperplane to a timelike hypersurface U^{n} at any point intersects the light cone C_{x} along an (n−1)-dimensional cone. This implies that on U^{n} a pseudo-Riemannian metric of Lorentzian signature (n−1, 1) is induced. Finally, the tangent hyperplane to a lightlike hypersurface U^{n} at any point is tangent to the light cones C_{x}. This implies that on U^{n} a degenerate Riemannian metric signature (n − 1, 0) is induced.

On spacelike and timelike hypersurfaces of a manifold of Lorentzian signature, an invariant normalization and an affine Levi-Civita connection are induced by a first-order neighborhood while on lightlike hypersurfaces one should use differential neighborhoods of higher order to construct an invariant normalization and an affine connection,

AMS Subject Classification (1991): Primary 53B20 This research was partially supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach).

From the point of view of physics lightlike hypersurfaces are of great importance since they are models of different types of horizons studied in general relativity: event horizons, Cauchy’s horizons, Kruskal’s horizons (see [8] and [13]). This is the reason that the study of geometric structure of lightlike hypersurfaces is of interest.

In the current paper we consider lightlike hypersurfaces in the de Sitter space (a pseudo-Riemannian space of Lorentzian signature and constant positive curvature), study their geometric structure, and prove that there are singular points and singular submanifolds on them.

The de Sitter space S_{1}^{n+1} admits a realization on the exterior of an ndimensional oval hyperquadric Q^{n} of a projective space P^{n+1}. Thus the de Sitter space is isometric to a pseudoelliptic space, S_{1}^{n+1} ∼ ext Q^{n}. Since the interior of the hyperquadric Q^{n} is isometric to the hyperbolic geometry of the Lobachevsky space H^{n+1}, H^{n+1} ∼ int Q^{n} and the geometry of Q^{n} itself is equivalent to that of an n-dimensional conformal space C^{n}, C^{n} ∼ Q^{n}, the groups of motions of these three spaces are isomorphic to each other and are isomorphic to the group SO(n + 2, 1) of rotations of a pseudo-Euclidean space R_{1}^{n+2} of Lorentzian signature. This allows us to apply the apparatus developed in the book [4] for the conformal space C^{n} to the study of the de Sitter space.

Note also that the geometry of lightlike hypersurfaces on pseudo-Riemannian manifolds of different signatures was the subject of many journal papers and even two books [9] and [12]. However, the geometry of lightlike hypersurfaces in the de Sitter space was not studied in spite of the fact that this geometry has many interesting geometric features.

In the present paper we study the geometry of the de Sitter space S_{1}^{n+1} using its connection with the geometry of the conformal space. We prove that the geometry of lightlike hypersurfaces of the space S_{1}^{n+1} is directly connected with the geometry of hypersurfaces of the conformal space C^{n}. The latter was studied in detail in the papers of the first author (see [1], [2]) and also in the book [4]. This simplifies the study of lightlike hypersurfaces of the de Sitter space S_{1}^{n+1} and makes possible to apply for their consideration the apparatus constructed in the conformal theory.

In Section 1 we study the geometry of the de Sitter space and its connection with the geometry of the conformal space. Next we study lightlike hypersurfaces U^{n} in the space S_{1}^{n+1}, investigate their structure, and prove that such a hypersurface is tangentially degenerate of rank r ≤ n−1. Its rectilinear or plane generators form an isotropic fibre bundle on U^{n}.

In Sections 2–5 we investigate lightlike hypersurfaces U^{n} of maximal rank, and for their study we use the relationship between the geometry of such hypersurfaces and the geometry of hypersurfaces of the conformal space. For a lightlike hypersurface, we construct the fundamental quadratic forms and connections determined by a normalization of a hypersurface by means of a distribution (the screen distribution) which is complementary to the isotropic distribution. The screen distribution plays an important role in the book [9] since it defines a connection on a lightlike hypersurface U^{n}, and it appears to be important for applications. We prove that the screen distribution on a lightlike hypersurface can be constructed invariantly by means of quantities from a third-order differential neighborhood, that is, such a distribution is intrinsically connected with the geometry of a hypersurface.

In Section 5 we study singular points of a lightlike hypersurface in the de Sitter space S_{1}^{n+1}, classify them, and describe the structure of hypersurfaces carrying singular points of different types. Moreover, we establish the connection of this classification with that of canal hypersurfaces of the conformal space.

The principal method of our investigation is the method of moving frames and exterior differential forms in the form in which it is presented in the books [3] and [4]. All functions considered in the paper are assumed to be real and differentiable, and all manifolds are assumed to be smooth with the possible exception of some isolated singular points and singular submanifolds.

1. The de Sitter space. 1. In a projective space P^{n+1} of dimension n + 1 we consider an oval hyperquadric Q^{n}. Let x be a point of the space P^{n+1} with projective coordinates (x^{0}, x^{1},…, x^{n+1}). The hyperquadric Q^{n} is determined by the equations

(x, x) := g_{ξη}x^{ξ}x^{η} = 0, ξ, η = 0,…, n + 1,  (1)

whose left-hand side is a quadratic form (x, x) of signature (n + 1, 1). The hyperquadric Q^{n} divides the space P^{n+1} into two parts, external and internal. Normalize the quadratic form (x, x) in such a way that for the points of the external part the inequality (x, x) > 0 holds. This external domain is a model of the de Sitter space S_{1}^{n+1} (see [15]). We will identify the external domain of Q^{n} with the space S_{1}^{n+1}. The hyperquadric Q^{n} is the absolute of the space S_{1}^{n+1}.

On the hyperquadric Q^{n} of the space P^{n+1} the geometry of a conformal space C^{n} is realized. The bijective mapping C^{n} ↔ Q^{n} is called the Darboux mapping, and the hyperquadric Q^{n} itself is called the Darboux hyperquadric.

Under the Darboux mapping to hyperspheres of the space C^{n} there correspond cross-sections of the hyperquadric Q^{n} by hyperplanes ξ. But to a hyperplane ξ there corresponds a point x that is polar-conjugate to ξ with respect to Q^{n} and lies outside of Q^{n}, that is, a point of the space S_{1}^{n+1}. Thus to hyperspheres of the space C^{n} there correspond points of the space S_{1}^{n+1}.

Let x be an arbitrary point of the space S_{1}^{n+1}. The tangent lines from the point x to the hyperquadric Q^{n} form a second-order cone C_{x} with vertex at the point x. This cone is called the isotropic cone. For spacetime whose model is the space S_{1}^{n+1} this cone is the light cone, and its generators are lines of propagation of light impulses whose source coincides with the point x.

The cone C_{x} separates all straight lines passing through the point x into spacelike (not having common points with the hyperquadric Q^{n}), timelike (intersecting Q^{n} in two different points), and lightlike (tangent to Q^{n}). The lightlike straight lines are generators of the cone C_{x}.

To a spacelike straight line l ⊂ S_{1}^{n+1} there corresponds an elliptic pencil of hyperspheres in the conformal space C^{n}. All hyperspheres of this pencil pass through a common (n − 2)-sphere S^{n−2} (the center of this pencil). The sphere S^{n−2} is the intersection of the hyperquadric Q^{n} and the (n − 1)-dimensional subspace of the space P^{n+1} which is polar-conjugate to the line l with respect to the hyperquadric Q^{n}.

To a timelike straight line l ⊂ S_{1}^{n+1} there corresponds a hyperbolic pencil of hyperspheres in the space C^{n}. Two arbitrary hyperspheres of this pencil do not have common points, and the pencil contains two hyperspheres of zero radius which correspond to the points of intersection of the straight line l and the hyperquadric Q^{n}.

Finally, to a lightlike straight line l ⊂ S_{1}^{n+1} there corresponds a parabolic pencil of hyperspheres in the space C^{n} consisting of hyperspheres tangent one to another at a point that is a unique hypersphere of zero radius belonging to this pencil.

Hyperplanes of the space S_{1}^{n+1} are also divided into three types. Spacelike hyperplanes do not have common points with the hyperquadric Q^{n}; a timelike hyperplane intersects Q^{n} along a real hypersphere; and lightlike hyperplanes are tangent to Q^{n}. Subspaces of any dimension r, 2 ≤ r ≤ n − 1, can be also classified in a similar manner.

Let us apply the method of moving frames to study some questions of differential geometry of the space S_{1}^{n+1}. With a point x ∈ S_{1}^{n+1} we associate a family of projective frames {A_{0}, A_{1},…, A_{n+1}}. However, in order to apply formulas derived in the book [4], we will use the notations used in this book. Namely, we denote by A_{n} the vertex of the moving frame which coincides with the point x, A_{n} = x; we locate the vertices A_{0}, A_{i}, and A_{n+1} at the hyperplane ξ which is polar conjugate to the point x with respect to the hyperquadric Q^{n}, and we assume that the points A_{0} and A_{n+1} lie on the hypersphere S^{n−1} = Q^{n} ∩ ξ, and the points A_{i} are polar-conjugate to the straight line A_{0}A_{n+1} with respect to S^{n−1}. Since (x, x) > 0, we can normalize the point A_{n} by the condition (A_{n}, A_{n}) = 1. The points A_{0} and A_{n+1} are not polar-conjugate with respect to the hyperquadric Q^{n}. Hence we can normalize them by the condition (A_{0}, A_{n+1}) = −1. As a result, the matrix of scalar products of the frame elements has the form

(A_{ξ}, A_{η}) = ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & g_{ij} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ) , i, j = 1,…, n − 1,  (2)

and the quadratic form (x, x) takes the form

(x, x) = g_{ij}x^{i}x^{j} + (x^{n})^{2} − 2x^{0}x^{n+1}.  (3)

The quadratic form g_{ij}x^{i}x^{j} occurring in (3) is positive definite.

The equations of infinitesimal displacement of the conformal frame {A_{ξ}}, ξ = 0, 1,…, n + 1, we have constructed have the form

dA_{ξ} = ω_{ξ}^{η}A_{η}, ξ, η = 0, 1,…, n + 1,  (4)

where by (2), the 1-forms ω_{ξ}^{η} satisfy the following Pfaffian equations:

\begin{matrix}ω_{0}^{n+1} = ω_{n+1}^{0} = 0, & ω_{0}^{0} + ω_{n+1}^{n+1} = 0, \\ ω_{i}^{n+1} = g_{ij}ω_{0}^{j}, & ω_{i}^{0} = g_{ij}ω_{n+1}^{j}, \\ ω_{n}^{n+1} − ω_{0}^{n} = 0, & ω_{n}^{0} − ω_{n+1}^{n} = 0, \\ g_{ij}ω_{n}^{j} + ω_{i}^{n} = 0, & ω_{n}^{n} = 0, \\ dg_{ij} = g_{jk}ω_{i}^{k} + g_{ik}ω_{j}^{k}. \end{matrix}  (5)

These formulas are precisely the formulas derived in the book [4] (see p. 32) for the conformal space C^{n}.

It follows from (4) that

dA_{n} = ω_{n}^{0}A_{0} + ω_{n}^{i}A_{i} + ω_{n}^{n+1}A_{n+1}.  (6)

The differential dA_{n} belong to the tangent space T_{x}(S_{1}^{n+1}), and the 1-forms ω_{n}^{0}, ω_{n}^{i}, and ω_{n}^{n+1} form a coframe of this space. The total number of these forms is n + 1, and this number coincides with the dimension of T_{x}(S_{1}^{n+1}). The scalar square of the differential dA_{n} is the metric quadratic form g̃ on the manifold S_{1}^{n+1}. By (2), this quadratic form g̃ can be written as

g̃ = (dA_{n}, dA_{n}) = g_{ij}ω_{n}^{i}ω_{n}^{j} − 2ω_{n}^{0}ω_{n}^{n+1}.

Since the first term of this expression is a positive definite quadratic form, the form g̃ is of Lorentzian signature (n, 1). The coefficients of the form g̃ produce the metric tensor of the space S_{1}^{n+1} whose matrix is obtained from the matrix (2) by deleting the nth row and the nth column.

The quadratic form g̃ defines on S_{1}^{n+1} a pseudo-Riemannian metric of signature (n, 1). The isotropic cone defined in the space T_{x}(S_{1}^{n+1}) by the equation g̃ = 0 coincides with the cone C_{x} that we defined earlier in the space S_{1}^{n+1} geometrically.

The 1-forms ω_{ξ}^{η} occurring in equations (4) satisfy the structure equations of the space C^{n}:

dω_{ξ}^{η} = ω_{ξ}^{ζ} ∧ ω_{ζ}^{η},  (7)

which are obtained by taking exterior derivatives of equations (4) and which are conditions of complete integrability of (4). The forms ω_{ξ}^{η} are invariant forms of the fundamental group PO(n + 2, 1) of transformations of the spaces H^{n+1}, C^{n}, and S_{1}^{n+1} which is locally isomorphic to the group SO(n + 2, 1).

2. Lightlike hypersurfaces in the de Sitter space. A hypersurface U^{n} in the de Sitter space S_{1}^{n+1} is said to be lightlike if all its tangent hyperplanes are lightlike, that is, they are tangent to the hyperquadric Q^{n} which is the absolute of the space S_{1}^{n+1}.

Denote by x an arbitrary point of the hypersurface U^{n}, by η the tangent hyperplane to U^{n} at the point x, η = T_{x}(U^{n}), and by y the point of tangency of the hyperplane η with the hyperquadric Q^{n}. Next, as in Section 1, denote by ξ the hyperplane which is polar-conjugate to the point x with respect to the hyperquadric Q^{n}, and associate with a point x a family of projective frames such that x = A_{n}, y = A_{0}, the points A_{i}, i = 1,…, n − 1, belong to the intersection of the hyperplanes ξ and η, A_{i} ∈ ξ ∩ η, and the point A_{n+1}, as well as the point A_{0}, is the intersection point of the quadric ξ ∩ Q^{n} and the straight line that is polar-conjugate to the (n − 2)-dimensional subspace spanned by the points A_{i}. In addition, we normalize the frame vertices in the same way as this was done in Section 1. Then the matrix of scalar products of the frame elements has the form (2), and the components of infinitesimal displacements of the moving frame satisfy the Pfaffian equations (5).

Since the hyperplane η is tangent to the hypersurface U^{n} at the point x = A_{n} and does not contain the point A_{n+1}, the differential of the point x = A_{n} has the form

dA_{n} = ω_{n}^{0}A_{0} + ω_{n}^{i}A_{i},  (8)

the following equation holds:

ω_{n}^{n+1} = 0,  (9)

and the forms ω_{n}^{0} and ω_{n}^{i} are basis forms of the hypersurface U^{n}. By relations (5), it follows from equation (8) that

ω_{0}^{n} = 0  (10)

and

dA_{0} = ω_{0}^{0}A_{0} + ω_{0}^{i}A_{i}.  (11)

Taking exterior derivative of equation (9), we obtain

ω_{n}^{i} ∧ ω_{i}^{n+1} = 0.  (12)

Since the forms ω_{n}^{i} are linearly independent, by Cartan’s lemma, we find from (12) that

ω_{i}^{n+1} = ν_{ij}ω_{n}^{j}, ν_{ij} = ν_{ji}.  (13)

Applying an appropriate formula from (5), we find that

ω_{0}^{i} = g^{ij}ω_{j}^{n+1} = g^{ik}ν_{kj}ω_{n}^{j},  (14)

where (g^{ij}) is the inverse matrix of the matrix (g_{ij}).

Now formulas (8) and (11) imply that for ω_{n}^{i} = 0, the point A_{n} of the hypersurface U^{n} moves along the isotropic straight line A_{n}A_{0}, and hence U^{n} is a ruled hypersurface. In what follows, we assume that the entire straight line A_{n}A_{0} belongs to the hypersurface U^{n}. Thus the following theorem holds:

Theorem 1. A lightlike hypersurface U^{n} is the image of the direct product M^{n−1} × l of a differentiable manifold M^{n−1} and a projective line l under the mapping f : M^{n−1} × l → P^{n+1} into a projective space P^{n+1}: U^{n} = f(M^{n−1} × l) which sends the straight line l to the straight line A_{n}A_{0} ∈ P^{n+1}.

Precisely this mapping is the subject of this paper.

In addition, formulas (8) and (11) show that at any point of a generator of the hypersurface U^{n}, its tangent hyperplane is fixed and coincides with the hyperplane η. Thus U^{n} is a tangentially degenerate hypersurface.

We recall that the rank of a tangentially degenerate hypersurface is the number of parameters on which the family of its tangent hyperplanes depends (see, for example, [3, p. 113]). From relations (8) and (11) it follows that the tangent hyperplane η of the hypersurface U^{n} along its generator A_{n}A_{0} is determined by this generator and the points A_{i}, η = A_{n} ∧ A_{0} ∧ A_{1} ∧… ∧ A_{n−1}. The displacement of this hyperplane is determined by the differentials (8), (11), and

dA_{i} = ω_{i}^{0}A_{0} + ω_{i}^{j}A_{j} + ω_{i}^{n}A_{n} + ω_{i}^{n+1}A_{n+1}.

But by (5), ω_{i}^{n} = −g_{ij}ω_{n}^{j}, and the forms ω_{i}^{n+1} are expressed according to formulas (13). ¿From formulas (13) and (14) it follows that the rank of a tangentially degenerate hypersurface U^{n} is determined by the rank of the matrix (ν_{ij}) in terms of which the 1-forms ω_{i}^{n+1} and ω_{0}^{i} are expressed. But by (11) and (14) the dimension of the submanifold V described by the point A_{0} on the hyperquadric Q^{n} is also equal to the rank of the matrix (ν_{ij}). Thus we have proved the following result:

Theorem 2. A lightlike hypersurface of the de Sitter space S_{1}^{n+1} is a ruled tangentially degenerate hypersurface whose rank is equal to the dimension of the submanifold V described by the point A_{0} on the hyperquadric Q^{n}.

Denote the rank of the tensor ν_{ij} and of the hypersurface U^{n} by r. In this and next sections we will assume that r = n−1. The case r < n−1 was considered by the authors in [6].

For r = n − 1, the hypersurface U^{n} carries an (n − 1)-parameter family of isotropic rectilinear generators l = A_{n}A_{0} along which the tangent hyperplane T_{x}(U^{n}) is fixed. From the point of view of physics, the isotropic rectilinear generators of a lightlike hypersurface U^{n} are trajectories of light impulses, and the hypersurface U^{n} itself represents a light flux in spacetime.

Since rank(ν_{ij}) = n − 1, the submanifold V described by the point A_{0} on the hyperquadric Q^{n} has dimension n − 1, that is, V is a hypersurface. We denote it by V^{n−1}. The tangent subspace T_{A_{0}}(V^{n−1}) to V^{n−1} is determined by the points A_{0}, A_{1},…, A_{n−1}. Since (A_{n}, A_{i}) = 0, this tangent subspace is polar-conjugate to the rectilinear generator A_{0}A_{n} of the lightlike hypersurface U^{n}.

The submanifold V^{n−1} of the hyperquadric Q^{n} is the image of a hypersurface of the conformal space C^{n} under the Darboux mapping. We will denote this hypersurface also by V^{n−1}. In the space C^{n}, the hypersurface V^{n−1} is defined by equation (10) which by (5) is equivalent to equation (9) defining a lightlike hypersurface U^{n} in the space S_{1}^{n+1}. To the rectilinear generator A_{n}A_{0} of the hypersurface U^{n} there corresponds a parabolic pencil of hyperspheres A_{n} + sA_{0} tangent to the hypersurface V^{n−1} (see [4, p. 40]). Thus the following theorem is valid:

Theorem 3. There exists a one-to-one correspondence between the set of hypersurfaces of the conformal space C^{n} and the set of lightlike hypersurfaces of the maximal rank r = n − 1 of the de Sitter space S_{1}^{n+1}. To pencils of tangent hyperspheres of the hypersurface V^{n−1} there correspond isotropic rectilinear generators of the lightlike hypersurface U^{n}.

Note that for lightlike hypersurfaces of the four-dimensional Minkowski space M^{4} the result similar to the result of Theorem 2 was obtained in [11].

3. The fundamental forms and connections on a lightlike hypersurface of the de Sitter space. The first fundamental form of a lightlike hypersurface U^{n} of the space S_{1}^{n+1} is a metric quadratic form. It is defined by the scalar square of the differential dx of a point of this hypersurface. Since we have x = A_{n}, by (8) and (2) this scalar square has the form

(dA_{n}, dA_{n}) = g_{ij}ω_{n}^{i}ω_{n}^{j} = g  (15)

and is a positive semidefinite differential quadratic form of signature (n − 1, 0). It follows that the system of equations ω_{n}^{i} = 0 defines on the hypersurface U^{n} a fibration of isotropic lines which, as we showed in Section 2, coincide with rectilinear generators of this hypersurface.

The second fundamental form of a lightlike hypersurface U^{n} determines its deviation from the tangent hyperplane η. To find this quadratic form, we compute the part of the second differential of the point A_{n} which does not belong to the tangent hyperplane η = A_{0} ∧ A_{1} ∧… ∧ A_{n}:

d^{2}A_{n} ≡ ω_{n}^{i}ω_{i}^{n+1}A_{n+1} (mod η).

This implies that the second fundamental form can be written as

b = ω_{n}^{i}ω_{i}^{n+1} = ν_{ij}ω_{n}^{i}ω_{n}^{j},  (16)

where we used expression (13) for the form ω_{i}^{n+1}. Since we assumed that rank(ν_{ij}) = n − 1, the rank of the quadratic form (16) as well as the rank of the form g is equal to n − 1. The nullspace of this quadratic form (see [14, p. 53]) is again determined by the system of equations ω_{n}^{i} = 0 and coincides with the isotropic direction on the hypersurface U^{n}. The reduction of the rank of the quadratic form b is connected with the tangential degeneracy of the hypersurface U^{n}. The latter was noted in Theorem 2.

On a hypersurface V^{n−1} of the conformal space C^{n} that corresponds to a lightlike hypersurface U^{n} ⊂ S_{1}^{n+1}, the quadratic forms (15) and (16) define the net of curvature lines, that is, an orthogonal and conjugate net.

To find the connection forms of the hypersurface U^{n}, we find exterior derivatives of its basis forms ω_{n}^{0} and ω_{n}^{i}:

\begin{aligned}dω_{n}^{0} & = ω_{n}^{0} ∧ ω_{0}^{0} + ω_{n}^{i} ∧ ω_{i}^{0}, \\ dω_{n}^{i} & = ω_{n}^{0} ∧ ω_{0}^{i} + ω_{n}^{j} ∧ ω_{j}^{i}. \end{aligned}  (17)

This implies that the matrix 1-form

ω = ( \begin{matrix}ω_{0}^{0} & ω_{i}^{0} \\ ω_{0}^{i} & ω_{j}^{i} \end{matrix} )  (18)

defines a torsion-free connection on the hypersurface U^{n}. To clarify the properties of this connection, we find its curvature forms. Taking exterior derivatives of the forms (18) and applying equations (5), (7), (9), and (10), we obtain

\begin{matrix}Ω_{0}^{0} = dω_{0}^{0} − ω_{0}^{i} ∧ ω_{i}^{0} = 0, \\ Ω_{0}^{i} = dω_{0}^{i} − ω_{0}^{0} ∧ ω_{0}^{i} − ω_{0}^{j} ∧ ω_{j}^{i} = 0, \\ Ω_{i}^{0} = dω_{i}^{0} − ω_{i}^{0} ∧ ω_{0}^{0} − ω_{i}^{j} ∧ ω_{j}^{0} = −g_{ij}ω_{n}^{j} ∧ ω_{n}^{0}, \\ Ω_{j}^{i} = dω_{j}^{i} − ω_{j}^{0} ∧ ω_{0}^{i} − ω_{j}^{k} ∧ ω_{k}^{i} − ω_{j}^{n+1} ∧ ω_{n+1}^{i} = −g_{jk}ω_{n}^{k} ∧ ω_{n}^{i}. \end{matrix}  (19)

In these formulas the forms ω_{j}^{n+1} and ω_{0}^{i} are expressed in terms of the basis forms ω_{n}^{i}, and the forms ω_{0}^{j}, ω_{j}^{i}, and ω_{i}^{0} are fiber forms. If the principal parameters are fixed, then these fiber forms are invariant forms of the group G of admissible transformations of frames associated with a point x = A_{n} of the hypersurface U^{n}, and the connection defined by the form (18) is a G-connection.

To assign an affine connection on the hypersurface U^{n}, it is necessary to make a reduction of the family of frames in such a way that the forms ω_{i}^{0} become principal. Denote by δ the symbol of differentiation with respect to the fiber parameters, that is, for a fixed point x = A_{n} of the hypersurface U^{n}, and by π_{η}^{ξ} the values of the 1-forms ω_{η}^{ξ} for a fixed point x = A_{n}, that is, π_{η}^{ξ} = ω_{η}^{ξ}(δ). Then we obtain

π_{n}^{0} = 0, π_{n}^{i} = 0, π_{i}^{n} = 0, π_{i}^{n+1} = 0.

It follows

δA_{i} = π_{i}^{0}A_{0} + π_{i}^{j}A_{j}.  (20)

The points A_{0} and A_{i} determine the tangent subspace to the submanifold V^{n−1} described by the point A_{0} on the hyperquadric Q^{n}. If we fix an (n − 2)- dimensional subspace ζ not containing the point A_{0} in this tangent subspace and place the points A_{i} into ζ, then we obtain π_{i}^{0} = 0. This means that the forms ω_{i}^{0} become principal, that is,

ω_{i}^{0} = μ_{ij}ω_{n}^{j} + μ_{i}ω_{n}^{0},  (21)

and as a result, an affine connection arises on the hypersurface U^{n}.

We will call the subspace ζ ⊂ T_{A_{0}}(V^{n−1}) the normalizing subspace of the lightlike hypersurface U^{n}. We have proved the following result:

Theorem 4. If in every tangent subspace T_{A_{0}}(V^{n−1}) of the submanifold V^{n−1} associated with a lightlike hypersurface U^{n}, V^{n−1} ⊂ Q^{n}, a normalizing (n−2)- dimensional subspace ζ not containing the point A_{0} is assigned, then there arises a torsion-free affine connection on U^{n}.

The last statement follows the first two equations of (19).

The constructed above fibration of normalizing subspaces ζ defines a distribution Δ of (n − 1)-dimensional elements on a lightlike hypersurface U^{n}. In fact, the point x = A_{n} of the hypersurface U^{n} along with the subspace ζ = A_{1} ∧… ∧ A_{n−1} define the (n − 1)-dimensional subspace which is complementary to the straight line A_{n}A_{0} and lies in the tangent subspace η of the hypersurface U^{n}. Following the book [9], we will call this subspace the screen, and the distribution Δ the screen distribution. Since at the point x the screen is determined by the subspace A_{n}A_{1} … A_{n−1}, the differential equations of the screen distribution has the form

ω_{n}^{0} = 0.  (22)

But by (21)

dω_{n}^{0} = ω_{n}^{i} ∧ (μ_{ij}ω_{n}^{j} + μ_{i}ω_{n}^{0}).

Hence the screen distribution is integrable if and only if the tensor μ_{ij} is symmetric. Thus we arrived at the following result:

Theorem 5. The fibration of normalizing subspaces ζ defines a screen distribution Δ of (n − 1)-dimensional elements on a lightlike hypersurface U^{n}. This distribution is integrable if and only if the tensor μ_{ij} defined by equation (21) is symmetric.

Note that the configurations similar to that described in Theorem 5 occurred in the works of the Moscow geometers published in the 1950s. They were called the one-side stratifiable pairs of ruled surfaces (see [10, §30] or [3, p. 187]).

4. An invariant normalization of lightlike hypersurfaces of the de Sitter space. In [1] (see also [4, Ch. 2]) an invariant normalization of a hypersurfaces V^{n−1} of the conformal space C^{n} was constructed. By Theorem 3, this normalization can be interpreted in terms of the geometry of the de Sitter space S_{1}^{n+1}.

Taking exterior derivative of equations (10) defining the hypersurface V^{n−1} in the conformal space C^{n}, we obtain

ω_{i}^{n} ∧ ω_{0}^{i} = 0,

from which by linear independence of the 1-forms ω_{0}^{i} on V^{n−1} and Cartan’s lemma we find that

ω_{i}^{n} = λ_{ij}ω_{0}^{j}, λ_{ij} = λ_{ji}.  (23)

Here and in what follows we retain the notations used in the study of the geometry of hypersurfaces of the conformal space C^{n} in the book [4].

It is not difficult to find relations between the coefficients ν_{ij} in formulas (13) and λ_{ij} in formulas (23). Substituting the values of the forms ω_{i}^{n} and ω_{0}^{j} from (5) into (23), we find that

−g_{ij}ω_{n}^{j} = λ_{ij}g^{jk}ω_{k}^{n+1}.

Solving these equations for ω_{k}^{n+1}, we obtain

ω_{i}^{n+1} = −g_{ik}λ̃^{kl}g_{lj}ω_{n}^{j},

where (λ̃^{kl}) is the inverse matrix of the matrix (λ_{ij}). Comparing these equations with equations (13), we obtain

ν_{ij} = −g_{ik}λ̃^{kl}g_{lj}.  (24)

Of course, in this computation we assumed that the matrix (λ_{ij}) is nondegenerate.

Let us clarify the geometric meaning of the vanishing of det(λ_{ij}). To this end, we make an admissible transformation of the moving frame associated with a point of a lightlike hypersurface U^{n} by setting

Â_{n} = A_{n} + sA_{0}.  (25)

The point Â_{n} as the point A_{n} lies on the rectilinear generator A_{n}A_{0}. Differentiating this point and applying formulas (8) and (11), we obtain

dÂ_{n} = (ds + sω_{0}^{0} + ω_{n}^{0})A_{0} + (ω_{n}^{i} + sω_{0}^{i})A_{i}.  (26)

It follows that in the new frame the forms ω_{n}^{i} become

ω_{n}^{i} = ω_{n}^{i} + sω_{0}^{i}.

By (5) and (23), it follows that

ω_{n}^{i} = −g^{ik}(λ_{kj} − sg_{kj})ω_{0}^{j}.

This implies that in the new frame the quantities λ_{ij} become

λ̂_{ij} = λ_{ij} − sg_{ij}.  (27)

Consider also the matrix (λ̂_{j}^{i}) = (g^{ik}λ̂_{kj}). Since g_{ij} is a nondegenerate tensor, the matrices (λ̂_{j}^{i}) and (λ̂_{ij}) have the same rank ρ ≤ n − 1.

¿From equation (26) it follows that

dÂ_{n} = (ds + sω_{0}^{0} + ω_{n}^{0})A_{0} − λ̂_{j}^{i}A_{i}ω_{0}^{j}.

The differential dÂ_{n} is the differential of the mapping f : M^{n−1} × l → P^{n+1} which was considered in Theorem 1. The linearly independent forms ω_{0}^{i} are basis forms on the manifold M^{n−1}, and the form ω_{n}^{0} = ds+sω_{0}^{0}+ω_{n}^{0} containing a nonhomogeneous parameter s of the projective line l is a basis form on this line. Thus the matrix

( \begin{matrix}1 & 0 \\ 0 & λ̂_{j}^{i} \end{matrix} )  (28)

is the Jacobi matrix of this mapping. Hence the tangent subspace to the hypersurface U^{n} at the point Â_{n} is determined by the points Â_{n}, A_{0}, and λ̂_{j}^{i}A_{i}. At the points, at which the rank ρ of the matrix (λ̂_{j}^{i}) is equal to n − 1, ρ = n − 1, the tangent subspace to the hypersurface U^{n} has dimension n, and such points are regular points of the hypersurface. The points, at which the rank ρ of the matrix (λ̂_{j}^{i}) is reduced, are singular points of the hypersurface U^{n}. The coordinates of singular points are defined by the condition det(λ̂_{j}^{i}) = 0 which by (27) is equivalent to the equation

det(λ_{ij} − sg_{ij}) = 0,  (29)

the characteristic equation of the matrix (λ_{ij}) with respect to the tensor g_{ij}. The degree of this equation is equal to n − 1.

In particular, if A_{n} is a regular point of the hypersurface U^{n}, then the matrix (λ_{ij}) is nondegenerate, and equation (24) holds. On the other hand, if A_{n} is a singular point of U^{n}, then equation (24) is meaningless.

Since the matrix (λ_{ij}) is symmetric and the matrix (g_{ij}) defines a positive definite form of rank n−1, equation (29) has n−1 real roots if each root is counted as many times as its multiplicity. Thus on a rectilinear generator A_{n}A_{0} of a lightlike hypersurface U^{n} there are n − 1 real singular points.

By Vieta’s theorem, the sum of the roots of equation (29) is equal to the coefficient in s^{n−2}, and this coefficient is λ_{ij}g^{ij}. Consider the quantity

λ = \frac{1}{n − 1}λ_{ij}g^{ij},  (30)

which is the arithmetic mean of the roots of equation (29). This quantity λ allows us to construct new quantities

a_{ij} = λ_{ij} − λg_{ij}.  (31)

It is easy to check that the quantities a_{ij} do not depend on the location of the point A_{n} on the straight line A_{n}A_{0}, that is, a_{ij} is invariant with respect to the transformation of the moving frame defined by equation (25). Thus the quantities a_{ij} form a tensor on the hypersurface U^{n} defined in its second-order neighborhood. This tensor satisfies the condition

a_{ij}g^{ij} = 0,  (32)

that is, it is apolar to the tensor g_{ij}.

On the straight line A_{n}A_{0} we consider a point

C = A_{n} + λA_{0}.  (33)

It is not difficult to check that this point remains also fixed when the point A_{n} moves along the straight line A_{n}A_{0}. Since λ is the arithmetic mean of the roots of equation (29) defining singular points on the straight line A_{n}A_{0}, the point C is the harmonic pole (see [7]) of the point A_{0} with respect to these singular points. In particular, for n = 3, the point C is the fourth harmonic point to the point A_{0} with respect to two singular points of the rectilinear generator A_{3}A_{0} of the lightlike hypersurface U^{3} of the de Sitter space S_{1}^{4}.

In the conformal theory of hypersurfaces, to the point C there corresponds a hypersphere which is tangent to the hypersurface at the point A_{0}. This hypersphere is called the central tangent hypersphere (see [4, pp. 41–42]). Since

(d^{2}A_{0}, C) = a_{ij}ω_{0}^{i}ω_{0}^{j},  (34)

the cone

a_{ij}ω_{0}^{i}ω_{0}^{j} = 0

with vertex at the point A_{0} belonging to the tangent subspace T_{A_{0}}(V^{n−1}) contains the directions along which the central hypersphere has a second-order tangency with the hypersurface V^{n−1} at the point A_{0}. From the apolarity condition (33) it follows that it is possible to inscribe an orthogonal (n − 1)-hedron with vertex at A_{0} into the cone defined by equation (34).

Now we can construct an invariant normalization of a lightlike hypersurface U^{n} of the de Sitter space S_{1}^{n+1}. To this end, first we repeat some computations from Ch. 2 of [4].

Taking exterior derivatives of equations (23) and applying Cartan’s lemma, we obtain the equations

∇λ_{ij} + λ_{ij}ω_{0}^{0} + g_{ij}ω_{n}^{0} = λ_{ijk}ω_{0}^{k},  (35)

where

∇λ_{ij} = dλ_{ij} − λ_{ik}ω_{j}^{k} − λ_{kj}ω_{i}^{k},

and the quantities λ_{ijk} are symmetric with respect to all three indices. Equations (35) confirm one more time that the quantities λ_{ij} do not form a tensor and depend on a location of the point A_{n} on the straight line A_{n}A_{0}. This dependence is described by a closed form relation (27). From formulas (35) it follows that the quantity λ defined by equations (30) satisfy the differential equation

dλ + λω_{0}^{0} + ω_{n}^{0} = λ_{k}ω_{0}^{k},  (36)

where

λ_{k} = \frac{1}{n − 1}g^{ij}λ_{ijk}

(see formulas (2.1.36) and (2.1.37) in the book [4]). The quantities λ_{k} as well as the quantities λ_{ijk} are determined by a third-order neighborhood of a generator A_{0}A_{n} of a lightlike hypersurface U^{n} ⊂ S_{1}^{n+1}.

The point C lying on the rectilinear generator A_{n}A_{0} of the hypersurface U^{n} describes a submanifold W ⊂ U^{n} when A_{n}A_{0} moves. Let us find the tangent subspace to U^{n} at the point C. Differentiating equation (33) and applying formulas (8) and (11), we obtain

dC = (dλ + λω_{0}^{0} + ω_{n}^{0})A_{0} + (ω_{n}^{i} + λω_{0}^{i})A_{i}.

By (5), (23), (30), and (36), it follows that

dC = (λ_{i}A_{0} − g^{jk}a_{ki}A_{j})ω_{0}^{i}.  (37)

Define the affinor

a_{j}^{i} = g^{ik}a_{kj},  (38)

whose rank coincides with the rank of the tensor a_{ij}. Then equation (37) takes the form

dC = (λ_{i}A_{0} − a_{i}^{j}A_{j})ω_{0}^{i}.

The points

C_{i} = λ_{i}A_{0} − a_{i}^{j}A_{j}  (39)

together with the point C define the tangent subspace to the submanifold W described by the point C on the hypersurface U^{n}.

If the point C is a regular point of the rectilinear generator A_{n}A_{0} of the hypersurface U^{n}, then the rank of the tensor a_{ij} defined by equations (30) as well as the rank of the affinor a_{j}^{i} is equal to n − 1. As a result, the points C_{i} are linearly independent and together with the point C define the (n − 1)-dimensional tangent subspace T_{C}(W), and the submanifold W itself has dimension n−1, dim W = n−1.

The points C_{i} also belong to the tangent subspace T_{A_{0}}(V^{n−1}) and define the (n − 2)-dimensional subspace ζ = T_{A_{0}}(V^{n−1}) ∩ T_{C}(W) in it. This subspace is a normalizing subspace. Since such a normalizing subspace is defined in each tangent subspace T_{A_{0}}(V^{n−1}) of the hypersurface V^{n−1} ⊂ Q^{n}, there arises the fibration of these subspaces which by Theorem 4 defines an invariant affine connection on the lightlike hypersurface U^{n}. The subspace T_{C}(W) is determined by a third-order neighborhood of a generator A_{0}A_{n} of the hypersurface U^{n}.

Thus we proved the following result:

Theorem 6. If the tensor a_{ij} defined by formula (40) on a lightlike hypersurface U^{n} ⊂ S_{1}^{n+1} is nondegenerate, then it is possible to construct the invariant normalization of U^{n} by means of the (n − 2)-dimensional subspaces

ζ = C_{1} ∧ C_{2} ∧… ∧ C_{n−1}.

This normalization induces on U^{n} an invariant affine connection intrinsically connected with the geometry of this hypersurface. The normalization as well as the induced affine connection are determined by a third-order neighborhood of a generator A_{0}A_{n} of the hypersurface U^{n}.

Theorem 5 implies that the invariant normalization we have constructed defines on U^{n} an invariant screen distribution Δ which is also intrinsically connected with the geometry of the hypersurface U^{n}; here Δ_{x} = x ∧ ξ, x ∈ A_{n}A_{0}.

Note that for the hypersurface V^{n−1} of the conformal space C^{n} a similar invariant normalization was constructed as far back as 1953 (see [1] and also [4, Ch. 2]). In the present paper we gave a new geometric meaning of this invariant normalization.

5. Singular points of lightlike hypersurfaces of the de Sitter space. As we indicated in Section 4, the points

z = A_{n} + sA_{0}  (40)

of the rectilinear generator A_{n}A_{0} of the lightlike hypersurface U^{n} are singular if their nonhomogeneous coordinate s satisfies the equation

det(λ_{ij} − sg_{ij}) = 0,  (41)

(In these points the Jacobian of the mapping f : M^{n−1} × l → P^{n+1}, which is equal to the determinant of the matrix (28), vanishes.) We will investigate in more detail the structure of a lightlike hypersurface U^{n} in a neighborhood of its singular point.

Equation (41) is the characteristic equation of the matrix (λ_{ij}) with respect to the tensor (g_{ij}). The degree of this equation is n − 1, and since the matrix (λ_{ij}) is symmetric and the matrix (g_{ij}) is also symmetric and positive definite, then according to the well-known result of linear algebra, all roots of this equation are real, and the matrices (λ_{ij}) and (g_{ij}) can be simultaneously reduced to a diagonal form.

Denote the roots of the characteristic equation by s_{h}, h = 1, 2,…, n − 1, and denote the corresponding singular points of the rectilinear generator A_{n}A_{0} by

B_{h} = A_{n} + s_{h}A_{0}.  (42)

These singular points are called foci of the rectilinear generator A_{n}A_{0} of a lightlike hypersurface U^{n}.

It is clear from (42) that the point A_{0} is not a focus of the rectilinear generator A_{n}A_{0}. This is explained by the fact that by our assumption rank(ν_{ij}) = n − 1, and by (14), on the hyperquadric Q^{n} the point A_{0} describes a hypersurface V^{n−1} which is transversal to the straight lines A_{0}A_{n}.

In the conformal theory of hypersurfaces, to the singular points B_{h} there correspond the tangent hyperspheres defining the principal directions at a point A_{0} of the hypersurface V^{n−1} of the conformal space C^{n} (see [4, p. 56]).

We will construct a classification of singular points of a lightlike hypersurface U^{n} of the space S_{1}^{n+1}. We will use some computations that we made while constructing a classification of canal hypersurfaces in [5].

Suppose first that B_{1} = A_{n} + s_{1}A_{0} be a singular point defined by a simple root s_{1} of characteristic equation (41), s_{1} ≠ s_{h}, h = 2,…, n − 1. For this singular point we have

dB_{1} = (ds_{1} + s_{1}ω_{0}^{0} + ω_{n}^{0})A_{0} − λ̂_{j}^{i}ω_{0}^{j}A_{i},  (43)

where

λ̂_{j}^{i} = g^{ik}(λ_{kj} − s_{1}g_{kj})  (44)

is a degenerate symmetric affinor having a single null eigenvalue. The matrix of this affinor can be reduced to a quasidiagonal form

(λ̂_{j}^{i}) = ( \begin{matrix}0 & 0 \\ 0 & λ̂_{q}^{p} \end{matrix} ) ,  (45)

where p, q = 2,…, n − 1, and (λ̂_{q}^{p}) is a nondegenerate symmetric affinor. The matrices (g_{ij}) and (λ_{ij} − s_{1}g_{ij}) are reduced to the forms

( \begin{matrix}1 & 0 \\ 0 & g_{pq} \end{matrix} ) and ( \begin{matrix}0 & 0 \\ 0 & λ̂_{pq} \end{matrix} ) ,

where (λ̂_{pq}) = (λ_{pq} − s_{1}g_{pq}) is a nondegenerate symmetric matrix.

Since the point B_{1} is defined invariantly on the generator A_{n}A_{0}, then it will be fixed if ω_{0}^{i} = 0. Thus it follows from (43) that

ds_{1} + s_{1}ω_{0}^{0} + ω_{n}^{0} = s_{1i}ω^{i};  (46)

here and in what follows ω^{i} = ω_{0}^{i}. By (45) and (46) relation (43) takes the form

dB_{1} = s_{11}ω^{1}A_{0} + (s_{1p}A_{0} − λ̂_{p}^{q}A_{q})ω^{p}.  (47)

Here the points C_{p} = s_{1p}A_{0} − λ̂_{p}^{q}A_{q} are linearly independent and belong to the tangent subspace T_{A_{0}}(V^{n−1}).

Consider the submanifold F_{1} described by the singular point B_{1} in the space S_{1}^{n+1}. This submanifold is called the focal manifold of the hypersurface U^{n}. Relation (47) shows that two cases are possible:

1) s_{11} ≠ 0. In this case the submanifold F_{1} is of dimension n−1, and its tangent subspace at the point B_{1} is determined by the points B_{1}, A_{0}, and C_{p}. This subspace contains the straight line A_{n}A_{0}, intersects the hyperquadric Q^{n}, and thus it, as well as the submanifold F_{1} itself, is timelike. For ω^{p} = 0, the point B_{1} describes a curve γ on the submanifold F_{1} which is tangent to the straight line B_{1}A_{0} coinciding with the generator A_{n}A_{0} of the hypersurface U^{n}. The curve γ is an isotropic curve of the de Sitter space S_{1}^{n+1}. Thus on F_{1} there arises a fibre bundle of focal lines. The hypersurface U^{n} is foliated into an (n − 2)-parameter family of torses for which these lines are edges of regressions. The points B_{1} are singular points of a kind which is called a fold. If the characteristic equation (41) has distinct roots, then an isotropic rectilinear generator l of a lightlike hypersurface U^{n} carries n − 1 distinct foci B_{h}, h = 1,…, n − 1. If for each of these foci the condition of type s_{11} ≠ 0 holds, then each of them describes a focal submanifold F_{h}, carrying conjugate net. Curves of one family of this net are tangent to the straight lines l, and this family is isotropic. On the hypersurface V^{n−1} of the space C^{n} = Q^{n} described by the point A_{0}, to these conjugate nets there correspond the net of curvature lines.

2) s_{11} = 0. In this case relation (47) takes the form

dB_{1} = (s_{1p}A_{0} − λ̂_{p}^{q}A_{q})ω^{p},  (48)

and the focal submanifold F_{1} is of dimension n − 2. Its tangent subspace at the point B_{1} is determined by the points B_{1} and C_{p}. An arbitrary point z of this subspace can be written in the form

z = z^{n}B_{1} + z^{p}C_{p} = z^{n}(A_{n} + s_{1}A_{0}) + z^{p}(s_{1p}A_{0} − λ̂_{p}^{q}A_{q}).

Substituting the coordinates of this point into relation (3), we find that

(z, z) = g_{rs}λ̂_{p}^{r}λ̂_{q}^{s}z^{p}z^{q} + (z^{n})^{2} > 0.

It follows that the tangent subspace T_{B_{1}}(F_{1}) does not have common points with the hyperquadric Q^{n}, that is, it is spacelike. Since this takes place for any point B_{1} ∈ F_{1}, the focal submanifold F_{1} is spacelike.

For ω^{p} = 0, the point B_{1} is fixed. The subspace T_{B_{1}}(F_{1}) will be fixed too. On the hyperquadric Q^{n}, the point A_{0} describes a curve q which is polar-conjugate to T_{B_{1}}(F_{1}). Since dim T_{B_{1}}(F_{1}) = n − 2, the curve q is a conic, along which the two-dimensional plane polar-conjugate to the subspace T_{B_{1}}(F_{1}) with respect to the hyperquadric Q^{n}, intersects Q^{n}. Thus for ω^{p} = 0, the rectilinear generator A_{n}A_{0} of the hypersurface U^{n} describes a twodimensional second-order cone with vertex at the point B_{1} and the directrix q. Hence in the case under consideration a lightlike hypersurface U^{n} is foliated into an (n−2)-parameter family of second-order cones whose vertices describe the (n − 2)-dimensional focal submanifold F_{1}, and the points B_{1} are conic singular points of the hypersurface U^{n}. The hypersurface V^{n−1} of the conformal space C^{n} corresponding to such a lightlike hypersurface U^{n} is a canal hypersurface which envelops an (n − 2)- parameter family of hyperspheres. Such a hypersurface carries a family of cyclic generators which depends on the same number of parameters. Such hypersurfaces were investigated in detail in [5]. Further let B_{1} be a singular point of multiplicity m, where m ≥ 2, of a

rectilinear generator A_{n}A_{0} of a lightlike hypersurface U^{n} of the space S_{1}^{n+1} defined by an m-multiple root of characteristic equation (41). We will assume that

s_{1} = s_{2} = … = s_{m} := s_{0}, s_{0} ≠ s_{p},  (49)

and also assume that a, b, c = 1,…, m and p, q, r = m + 1,…, n − 1. Then the matrices (g_{ij}) and (λ_{ij}) can be simultaneously reduced to quasidiagonal forms

( \begin{matrix}g_{ab} & 0 \\ 0 & g_{pq} \end{matrix} ) and ( \begin{matrix}s_{0}g_{ab} & 0 \\ 0 & λ_{pq} \end{matrix} ) .

We also construct the matrix (λ̂_{ij}) = (λ_{ij} − s_{0}g_{ij}). Then

(λ̂_{ij}) = ( \begin{matrix}0 & 0 \\ 0 & λ̂_{pq} \end{matrix} ) ,  (50)

where λ̂_{pq} = λ_{pq} − s_{0}g_{pq} is a nondegenerate matrix of order n − m − 1. By relations (50) and formulas (5) and (23) we have

ω_{a}^{n} − s_{0}ω_{a}^{n+1} = 0,  (51)
ω_{p}^{n} − s_{0}ω_{p}^{n+1} = λ̂_{pq}ω^{q}.  (52)

Taking exterior derivative of equation (51) and applying relation (52), we find that

λ̂_{pq}ω_{a}^{p} ∧ ω^{q} + g_{ab}ω^{b} ∧ (ds_{0} + s_{0}ω_{0}^{0} + ω_{n}^{0}) = 0.  (53)

It follows that the 1-form ds_{0} + s_{0}ω_{0}^{0} + ω_{n}^{0} can be expressed in terms of the basis forms. We write these expressions in the form

ds_{0} + s_{0}ω_{0}^{0} + ω_{n}^{0} = s_{0c}ω^{c} + s_{0q}ω^{q}.  (54)

Substituting this decomposition into equation (53), we find that

(λ̂_{pq}ω_{a}^{p} + g_{ab}s_{0q}ω^{b}) ∧ ω^{q} + g_{ab}s_{0c}ω^{b} ∧ ω^{c} = 0.  (55)

The terms in the left hand side of (55) do not have similar terms. Hence both terms are equal to 0. Equating to 0 the coefficients of the summands of the second term, we find that

g_{ab}s_{0c} = g_{ac}s_{0b}.  (56)

Contracting this equation with the matrix (g^{ab}) which is the inverse matrix of the matrix (g_{ab}), we obtain

ms_{0c} = s_{0c}.

Since m ≥ 2, it follows that

s_{0c} = 0,

and relation (54) takes the form

ds_{0} + s_{0}ω_{0}^{0} + ω_{n}^{0} = s_{0p}ω^{p}.  (57)

For the singular point of multiplicity m of the generator A_{n}A_{0} in question the equation (43) can be written in the form

dB_{1} = (ds_{0} + s_{0}ω_{0}^{0} + ω_{n}^{0})A_{0} − λ̂_{q}^{p}ω_{0}^{q}A_{p}.

Substituting decomposition (57) in the last equation, we find that

dB_{1} = (s_{0p}A_{0} − λ̂_{p}^{q}A_{q})ω_{0}^{p}.  (58)

This relation is similar to equation (48) with the only difference that in (48) we had p, q = 2,…, n − 1, and in (58) we have p, q = m + 1,…, n − 1. Thus the point B_{1} describes now a spacelike focal manifold F_{1} of dimension n − m − 1. For ω_{0}^{p} = 0, the point B_{1} is fixed, and the point A_{0} describes an m-dimensional submanifold on the hyperquadric Q^{n} which is a cross-section of Q^{n} by an (m + 1)-dimensional subspace that is polar-conjugate to the (n − m − 1)-dimensional subspace tangent to the submanifold F_{1}.

The point B_{1} is a conic singular point of multiplicity m of a lightlike hypersurface U^{n}, and this hypersurface is foliated into an (n − m − 1)-parameter family of (m + 1)-dimensional second-order cones circumscribed about the hyperquadric Q^{n}. The hypersurface V^{n−1} of the conformal space C^{n} that corresponds to such a hypersurface U^{n} is an m-canal hypersurface (i.e., the envelope of an (n − m − 1)-parameter family of hyperspheres), and it carries an m-dimensional spherical generators.

Note also an extreme case when the rectilinear generator A_{n}A_{0} of a lightlike hypersurface U^{n} carries a single singular point of multiplicity n − 1. As follows from our consideration of the cases m ≥ 2, this singular point is fixed, and the hypersurface U^{n} become a second-order hypercone with vertex at this singular point which is circumscribed about the hyperquadric Q^{n}. This hypercone is the isotropic cone of the space S_{1}^{n+1}. The hypersurface V^{n−1} of the conformal space C^{n} that corresponds to such a hypersurface U^{n} is a hypersphere of the space C^{n}.

The following theorem combines the results of this section:

Theorem 7. A lightlike hypersurface U^{n} of maximal rank r = n−1 of the de Sitter space S_{1}^{n+1} possesses n − 1 real singular points on each of its rectilinear generators if each of these singular points is counted as many times as its multiplicity. The simple singular points can be of two kinds: a fold and conic. In the first case the hypersurface U^{n} is foliated into an (n − 2)-parameter family of torses, and in the second case it is foliated into an (n−2)-parameter family of second-order cones. The vertices of these cones describe the (n − 2)-dimensional spacelike submanifold in the space S_{1}^{n+1}. All multiple singular points of a hypersurface U^{n} are conic. If a rectilinear generator of a hypersurface U^{n} carries a singular point of multiplicity m, 2 ≤ m ≤ n − 1, then the hypersurface U^{n} is foliated into an (n − m − 1)- parameter family of (m + 1)-dimensional second-order cones. The vertices of these cones describe the (n−m−1)-dimensional spacelike submanifold in the space S_{1}^{n+1}. The hypersurface V^{n−1} of the conformal space C^{n} corresponding to a lightlike hypersurface U^{n} with singular points of multiplicity m is a canal hypersurface which envelops an (n − m − 1)-parameter family of hyperspheres and has m-dimensional spherical generators.

Since lightlike hypersurfaces U^{n} of the de Sitter space S_{1}^{n+1} represent a light flux (see Section 2), its focal submanifolds have the following physical meaning. If one of them is a lighting submanifold, then others will be manifolds of concentration of a light flux. Intensity of concentration depends on multiplicity of a focus describing this submanifold.

In the extreme case when an isotropic rectilinear generator l = A_{n}A_{0} of a hypersurface U^{n} carries one (n − 1)-multiple focus, the hypersurfaces U^{n} degenerates into the light cone generated by a point source of light. This cone represents a radiating light flux.

If each isotropic generator l ⊂ U^{n} carries two foci B_{1} and B_{2} of multiplicities m_{1} and m_{2}, m_{1} + m_{2} = n − 1, m_{1} > 1, m_{2} > 1, then these foci describe spacelike submanifolds F_{1} and F_{2} of dimension n − m_{1} − 1 and n − m_{2} − 1, respectively. If one of these submanifolds is a lighting submanifold, then on the second one a light flux is concentrated.

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Department of Mathematics (Received 19 01 1998) Jerusalem College of Technology - Mahon Lev 21 Havaad Haleumi St., POB 16031 Jerusalem 91160 Israel

akivis@math.jct.ac.il

Department of Mathematics New Jersey Institute of Technology University Heights, Newark, NJ 07102 USA

vlgold@numerics.njit.edu