DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 161–170

THE GEOMETRY OF LIGHTLIKE HYPERSURFACES ON MANIFOLDS ENDOWED WITH A CONFORMAL STRUCTURE

OF LORENTZIAN SIGNATURE

MAKS A. AKIVIS AND VLADISLAV V. GOLDBERG

Abstract. The authors study the geometry of lightlike hypersurfaces V on a manifold (M, c) endowed with a pseudoconformal structure c of Lorentzian signature. They find singular points and singular submanifolds of V and use them to construct a conformally invariant normalization of V intrinsically connected with its geometry. This normalization is constructed by means of the elements of a fourth-order neighborhood of a point x ∈ V and induces a torsion-free affine connection on V which is also intrinsically connected with the geometry of V.

1. The pseudo-Riemannian manifolds (M, g), dimM = 4, sign g = (3, 1), is a model of spacetime of general relativity. For dimM = n > 4 and sign g = (n−1, 1), we obtain natural generalizations of this model. Lightlike hypersurfaces in (M, g) are models of different types of horizons separating domains of (M, g) with different physical properties.

Many events and objects of general relativity are invariant under conformal transformations of the metric g. Hence it is appropriate to study them on a manifold (M, g), dimM = n, endowed with a conformal structure c = CO(n−1, 1) of Lorentzian signature, c = {σg}, σ = σ(x) ≠ 0, x ∈ M.

A hypersurface on (M, g) is called lightlike if it is tangent to the light cone C_{x} at each of its points x. Since the light cones on (M, g) are conformally invariant, then the same is true for a lightlike hypersurface. Thus it is appropriate to study lightlike hypersurfaces on the manifold (M, c).

In the present paper we study singular points and singular submanifolds of a lightlike hypersurface V ⊂ (M, c) and use them to construct a conformally invariant normalization of V intrinsically connected with V. This normalization induces a torsion-free affine connection on V which is also intrinsically connected with the geometry of V.

1991 Mathematics Subject Classification. 53A30, 53B25.

Key words and phrases. Pseudoconformal structure, Lorentzian signature, lightlike hypersurface, isotropic geodesics, singular point, invariant normalization, affine connection.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The results and methods of this paper can be used for studying lightlike hypersurfaces in the pseudo-Riemannian space (M, g) of Lorentzian signature, the pseudoconformal space C_{1}^{n}, and the Minkowski space R_{1}^{n}.

Lightlike hypersurface V on a manifold (M, g) of Lorentzian signature and in the space R_{1}^{n} were considered by many authors (see the books [DB 96] and [K 96] where one can find further references). However, an invariant normalization intrinsically connected with the geometry of V was not constructed in all these papers.

As we show in the present paper (see also our papers [AG 98a] and [AG 98b]), such normalizations can be constructed only by means of elements of a neighborhood of a point x ∈ V of order not less than three. But in the papers mentioned above a normalization was constructed by means of some normalizing elements not intrinsically connected with the geometry of V, and also by means of some elements defined in neighborhoods of the first and second order.

The more detailed proofs of the results of this paper will appear in [AG t. a.].

2. A conformal structure (M, c) of Lorentzian signature on a manifold M is given by a relatively invariant differential quadratic form g, sign g = (n − 1, 1). The tangent space T_{x}(M, c) to this manifold can be compactified in such a way that it will be endowed with the structure of a pseudoconformal space (C_{1}^{n})_{x} of the same signature (n − 1, 1) (see [AG 96], Ch. 1).

Under the Darboux mapping, to the conformal space (C_{1}^{n})_{x} there corresponds a hyperquadric Q_{1}^{n} of a projective space P_{x}^{n+1} , Q_{1}^{n} ⊂ P_{x}^{n+1} . The equation of this hyperquadric is (x, x) = 0.

In the space P_{x}^{n+1} , we consider a family of projective frames R_{x} = {A_{0}, A_{i}, A_{n+1} }, i = 1, …, n, consisting of the points A_{0} = x and A_{n+1} = y lying on the hyperquadric Q_{1}^{n}, so that

(A_{0}, A_{0}) = 0, (A_{n+1} , A_{n+1} ) = 0 , (A_{0}, A_{n+1} ) ≠ 0 , (1)

and points A_{i} ∈ T_{x} ∩ T_{y}, where T_{x} and T_{y} are the subspaces tangent to Q_{1}^{n} at the points x and y. Besides relations (1), elements of this frame satisfy the relations

(A_{0}, A_{i}) = (A_{n+1} , A_{i}) = 0, (A_{i}, A_{j}) = g_{ij} , (2)

where det(g_{ij}) ≠ 0, sign g = (n − 1, 1). The equation of the hyperquadric Q_{1}^{n} with respect to this frame has the form

g_{ij}x^{i}x^{j} − 2x^{0}x^{n} = 0 .

The isotropic cone C_{x} with vertex x is the intersection C_{x} = Q_{1}^{n} ∩ T_{x} and is defined by the equation g = g_{ij}ξ^{i}ξ^{j} = 0, ξ ∈ T_{x}. The group G = SO(n−1, 1)×H, where H = R^{+} , leaves the cone C_{x} ⊂ T_{x} invariant.

The equations of infinitesimal displacement of the frame R_{x} on (M, c) have the form

\{ \begin{matrix}dA_{0} = ω_{0}^{0}A_{0} & +ω_{0}^{i}A_{i} , \\ dA_{i} = ω_{i}^{0}A_{0} & +ω_{i}^{j}A_{j} + ω_{i}^{n+1} A_{n+1} , \\ dA_{n+1} = & ω_{n+1}^{i} A_{i} + ω_{n+1}^{n+1} A_{n+1} . \end{matrix} (3)

The 1-forms ω_{0}^{i} = ω^{i} are basis forms of the manifold M. By conditions (1) and (2), the forms occurring in (3) satisfy the equations

\{ \begin{matrix}ω_{i}^{n+1} = g_{ij}ω_{0}^{j} , ω_{n+1}^{i} = g^{ij}ω_{j}^{0} , ω_{n+1}^{n+1} = −ω_{0}^{0} , \\ dg_{ij} − g_{ik}ω_{j}^{k} − g_{kj}ω_{i}^{k} = 0 , \end{matrix} (4)

where g^{ik}g_{kj} = δ_{j}^{i}. Note that the quantities g_{ij} and the forms ω^{i} are defined in a first-order neighborhood of a point x ∈ V, the forms ω_{0}^{0} and ω_{j}^{i} in its second-order neighborhood, and the forms ω_{j}^{0} in its third-order neighborhood.

3. The independent 1-forms occurring in equations (3) satisfy the following structure equations (see [AG 96], Section 4.1):

\{ \begin{matrix}dω^{i} = ω_{0}^{0} ∧ ω^{i} + ω^{j} ∧ ω_{j}^{i} , \\ dω_{0}^{0} = ω^{i} ∧ ω_{i}^{0}, \\ dω_{j}^{i} = ω_{j}^{0} ∧ ω^{i} + ω_{j}^{k} ∧ ω_{k}^{i} + g_{jk}ω^{k} ∧ g^{il}ω_{l}^{0} + C_{jkl}^{i}ω^{k} ∧ ω^{l} , \\ dω_{i}^{0} = ω_{i}^{0} ∧ ω_{0}^{0} + ω_{i}^{j} ∧ ω_{j}^{0} + C_{ijk}ω^{j} ∧ ω^{k} . \end{matrix} (5)

Here the quantities C_{jkl}^{i} form the tensor of conformal curvature of the manifold (M, c) which is also called the Weyl tensor. It is defined in a third-order neighborhood of a point x ∈ V. The quantities C_{ijk} are defined in its fourth-order neighborhood and do not form a tensor. Let C = (C_{jkl}^{i}, C_{ijk}). For C = 0, a manifold (M, c) is conformally flat. Its structure is equivalent to that of the pseudoconformal space C_{1}^{n}.

For n ≥ 4, the condition C_{jkl}^{i} = 0 implies C_{ijk} = 0, and a manifold (M, c) is equivalent to the space C_{1}^{n}. For n = 3, the Weyl tensor C_{jkl}^{i} = 0, and the quantities C_{ijk} constitute the tensor of conformal curvature. We will not consider this case in the current paper.

The components of the Weyl tensor and the quantities C_{ijk} satisfy the conditions

\{ \begin{matrix}C_{ijkl} = g_{im}C_{jkl}^{m} , \\ C_{ijkl} = −C_{jikl} = −C_{ijlk}, C_{ijkl} = C_{klij} , \\ C_{ijkl} + C_{iklj} + C_{iljk} = 0, C_{jki}^{i} = 0 , \\ C_{ijk} = −C_{ikj}. \end{matrix} (6)

For C = (C_{jkl}^{i}, C_{ijk}) = 0, equations (3) are completely integrable and define a conformal frame bundle in the space C_{1}^{n}. For C ≠ 0, they are not completely integrable, but they can be integrated along any smooth curve x = x(t). As a result, we obtain a development of the manifold (M, c) into C_{1}^{n} along the curve x = x(t).

Isotropic geodesics are invariant under a conformal transformation of the metric g (see [AG 96], pp. 138–140). Thus they can be considered on a manifold (M, c) endowed with a conformal structure. Under the development of (M, c) into C_{1}^{n} along an isotropic geodesic, to this isotropic geodesic there corresponds a rectilinear generator of the hyperquadric Q_{1}^{n} ∼ (C_{1}^{n})_{x}.

4. A hypersurface V ⊂ (M, c), dimV = n − 1, is called lightlike if it is tangent to the light cone C_{x} at each point x ∈ V. With each point x ∈ V we associate a frame such that x = A_{0}, the straight line A_{0}A_{1} ⊂ T_{x}(V) is tangent to V along the isotropic direction, and A_{a} ⊂ T_{x}(V), a = 2, …, n − 1. The subspace S_{x} = A_{0} ∧ A_{2} ∧ … ∧ A_{n−1} ⊂ T_{x}(V), dimS_{x} = n − 2, is called the screen subspace of V. We will take a point A_{n} ∈ T_{x}(V) in such a way that the straight line A_{0}A_{n} is a second isotropic straight line that is conjugate to the screen subspace S_{x}. This line A_{0}A_{n} is called a normalizing straight line. We normalize the points A_{1} and A_{n} by the condition (A_{1}, A_{n}) = −1.

In this frame we have

(g_{ij}) = (\begin{matrix}0 & 0 & −1 \\ 0 & g_{ab} & 0 \\ −1 & 0 & 0 \end{matrix}) , i, j = 1, …, n; a, b = 2, …, n − 2 , (7)

and the equation of the isotropic cone C_{x} takes the form

g_{ab}ξ^{a}ξ^{b} − 2ξ^{1}ξ^{n} = 0, g_{ab}ξ^{a}ξ^{b} > 0 ,

where ξ = {ξ^{a}} ≠ 0. From (3) and (7) it follows that

\{ \begin{matrix}ω_{1}^{n} = ω_{n}^{1} = 0, & ω_{1}^{1} = −ω_{n}^{n} , \\ ω_{a}^{1} = g_{ab}ω_{n}^{b}, & ω_{a}^{n} = g_{ab}ω_{1}^{b} , \\ dg_{ab} − g_{ac}ω_{b}^{c} − g_{cb}ω_{a}^{c} = 0 . \end{matrix} (8)

A frame fiber bundle associated with a hypersurface V we have constructed is a first-order frame bundle. With respect to this frame bundle the hypersurface V is defined by the Pfaffian equation

ω_{0}^{n} = 0. (9)

Thus the first equation of (3) becomes

dA_{0} = ω_{0}^{0}A_{0} + ω_{0}^{1}A_{1} + ω_{0}^{a}A_{a} , (10)

where the forms ω^{1} and ω^{a} are basis forms of the hypersurface V.

On the hypersurface V, the first fundamental form takes the form

g|_{V} = (dA_{0}, dA_{0}) = g_{ab}ω^{a}ω^{b} ,

and sign g|_{V} = (n − 2, 0). The equations

ω^{a} = 0

define a foliation of isotropic lines on V.

Differential prolongation of equation (9) leads us to the following equations

ω_{a}^{n} = λ_{ab}ω^{b}, λ_{ab} = λ_{ba} . (11)

The investigation of equation (9) and its differential prolongation (11) leads to the following result.

Theorem 1 Lightlike hypersurfaces V on a manifold (M, c) endowed with a conformal structure exist, and the solution of a system defining such hypersurfaces depends on one function of n − 2 variables.

Note that by means of equations of (4) and formulas (8), equations (11) are equivalent to the equations

ω_{1}^{a} = λ_{b}^{a}ω^{b}, λ_{b}^{a} = g^{ac}λ_{cb} , (12)

and it follows from (4) that

ω_{1}^{n+1} = 0 . (13)

Equations (8) and (13) imply that

dA_{1} = ω_{1}^{0}A_{0} + ω_{1}^{1}A_{1} + ω_{1}^{a}A_{a} , (14)

Equations (10), (12), and (14) prove the following result.

Theorem 2

(a) V = f(M^{n−2} × l) where f is a differentiable mapping and l ∼ P^{1} is an isotropic geodesic on V.

(b) J = det(λ_{b}^{a} − sδ_{b}^{a}) is the Jacobian of the mapping f at the point z = A_{1} − sA_{0} ∈ l ⊂ V.

(c) The line l carries n−2 real singular points F_{a} = A_{1} −s_{a}A_{0} of the mapping f, where s_{a} are roots of the equation J = 0 (we count each singular point as many times as its multiplicity).

(d) Under the development of V into (Q_{1}^{n})_{x} ⊂ P^{n+1} , to the hyperplanes T_{z}(V), where z is a regular point of l, there corresponds the same space T_{l}, dim T_{l} = n − 1, tangent to (Q_{1}^{n})_{x} along l.

5. Consider

λ = \frac{1}{n − 2} ∑_{a} s_{a} = \frac{1}{n − 2}λ_{a}^{a} , (15)

where λ_{a}^{a} = tr (λ_{a}^{b}). The quantity λ satisfies the following differential equation:

dλ − λ(ω_{1}^{1} − ω_{0}^{0}) − ω_{1}^{0} = −μω^{1} − μ_{a}ω^{a} . (16)

The point H = A_{1} − λA_{0} is called the harmonic pole of the point x = A_{0} with respect to the singular points F_{a}. For n = 4, the point H satisfies the relation (A_{0}, H; F_{2}, F_{3}) = −1.

The quantities h_{ab} = λ_{ab}−λg_{ab} are components of a trace-free tensor of weight 1 on V : h_{ab}g^{ab} = 0. The tensor h_{ab} is the conformally invariant second fundamental tensor of V. The quadratic form h = h_{ab}ω^{a}ω^{b} is the conformally invariant second fundamental form of V.

The tensor h_{b}^{a} = g^{ac}h_{cb} = λ_{b}^{a} − λδ_{b}^{a} is called the conformal affinor of Burali-Forte (the conformal shape operator), tr (h_{b}^{a}) = h_{a}^{a} = 0.

The following theorem is valid.

Theorem 3

(a) The harmonic pole H is a regular point on a line l if and only if det (h_{b}^{a}) ≠ 0.

(b) If (h_{b}^{a}) = 0, then V is a totally umbilical hypersurface, and H is the only singular point on the isotropic geodesic l. H describes a line γ tangent to l.

(c) If (h_{b}^{a}) = 0 and the structure (M, c) is conformally flat, then H is fixed, and V coincides with the isotropic cone C_{x} ⊂ Q_{1}^{n} ∼ C_{1}^{n}.

6. As we noted earlier, the forms ω^{1} and ω^{a}, a = 2, …, n − 1, are the basis forms of the hypersurface V. By (5), their exterior differentials have the form

\{ \begin{matrix}dω^{1} = ω^{1} ∧ (ω_{1}^{1} − ω_{0}^{0}) + ω^{a} ∧ ω_{a}^{1} , \\ dω^{a} = ω^{1} ∧ ω_{1}^{a} + ω^{b} ∧ (ω_{b}^{a} − δ_{b}^{a}ω_{0}^{0}) . \end{matrix} (17)

Thus the matrix 1-form

ω = (\begin{matrix}ω_{1}^{1} − ω_{0}^{0} & ω_{a}^{1} \\ ω_{1}^{a} & ω_{b}^{a} − δ_{b}^{a}ω_{0}^{0} \end{matrix})

defines an affine structure on V. This matrix 1-form defines a torsion-free affine connection on V if and only if

dω + ω ∧ ω = Ω , (18)

where Ω is the curvature 2-form of this connection which is a linear combination of exterior products of the forms ω^{1} ∧ ω^{a} and ω^{a} ∧ ω^{b} (see [KN 63], Ch. III).

By differentiating the form ω, one can prove that condition (18) holds on V if and only if the forms ω_{1}^{0}, ω_{a}^{0}, ω_{n}^{0}, and ω_{n}^{a} = g^{ab}ω_{b}^{1} are linear combinations of the basis forms ω^{1} and ω^{a}. This leads to the following result.

Theorem 4 To define a torsion-free affine connection on a lightlike hypersurface V ⊂ (M, c), it is sufficient to assign on V a screen distribution S = ∪S_{x}, x ∈ V, and a field L of normalizing hyperplanes L_{x} ⊂ T_{x}(M), dimL_{x} = n − 1, x ∉ L_{x}.

7. We will make a reduction of the first-order frame bundle by imposing a point A_{1} with the harmonic pole H of the isotropic geodesic l, A_{1} = H. Then it follows from Section 5 that

λ = 0, λ_{ab} = h_{ab} , λ_{b}^{a} = h_{b}^{a}, ω_{1}^{a} = h_{b}^{a}ω^{b} . (19)

The constructed reduced frame bundle is a second-order frame bundle.

To construct normalizing elements, a screen subspace S_{x} and a normalizing hyperplane L_{x}, we consider the subspace R_{x} ⊂ T_{x}(V), dimR_{x} = n − 3, which is complementary to the straight line A_{0}A_{1} ⊂ (P^{n+1} )_{x}. The subspace R_{x} allows us to find a screen subspace S_{x} = x ∧ R_{x} and a complementary screen subspace S̃_{x} = H(x) ∧ R_{x}.

In R_{x} take the basis points

C_{a} = A_{a} + y_{a}A_{0} + z_{a}A_{1} , a = 2, …, n − 1 . (20)

The conditions for the subspace R_{x} to be invariant is

∇_{δ}y_{a} + π_{a}^{0} = 0 , ∇_{δ}z_{a} + z_{a}(π_{1}^{1} − π_{0}^{0}) + π_{a}^{1} = 0 . (21)

Here δ is the symbol of differentiation with respect to fiber parameters, π_{η}^{ξ} = ω_{η}^{ξ}(δ), and

∇_{δ}y_{a} = δy_{a} − y_{b}(π_{a}^{b} − δ_{a}^{b}π_{0}^{0}) , ∇_{δ}z_{a} = δz_{a} − z_{b}(π_{a}^{b} − δ_{a}^{b}π_{0}^{0}) .

The objects y_{a} and z_{a} are called the normalizing objects.

By relations (19), equations (16) take the form

ω_{1}^{0} = μω^{1} + μ_{a}ω^{a} , (22)

where μ = \frac{1}{n−2}h_{b}^{a}h_{a}^{b} is a relative invariant defined in a second-order neighborhood of a point x ∈ V, μ > 0 if (h_{a}^{b}) ≠ 0, and the quantities μ_{a} are defined in a third-order neighborhood of x ∈ V.

For ω^{1} = ω^{a} = 0, the quantities μ and μ_{a} satisfy the equations

δμ − 2μ(π_{1}^{1} − π_{0}^{0}) = 0 , (23)

and

∇_{δ}μ_{a} − μ_{a}(π_{1}^{1} − π_{0}^{0}) + h_{a}^{b}π_{b}^{0} − μπ_{a}^{1} = 0 . (24)

It follows from (23) that the quantity μ satisfies the equation

dμ − 2μ(ω_{1}^{1} − ω_{0}^{0}) = νω^{1} + ν_{a}ω^{a} , (25)

where the quantities ν and ν_{a} are defined in a third-order neighborhood of a point x ∈ V.

For ω^{1} = ω^{a} = 0, prolongation of equation (25) gives

\{ \begin{matrix}δν − 3ν(π_{1}^{1} − π_{0}^{0}) = 0 , \\ ∇_{δ}ν_{a} − 2ν_{a}(π_{1}^{1} − π_{0}^{0}) + 2μπ_{a}^{0} − (2μh_{a}^{b} + νδ_{a}^{b})π_{b}^{1} = 0 . \end{matrix} (26)

Equations (23) and (26) prove that the quantities μ and ν are relative invariants of weight 2 and 3, respectively.

8. We will construct now normalizing objects of a lightlike hypersurface V ⊂ (M, c) intrinsically connected with the geometry of V. Consider the objects

M_{a} = h_{a}^{b}μ_{b} + \frac{ν}{2μ}μ_{a} − \frac{1}{2}ν_{a} , N_{a} = −μμ_{a} + \frac{1}{2}h_{a}^{b}ν_{b} . (27)

For ω^{1} = ω^{a} = 0, they satisfy the equations

\{ \begin{matrix}∇_{δ}M_{a} − 2M_{a}(π_{1}^{1} − π_{0}^{0}) + H_{a}^{b}π_{b}^{0} = 0 , \\ ∇_{δ}N_{a} − 3N_{a}(π_{1}^{1} − π_{0}^{0}) − μH_{a}^{b}π_{b}^{1} = 0 , \end{matrix} (28)

where

H_{a}^{b} = h_{a}^{c}h_{c}^{b} + \frac{ν}{2μ}h_{a}^{b} − μδ_{a}^{b} (29)

is the tensor defined in a third-order neighborhood of x ∈ V. For ω^{1} = ω^{a} = 0, this tensor satisfies the equations

∇_{δ}H_{a}^{b} = 2H_{a}^{b}(π_{1}^{1} − π_{0}^{0}) , (30)

and hence it is a relative tensor of weight 2.

The following theorem can be proved by a direct calculation.

Theorem 5 (a) The tensor H_{a}^{b} is trace-free: H_{a}^{a} = 0.

(b) The tensor H_{a}^{b} can be reduced to a diagonal form simultaneously with the tensors g_{ab} and h_{a}^{b} and has the eigenvalues

H_{a} = h_{a}^{2} + \frac{ν}{2μ}h_{a} − μ ,

where h_{a} are the eigenvalues of the tensor h_{a}^{b}.

(c) If the tensor H_{a}^{b} is degenerate, then the relative invariants μ and ν are connected by an algebraic equation, and the hypersurface V is of a special type.

(d) For n = 4, the tensors H_{a}^{b} and h_{a}^{b} are proportional: H_{a}^{b} = \frac{ν}{2μ}h_{a}^{b}.

9. Suppose now that the tensor H_{a}^{b} is nondegenerate. Denote by H̃_{a}^{b} the inverse tensor of the tensor H_{a}^{b}. Then we can construct two more objects

P_{a} = H̃_{a}^{b}M_{b} , Q_{a} = \frac{1}{μ}H̃_{a}^{b}N_{b} .

By (27), (28), and (30), these objects satisfy the equations

∇_{δ}P_{a} + π_{a}^{0} = 0 , ∇_{δ}Q_{a} + Q_{a}(π_{1}^{1} − π_{0}^{0}) + π_{a}^{1} = 0 . (31)

Comparing (31) and (21), we see that the objects P_{a} and Q_{a} are normalizing objects, and the points

C_{a} = A_{a} + P_{a}A_{0} + Q_{a}A_{1}

define an invariant subspace R_{x} intrinsically connected with the geometry of the hypersurface V. Thus the following theorem is valid.

Theorem 6 If det(H_{b}^{a}) ≠ 0, then an invariant screen distribution S and an invariant complementary screen distribution S̃ that are intrinsically connected with the geometry of V ⊂ (M, c) are defined by elements of a third-order neighborhood of a point x ∈ V.

10. We will make a reduction of the second-order frame bundle by imposing points A_{a} with the points C_{a}. As a result, we obtain a third-order frame bundle in which

P_{a} = Q_{a} = 0 ⟹ M_{a} = N_{a} = 0 ⟹ μ_{a} = ν_{a} = 0 .

This and formulas (31) imply that the forms ω_{a}^{0} and ω_{a}^{1} are expressed in terms of the basis forms ω^{1} and ω^{a}:

\{ \begin{matrix}ω_{a}^{0} = σ_{a}ω^{1} + σ_{ab}ω^{b} , \\ ω_{a}^{1} = τ_{a}ω^{1} + τ_{ab}ω^{b} . \end{matrix} (32)

The coefficients σ_{a}, σ_{ab}, τ_{a}, and τ_{ab} are defined by elements of a fourth-order neighborhood of a point x ∈ V.

After the indicated reduction has been made, the screen subspace S_{x} = A_{0} ∧ A_{2} ∧ … ∧ A_{n−1} and the normalizing straight line A_{0}A_{n}, which is conjugate to the subspace S_{x}, with respect to the cone C_{x}, become intrinsically connected with the geometry of V.

In order to complete the construction of an invariant normalization of V, we need to construct an invariant point C_{n} on the isotropic normalizing straight line A_{0}A_{n} which together with H and R_{x} determines a normalizing hyperplane L_{x}.

The set of isotropic geodesics l̃ corresponding to the straight line A_{0}A_{n} on the manifold (M, c) form an isotropic congruence U. Each ray l̃ of this congruence carries n−2 singular points (see [AG 98a] and [AG 98b]). They are defined on l̃by the eigenvalues of the tensor τ_{ab}. As a normalizing point C_{n} we take the harmonic pole H̃ of the point x = A_{0} with respect these singular points. Since the tensor τ_{ab} is defined in a fourth-order neighborhood of a point x ∈ V, the point H̃ and consequently the hyperplane L_{x} are also defined in this neighborhood.

Theorem 7 A complete invariant normalization of a lightlike hypersurface V ⊂ (M, c) intrinsically connected with the geometry of V is defined in a fourth-order neighborhood of a point x ∈ V.

Corollary 8 The curvature tensor of the affine connection induced by the complete normalization indicated in Theorem 7 is defined in a fifth-order neighborhood of a point x ∈ V.

11. The following theorem is related to the integrability of the screen distributions.

Theorem 9 (a) The screen distributions S and S̃ are integrable simultaneously.

(b) The screen distributions S and S̃ are integrable if and only if τ_{ab} = τ_{ba}.

(c) If the screen distributions S and S̃ are integrable, then all singular points of the isotropic congruence U are real. The congruence U is foliated into a 1-parameter family of lightlike hypersurfaces.

References

[AG 96] Akivis, M. A., Goldberg, V. V., Conformal differential geometry and its generalizations,

John Wiley & Sons, New York, 1996, xiii+383 pp.

[AG 98a] Akivis, M. A., Goldberg, V. V., On geometry of hypersurfaces of pseudoconformal

spaces of Lorentzian signature, J. Geom. Phys. 26 (1998), no. 1–2, 112–126.

[AG 98b] Akivis, M. A., Goldberg, V. V., Singular points of lightlike hypersurfaces of the de

Sitter space, Publ. Inst. Math. (Beograd), 63 (77) (1998), 81–101.

[AG t. a.] Akivis, M. A., Goldberg, V. V., Lightlike hypersurfaces on manifolds endowed with a

conformal structure of Lorentzian signature, to appear.

[DB 96] Duggal, K. L., Bejancu, A., Lightlike submanifolds of semi-Riemannian manifolds and

applications, Kluwer Academic Publishers, Amsterdam, 1996, viii+300 pp.

[KN 63] Kobayashi, S., Nomizu, K., Foundations of differential geometry, Vol. 1. Wiley–

Interscience, New York, 1963, xi+329 pp.

[Ku 96] Kupeli, D. N., Singular semi-Riemannian geometry, Kluwer Academic Publishers, Dordrecht, 1996, x+177 pp.

M. A. Akivis, Department of Mathematics, Jerusalem College of Technology

– Mahon Lev, P. O. B. 16031, Jerusalem 91160, ISRAEL E-mail: akivis@avoda.jct.ac.il

V. V. Goldberg, Department of Mathematics, New Jersey Institute of Technology University Heights, Newark, NJ 07102, U.S.A. E-mail: vlgold@m.njit.edu