© Hindawi Publishing Corp. PSEUDOINVERSION OF DEGENERATE METRICS

Let (M,g) be a smooth manifold M endowed with a metric g. A 
large class of differential operators in 
differential geometry is intrinsically defined by means of the 
dual metric g∗ on the dual bundle 
TM∗ of 1-forms on M. If the metric g is (semi)-Riemannian, 
the metric g∗ is just the inverse of g. This 
paper studies the definition of the above-mentioned geometric 
differential operators in the case of manifolds 
endowed with degenerate metrics for which g∗ is not 
defined. We apply the theoretical results to Laplacian-type 
operator on a lightlike hypersurface to deduce a Takahashi-like 
theorem (Takahashi (1966)) for lightlike hypersurfaces in 
Lorentzian space ℝ1n


Introduction.
The introduction of a Riemannian metric on a smooth manifold gives rise to a series of differential operators which reveal deep relationships between the geometry and the topology of the manifold. If ∆ k is the Laplacian on k-forms, it is well known that, on a compact manifold, its spectrum {λ k i } contains topological and geometric information on the manifold. According to the Hodge decomposition theorem, the dimension of the kernel of ∆ k equals the kth Betti number so that the Laplacian determines Euler characteristic χ(M) of compact Riemannian manifolds (M, g).
Among usual differential operators on (M, g), the exterior derivative which takes k-forms to (k + 1)-forms is the only one defined in terms of the smooth structure of the manifold M. The others are defined by means of the metric g * on the dual bundle, which is the inverse of the given metric g, in semi-Riemannian case.
Many situations arise in mathematical physics where the metric is degenerated and it is not possible to define the inverse g * . A typical case is the one we have in the coupling of Einstein theory of gravity with both a quantum mechanics particle with spin and an electromagnetic field, that is, Einstein-Dirac-Maxwell (EDM) system. If, for instance, we consider Lorentz framed manifold M endowed with Eddington-Finkelstein metric (cf. Hawking and Ellis [12, page 150]) given in a coordinate system (u,r ,θ,φ), where u = t +r is an advanced null coordinate with r = dr 1 − 2m/r = r + 2m ln(r − 2m) (r > 2m), (1.2) the hypersurface M : u = constant is a degenerate hypersurface. Dirac operator can be written outside M and in its interior region. But it is not easy to match the two operators on M due to the fact that the inverse metric g * cannot be defined on M. Normalization problems and the geometry of those manifolds are considered in several papers (see [1,2,3,5,6,7,8,10,11,13,14] and references therein).
In the following, we consider a smooth (semi)-Riemannian manifold (M, g), its Levi-Civita connection ∇, its tangent (resp., cotangent) bundle T M (resp., T M * ), and Ᏺ(M) the space of smooth functions on M. For a vector bundle E, we denote by Γ (E) the space of its smooth sections. For a smooth function f in Ᏺ(M), the gradient of f is the vector field grad g f given by g grad g f ,X = X(f ) = df (X) (1.3) for any X in Γ (T M).
In local coordinates (x 1 ,...,x n ), the gradient of f is given by (1.4) where f i = ∂f /∂x i , E j = ∂/∂j, and g ij is the (i, j)-entry of the inverse g −1 of g.
The gradient can be defined using exterior differential d and the natural isomorphism # between the tangent bundle T M and its dual bundle T M * , by the composition is such that, for all Y ∈ Γ (T M), The inverse of # is the isomorphism = (#) −1 defined by such that The divergence of a vector field X is the codifferential of the dual 1-form X defined by (1.11) The covariant differentiation ∇ defined on vector fields can be extended to 1-forms by duality using isomorphism #: The Laplace-Beltrami operator ∆ g , whose physical and mathematical importance is well known, is defined on smooth functions by (1.13) that is, locally, (1.14) From (1.4) and (1.14), it is obvious that the entries g ij of the inverse of g ij need to be computed in local coordinates. So the following question seems to be coming out in a natural way: how can we define those differential operators on a lightlike manifold, that is, on a submanifold with a degenerate (noninvertible) metric g? This question will be answered by Proposition 3.1. First, note that (dx i ) and (∂ i ) are two local bases of T * M and the decomposition (1.15) holds. Hence, and then, (1.17) So inverting g, we get Thus, coefficients g ij appear as the entries of the matrix of transition from basis (∂ i ) to (dx k ).
Then, an important step in answering the above question should be the construction of the isomorphisms # and adapted to the case of lightlike manifolds.
Below, we discuss lightlike submanifolds of semi-Riemannian manifolds. We are particularly interested in lightlike hypersurfaces. One may proceed similarly in any other codimension. The paper is organized as follows. Section 2 deals with preliminaries and specifies our notations and some basic results. Section 3 is devoted to the definition and construction of the pseudoinverse of a degenerate metric g, and a concept of associate metric with the degenerate metric g is also introduced. In Section 4, the compatibility of this associate metric with respect to the connection induced by g is examined. The existence and the properties of the pseudoinverse allow the definition of a Laplacian ∆ g on smooth functions of (M, g). Actually, this ∆ g is a d'Alembertian since g is nondefinite. However, for the sake of simplicity, we call it Laplacian and we denote it by ∆ g . It is formally defined in the same way as the Laplace operator in Section 5. The Laplacian of lightlike hypersurfaces endowed with parallel screen distribution is then defined and its action on position vector field is expressed (see Theorem 5.2). Finally, in Section 6, we give an application which oddly reminds us of Takahashi theorem [15]. It is our Theorem 6.1.

Preliminaries on lightlike hypersurfaces.
Let M be a hypersurface of an (n + 2)-dimensional semi-Riemannian manifold (M, g). In the classical theory of nondegenerate hypersurface, the normal bundle has trivial intersection {0} with the tangent one and plays an important role in the introduction of main geometric objects. In case of degenerate hypersurface, the situation is totally different. The normal bundle T M ⊥ is a rank-one distribution over M : T M ⊥ ⊂ T M; hence, the metric g on M is degenerate, that is, noninvertible and of rank n. In fact, the following propositions is proved in [11] Proposition 2.1. Let (M, g) be a hypersurface of an (n + 2)-dimensional semi-Riemannian manifold (M,ḡ). Then the following assertions are equivalent: (i) M is a lightlike hypersurface ofM; (ii) g has a constant rank n on M; A complementary bundle of T M ⊥ in T M is a rank-n nondegenerate distribution over M. It is called a screen distribution in the tangent bundle and is often denoted by S(T M). In [11,Chapter 4], degenerate hypersurface (M, g) endowed with this specific distribution is denoted by (M, g, S(T M)) and is also called lightlike hypersurface. For the convenience of the reader, we summarize the properties of lightlike hypersurfaces which are appropriate to our purpose.
There exists a vector bundle tr(T M) of rank one over M such that for any nonzero section ξ of T M ⊥ on a coordinate neighbourhood ᐁ ⊂ M, there is a unique section N of tr(T M) defined on ᐁ such that Let ∇ and ∇ be the Levi-Civita connection of (M, g) and an induced connection on (M, g, S(T M)), respectively. It is known that ∇ is not unique. It depends on both g and the considered screen distribution.
With the decompositions into orthogonal direct sums we have Gauss and Weingarten formulae in the form 4) or equivalently, where we put and ∇ t is a linear connection on the lightlike transverse vector bundle tr(T M).
Let P denote the projection morphism of Γ (T M) on Γ (S(T M)) with respect to the decomposition (2.2). We have where ∇ * X P Y and A * U X belong to Γ (S(T M)), ∇ * and ∇ * t are linear connections on S(T M) and They are the second fundamental form and the shape operator of the screen distribution S(T M), respectively. Note that some authors called A * U (or A N ) the Burali-Forti affinor [3,9].
Define the following also on ᐁ: One can verify that ε(X) = −τ(X). Thus, locally, one has (2.14) We will make use of the following theorem. Then it is easy to see that the following relations hold: 3. Pseudoinverse of a degenerate metric. We start this section by the construction of the isomorphisms # and with respect to the noninvertible induced metric g.
First, we consider on M the 1-form η defined by where N is given by (2.1). Let P be the projection morphism of Γ (T M) on Γ (ST M) in (2.2). We have, for all X ∈ Γ (T M), and from (2.1) and (3.1), Now, we define by such that It is obvious that is linear.
As T M and T * M have the same finite-constant rank, it suffices to show that is injective, that is, X = 0 X = 0, to get the bijection of . Consider X ∈ Γ (T M) and assume X = 0. Then In particular, for any Y ∈ Γ (S(T M)), we have We denote by # its inverse isomorphism. Note that the above construction generalizes the one of nondegenerate case for, in the latter case, S(T M) coincides with T M. As a consequence, the 1-form η vanishes identically and P becomes the identity map on T M.
We now define the associate metricg of g. Put It is obvious thatg is bilinear in X and Y . Furthermore, from (3.5), we have that X (Y ) = Y (X), theng is symmetric. Its inverseg −1 , which we denote by g [·,·] , will be called the pseudoinverse of the degenerate metric g on M.
Note that in the case where g is nondegenerate, the associate metricg coincides with g and the pseudoinverse g [·,·] coincides with the dual g * (or g −1 ) of g.
In what follows, we use the following range of indices: Now, consider a local coordinate system (x 0 ,...,x n+1 ) adapted to decompositions (2.2) and (2.3), that is, is a local quasiorthogonal basis of T M |M such that 3), and (3.5), we have a standard computation that gives 15) and the first terms in (3.14) areg 00 andg 0i , respectively. Hence we havẽ We also have Consequently,g ij = g ij , i,j = 1,...,n.
(3.18) Thus, with respect to the quasiorthonormal basis (ξ, X 1 ,...,X n ,N), the matrices ofg and g [·,·] are given bỹ We have Hence, from (3.23), we infer that Let X be a smooth vector field defined on ᐁ ⊂ M. The divergence div g X of X with respect to the degenerate metric g is intrinsically defined by div g X = − n α=0 ε α X α ∇ Xα X .
In summary, we have proved the following proposition.  (M,ḡ). There exists an associate metricg and a pseudoinverse g [·] of g on M such that locally on ᐁ ⊂ M, the following holds: (i) for any smooth function f : ᐁ ⊂ M → R,
Note that if the section ξ is globally defined on M, then our result is also global.
4. Compatibility ofg with respect to the connection ∇ induced by g. It is well known that the induced metric g is not compatible with the induced connection ∇ in general, and this compatibility arises if and only if the lightlike hypersurface M is totally geodesic in M. This section deals with the compatibility of the associate metricg with respect to ∇. For Now use (2.9), (2.11), and (3.10) to obtain The above expression reduces to

(4.4)
Now, recall that since the connection * ∇ is compatible with g, the expression between brackets is zero, and therefore we have

Proof. It is well known that ∇ is compatible withg if and only if (∇ Xg )(Y , Z) = 0 for all X, Y , Z ∈ Γ (T M). Putting Y = Z = ξ in (4.5) gives τ(X) = 0 for all X ∈ Γ (T M), and setting Y = ξ yields B(X, P Z) = C(X, P Z) for all X, Z ∈ Γ (T M).
We derive from (4.5) the main covariant derivative formula (4.7) We discuss here the effect of the change of the screen distribution on the compatibility ofg with g. The first equation in (4.6) indicates that only some privileged changes in the screen distribution maintain the compatibility property of the associate metricg with g. For two screen distributions S(T M) and S(T M) on (M, g), with two orthogonal frame fields (W i ) i=1,...,n and (W i ) i=1,...,n , respectively, the relationship between the second fundamental forms C and C of the screen distributions S(T M) and S(T M) , respectively, is given by (see [4]) where W = i=n i=1 c i W i is the characteristic vector field of the screen change. Therefore, a change in screen distribution leaves the compatibility ofg with g invariant if and only if where ω is the dual 1-form of W defined by ω(·) = g(W , ·). From (3.1) and (3.10), ifg is associated to g on S(T M) andg is associated to g on S(T M) , we haveg where D is the bilinear form defined by Before starting the next section, it is important to stress that the 1-form τ of (4.6), defined in (2.6) and (2.8), depends on the normalization of the rankone subbundle Rad(T M), that is, on the choice of ξ ∈ Γ (T M ⊥ | ᐁ ). It has been proved in [11, page 99] that there exists a pair (a normalization) {ξ, N} on ᐁ such that the corresponding 1-form τ vanishes identically.
From now on we use this normalization. Then, relations (2.6) and (2.14) become∇ . We have The position vector field x can be written in R n+2 with respect to (ξ,N,X 1 ,...,X n ). As ε k = 1, k = 1,...,n, we have Let ∆ g denote the Laplace operator on functions with respect to the degenerate metric g. Then According to the above remark on the signature and (2.1), we have here ε α = 1 for all α. Using (4.7) yields (5.10) Hence X α ·g U A ,X α = X α · f A;α =: f A;αα . (5.11) Now we compute f A;α and f A;αα for A = 0,...,n+ 1 and α = 0,...,n: Note that as τ = 0 and S(T M) is parallel, N is a parallel vector field. Indeed, ∇ X N = −A N X + τ(X)N. It follows from (4.12) and Theorem 2.3 that Hence f 0;α = X α ,N , (5.14) Thus But a straightforward computation leads tō We also obtain where we have used * Therefore, x, X i X α · B X α ,X i , x, X i X α · B X α ,X i , f k;α = X α ,X k + x, B X α ,X k N , f k;αα = ∇ Xα X α ,X k + X α ,∇ Xα X k + X α · B X α ,X k x, N +B X α ,X k X α ,N .
We will make use of the following lemma. Proof. Clearly, B jk;i is symmetric with respect to j and k. It suffices to show its symmetry with respect to i and j. First, (5.28) Since S(T M) is parallel, the two last terms of this equality are zero. Hence, for all i, j, k = 1,...,n, Next, Gauss-Codazzi equation gives The ambient space is L n+2 := R n+2 1 , henceR = 0. As τ = 0, we get that is, Hence, using (2.15), we infer that n α=0 ε αg ∇ Xα U 0 ,X α = − n α=0 B X α ,X α . (5.34) We also compute similar sums for U n+1 and U k : x, X i X k · B X k ,X i . (5.40) If we now put x, X k X k · K    N. , endowed with a parallel screen distribution S(T M) and with symmetric Ricci tensor. Then its Laplacian with respect to the degenerate metric g is given by is called the mean curvature function of M.
6. An application. It is well known that minimal immersions of smooth manifolds M into Euclidean spheres are those immersions whose coordinate functions in the ambient Euclidian space are eigenfunctions of the Laplacian in the induced metric with eigenvalue λ = −dim(M). Then the following general problem comes out in a natural way: can we classify lightlike submanifolds by means of some Laplacian differential equation involving the isometric immersion?