Induced Riemannian Structures and Topology of Null Hypersurfaces in Lorentzian Manifold

In spite sharing common root, Lorentzian and Riemannian geometries diverge very fast. For example, in Lorentzian case, due to the causal character of three categorie of vector fields (namely, spacelike, timelike and null), the induced metric on a hypersurface is a nondegenerate metric tensor field or degenerate symmetric tensor field depending on whether the normal vector field is of the first two types or the third one. On no-degenerate hypersurfaces one can consider all the fundamental intrinsic and extrinsic geometric notions. In particular, a well defined (up to sign) of the unit orthogonal vector field is known to lead to a canonical splitting of the ambient tangent space into two factors: a tangent and an orthogonal one. Therefore by respective projections, one has fundamental equations such as the Gauss, the Codazzi, the Weingarten equations, along with the second fundamental form, shape operator, induced connection, etc. The case the normal vector field is null, the hypersurface is called null (or lightlike). Null hypersurfaces are then exclusive objects from Lorentzian manifolds, and have not Riemannian counterpart, making them interesting by their own from a geometric point of view, but also they are key objects for modern physics (quantum gravity effects). The geometry of null submanifolds is different and rather difficult since (contrary to the non-degenerate conterpart) the normal vector bundle intersects (non trivially) with the tangent bundle. Thus, one can not find natural projector (and hence there is no preferred induced connection such as Levi-Civita) to define induced geometric objects as usual. This degenerancy of the induced metric makes impossible to study them as part of standard submanifold theory, forcing to develop specific techniques and tools. For the most part, these tools are specific to a given problem, or sometimes with auxiliary non-canonical choices on which, unfortunately, depends the constructed null geometry. Indeed, Duggal and Bejancu introduced a non-degenerate screen distribution or equivalently a null transversal line vector bundle as we may see below so as to get a three factors splitting of the ambient tangent space and derive the main induced geometric objects such as second fundamental forms, shape operators, induced connection, curvature, etc. [1]. Unfortunately, the screen distribution is not unique and there is no preferred one in general. The least we can say is that for the above approach to be complete and consistent, we still need to build a distinguished normaization to accompany it. Most of the recent works of the first named author are indeed devoted to this normalization problem [2-4]. Given the collective expertise in Riemannian geometry, the ideal situation on could expect is that the developed tools could bring to a full reduction of problems in null geometry to purely Riemannian ones. In, the present first named author, after fixing a pair of normalization, constructed an associated Riemannian metric to the “normalized null structure” [5,6]. These ideas have been generalized and improved where authors used riggings defined on neighborhood of the null hypersurface. In the present paper, we first consider the associated Riemannian metric as [5] but arising from a null rigging defined on neighborhood of the null hypersurface, and establish links between the null geometry and basics invariants of the associated Riemannian metric. Also, note that one of the major issues of Riemannian geometry is how to obtain topological or differential properties of a manifold from some known properties of its curvatures. For example what can be said about a complete Riemannian manifold when some suitable estimates are know for the sectional or Ricci curvature? These considerations have been on much scrutinity with excellent results: Myers (compactness), Klingenberg (on the injectivity radius), Cheeger-Gromoll (splitting theorem), ShoenYau (3-manifolds that are diffeomorphic to the standard R3, Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. For further background on this problem we refer to the excellent texts [7-11]. Since in the present paper we have established links between the null geometry and basics invariants of the associated Riemannian metric, it is reasonnable to expect that the geometry of the null hypersurface provides insight informations on its topology. This constitutes our second and main goal. The plan of the article is as follows. Section (2) sets notations and definitions on riggings (normalizations) and review basics properties on null hypersurfaces. The associated Riemannian distance structure on the rigged (or normalized) null hypersurface are introduced and discussed. The relashionship between the null and the associated Riemannian geometry is considered in section (4) where we proceed to a connection of the main geometric objects (invariants) of both side involved in our analysis. In the last sections, thanks to some Riemannian comparison theorems we get some topological facts on the null hypersurfaces from its null geometry.


Introduction
In spite sharing common root, Lorentzian and Riemannian geometries diverge very fast. For example, in Lorentzian case, due to the causal character of three categorie of vector fields (namely, spacelike, timelike and null), the induced metric on a hypersurface is a nondegenerate metric tensor field or degenerate symmetric tensor field depending on whether the normal vector field is of the first two types or the third one. On no-degenerate hypersurfaces one can consider all the fundamental intrinsic and extrinsic geometric notions. In particular, a well defined (up to sign) of the unit orthogonal vector field is known to lead to a canonical splitting of the ambient tangent space into two factors: a tangent and an orthogonal one. Therefore by respective projections, one has fundamental equations such as the Gauss, the Codazzi, the Weingarten equations, along with the second fundamental form, shape operator, induced connection, etc. The case the normal vector field is null, the hypersurface is called null (or lightlike). Null hypersurfaces are then exclusive objects from Lorentzian manifolds, and have not Riemannian counterpart, making them interesting by their own from a geometric point of view, but also they are key objects for modern physics (quantum gravity effects). The geometry of null submanifolds is different and rather difficult since (contrary to the non-degenerate conterpart) the normal vector bundle intersects (non trivially) with the tangent bundle. Thus, one can not find natural projector (and hence there is no preferred induced connection such as Levi-Civita) to define induced geometric objects as usual. This degenerancy of the induced metric makes impossible to study them as part of standard submanifold theory, forcing to develop specific techniques and tools. For the most part, these tools are specific to a given problem, or sometimes with auxiliary non-canonical choices on which, unfortunately, depends the constructed null geometry. Indeed, Duggal and Bejancu introduced a non-degenerate screen distribution or equivalently a null transversal line vector bundle as we may see below so as to get a three factors splitting of the ambient tangent space and derive the main induced geometric objects such as second fundamental forms, shape operators, induced connection, curvature, etc. [1]. Unfortunately, the screen distribution is not unique and there is no preferred one in general. The least we can say is that for the above approach to be complete and consistent, we still need to build a distinguished normaization to accompany it. Most of the recent works of the first named author are indeed devoted to this normalization problem [2][3][4]. Given the collective expertise in Riemannian geometry, the ideal situation on could expect is that the developed tools could bring to a full reduction of problems in null geometry to purely Riemannian ones. In, the present first named author, after fixing a pair of normalization, constructed an associated Riemannian metric to the "normalized null structure" [5,6]. These ideas have been generalized and improved where authors used riggings defined on neighborhood of the null hypersurface. In the present paper, we first consider the associated Riemannian metric as [5] but arising from a null rigging defined on neighborhood of the null hypersurface, and establish links between the null geometry and basics invariants of the associated Riemannian metric. Also, note that one of the major issues of Riemannian geometry is how to obtain topological or differential properties of a manifold from some known properties of its curvatures. For example what can be said about a complete Riemannian manifold when some suitable estimates are know for the sectional or Ricci curvature? These considerations have been on much scrutinity with excellent results: Myers (compactness), Klingenberg (on the injectivity radius), Cheeger-Gromoll (splitting theorem), Shoen-Yau (3-manifolds that are diffeomorphic to the standard R 3 , Gromov's estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. For further background on this problem we refer to the excellent texts [7][8][9][10][11]. Since in the present paper we have established links between the null geometry and basics invariants of the associated Riemannian metric, it is reasonnable to expect that the geometry of the null hypersurface provides insight informations on its topology. This constitutes our second and main goal. The plan of the article is as follows. Section (2) sets notations and definitions on riggings (normalizations) and review basics properties on null hypersurfaces. The associated Riemannian distance structure on the rigged (or normalized) null hypersurface are introduced and discussed. The relashionship between the null and the associated Riemannian geometry is considered in section (4) where we proceed to a connection of the main geometric objects (invariants) of both side involved in our analysis. In the last sections, thanks to some Riemannian comparison theorems we get some topological facts on the null hypersurfaces from its null geometry.
M g ∇ denotes the connection on M induced from ∇ through the projection along the null rigging N, ∇ * denotes the Levi-Civita connection on the screen distribution (N) induced from∇ through the projection morphism P of Γ(TM) onto Γ((N)) with respect to the decomposition. Now the (0,2) tensor B N and C N are the second fundamental forms on TM and (N) respectively, A N and A ξ  are the shape operators on TM and (N) respectively and N a 1-form on TM defined by For the second fundamental B N and C N the following hold: It follows from eqn. (10) that integral curves of ξ are pregeodesicin both M and M as consider these integral curves to be geodesics which means that A null hypersurface M is called totally umbilical (resp. geodesic) if there exists a smooth function ρ on M such that at each p∈M and for all u,v∈T p M BN(u,v) p =(p)g p (u,v)resp B N vanishes identicallynon M). These are intrinsic notions on any null hypersurface in the following way. Note that N being a null rigging for M, a vector field  ( ) Then we have for details on changes in normalizations) which shows that total umbilicity and totally geodesibility are intrinsic properties for M [3]. The total umbilicity and the total geodesibility conditions for M can also be written respectively as . Also, the screen distribution (N) is totally umbilical (resp. totally geodesic) if for all X,Y∈Γ(TM) resp. C N =0) which is equivalent to . It is noteworthy

Riggins and Preliminaries of Null Hypersurfaces
Let ( , ) M g be a (n+2)-dimensional Lorentzian manifold and M a null hypersurface in M . This means that at each p∈M, the restriction for X∈T P M. Hence, in null setting, the normal bundle TM ⊥ of the null hypersurface M n+1 is a rank 1 vector subbundle of the tangent bundle TM, contrary to the classical theory of non-degenerate hypersurfaces for which the normal bundle has trivial intersection {0} with the tangent one plays an important role in the introduction of the main induced geometric objects on M. Let start with the usual tools involved in the study of such hypersurfaces according to Duggal [1]. They consist in fixing on the null hypersurface a geometric data formed by a lightlike section and sreen distribution. By screen distribution on M n+1 , we mean a complementary bundle of TM ⊥ in TM. In fact, there are infinitely many possibilities of choices for such a distribution provided the hypersurface M be paracompact, but each of them is canonically isomorphic to the factor vector bundle / TM TM ⊥ . For reasons that will become obious in few lines below, let denote such a distribution by S(N). We then have, Where orth ⊕ denotes orthogonal direct sum. From ref. [1], it is know that for a null hypersurface equipped with aa screen distribution, there exists a unique rank 1 vector subbundle ( ) tr TM of T M over M, such that for any non-zero section ξ of TM ⊥ on coordinate neighborhood  ⊂ M there exists a unique section N of tr(TM) on  satisfying: Then T M is decomposed as following: We call tr(TM) a (null) transversal vector bundle along M. In fact, from eqns. (2) and (3) one shows that, conversely, a choice of a transversal bundle tr(TM) determines uniquely the distribution  (N). A vector field N as in eqn. (2) is called a null transversal vector field of M. It is then noteworthy that the choice of null transversal vector field N along M determines both the null transversal vecto r bundle, the screen distribution  and a unique radical vector field, say satisfying eqn. (2). Now, to continue our discussion, we need to clarify the concept of rigging for our null hypersurface. We say that we have a null rigging in case the restriction of L to the null hypersurface is a null vector field. From now on we fix a null rigging N for M. In particular this rigging fixes a unique null vector field  to mention that the shape operators star A ξ and A N are (N)-valued. The induced connection ∇ is torsion-free, but not necessarily g-metric unless M be totally geodesic. In fact we have for all tangent vector fields X,Y and Z∈TM:

Let M be a null hypersurface of a Lorentzian manifold. A rigging for M is a vector field L defined on some open set containing M such that
Let denote by R and R the Riemann curvature tensors of ∇ and ∇, respectively. Recall the following Gauss-Codazzi equations for all , , The shape operator A ξ  is self-adjoint as the second fundamental form B N is symmetric. However, this is not the case for the operator A N as show in the following lemma.

Proof.
Recall that η=i*θ where = ,. N θ 〈 〉 taking the differential of θ and using the weingarten formula, we have for all X,Y∈ (TM) as anounced. In case the normalization is closed the (connection)1form τ N is related to the shape operator of M as follows. .
Proof. Assume η=i * θ closed and let X,Y be tangent vector fields to M. The condition . Then by the weingarten formula, we get In this relation, take Y=ξ to get which gives the desired formula as τ N (ξ)=0

Exemple 2.1
In the pseudo-Euclidean space . It is easy to check that such a hypersurface is null if and only if F is a solution of the partial differential equation (Which we assume from now on) and that the rank one normal bundle TM ⊥ is spanned by the global vector field Then, the vector field defined by is null rigging for the null Monge hypersurface M. For this null rigging, we have τ N =0Indeed, let ∇ denote the Levi-Civita connection of 2 n q R + and XΓ(TM) a tangent vector field. We have: Now, we show that the normalization given by eqn. (22) is closed (and in particular, the distribution (N) is integrable). The 1-forme θ metrically equivalent to N is given by and η=i*θ (i the inclusion map). Let X Y be two vector fields on M smoothly exetended in two vector fields on 2 n q R + (we also denote these extensions by X and Y). We have (as A ξ  is symmetric and τ N =0) which shows that η is closed. On can observe that this rigging induces a conformal screen (N) and that

Riemannian Distance Associated
We define the η-arc length of γ∈Ω p,q by From the above facts and using standard techniques as in Riemannian setting, and noting that a tangent vector X belongs to (N) if and only if η(X)=0, one gets the following lemma.
We derive the following result on the compatibility condition.

Theorem 3.1
Let (M,g,N) be a normalized null hypersurface of a pseudo-Riemannian manifold ( , ) M g [4]. The induced connexions ∇ and the Levi-Civita connexion ∇ η of the associate metric g η on (M,g,N) agree if and only if for all X,Z∈Γ(TM),

Definition 3.2
A normalized null hypersurface (M,g,N) of a pseudo-Riemannian manifold ( , ) M g is said to have a conformal screen if there exists a non vanishing smooth function ϕ on M such that This is equivalent to saying that for all tangent vector fields X and Y. The function ϕ is called the conformal factor. Theorem (3.1) asserts that the compatibility condition is fulfilled if and only if the normalization is screen conformal with constant conformal factor 1 and vanishing normalizing 1-form τ N .

Remark 3.1
Observe that in ambient Lorentzian case, the Riemannian distance that is  0 (1) coincides the unit bundle of M with respect to the associated Riemannian metric g η from the normalization. It also hods that for all

Relation Between the Null and Associated Riemannian Geometry
Connecting the covariant derivatives In order to relate the main geometric objects of both null and associated non-degenerate geometry on the null hypersurface, we first need to relate the covariant derivatives ∇ η and ∇. In this respect, we prove the following.
In particular for a closed normalization, and for all X∈Γ(TM), we set where , 〈 〉 stands for both g or g. Observe that, A remarkable fact is that due to the degenerancy of the induced metric g on the null hypersurface M, it is not possible to define the natural dual (musical) isomorphisms # and # batween the tangent vector bundle TM and its dual T*M following the usual Riemannian way. However, this construction is made possible by setting a rigging (normalization) N (we refer to ref. [5] for further details). Consider a normalized null hypersurface (M,g,N) and we define one form define by: Cleary, such a # is an isomorphism of Γ(TM) on to Γ(T * M) and can be used to generalize the usual non-degenerate theory. In the latter case,Γ((N)) coincides with Γ(TM), and as a consequence the 1-forme vanishes identically and the projection morphism P becomes the identity map on Γ(TM). Let # η denote the inverse of the isomorphism # given by eqn. (24). For X∈Γ(TM) (resp. # * )), w T M X η ∈ (resp. # w η ) is called the dual 1-form of X (resp. the dual vector field of w) with respect to the degenerate metric g. It follows from eqn. (24) that if w is a 1-form on M, we have for X∈Γ(TM), Define a (0,2)-tensor g η by Cleary, g η defines a non-degenerate metric on M which plays an important role in defining the usual differential operators gradient, divergence, Laplacien with respect to degenerate metric g on null hypersurface [5]. It is called the associate metric to g on ( , , ) M g N . Also, observe that innon-degenerate case, the two metrics g η and g coincide. The (0,2)-tenseur In particular, ( , ) = 1 g η ξ ξ and the last equality in eqn. (26) is telling us that the restrict to (N) the metrics g and g coincide. We know from eqn. (12) that the induced metric g is not compatible with the induced connection in general and this compatibility arises if and ony the null hypersurface M is totally geodesic in M . Let ∇ η denote the Levi-Civita connection of the nondegenerate associate metric g η on (M,g,N). We are now interested in characterizing the normalizations for which the Levi-Civita connection ∇ η of g agrees with the induced connection ∇ due to N, i.e ∇ η =∇ For this we recall the following.
Proof. Both connections∇ η and ∇ are torsion free. the we can write with  is a symmetric tensor. As ∇ η is g η -metric, we have Then using this and Lemma (3.2), we have

By circular permutation we get similar expression for
. Summing the first two expressions minus the last one leads to Now using eqn. (26) and (16), we get It follows the non-degenerancy of g η that which, using the fact that the operators A ξ gives the desired formulas. with

From above proposition (4.1), it follows that if the screen distribution is integrable (which is equivalent to the symmetry of C N on  (N)× (N)
we have for all X,Y∈ (N) Throughout, the normalization wil be assumed closed. Beyond its technical aspect this assumption guarentee integrability of thee screen distribution  (N).

Some curvature relations
In this section we relate various curvature tensors of the null geometry of (M,g,N) to those of the associated non-degenerate metric g η on M. Let R and R denote the Riemann curvature tensors of ∇ η and ∇ respectively. Using proposition (4.1) we prove the following. (M,g,N) be a closed normalized null hypersurface with rigged vector field ξ. Then, for all X,Y,ZΓ(TM), the following hold.

Remark 4.2
Note that by lemma (2.2), for a closed and conformal normalization (with factor ϕ we have In the following we let Ric η and Ric denote the Ricci curvature of ∇ η and ∇ respectively. Recall that this is a symmetric (0,2)-tensor on TM. unfortunately, the corresponding quantity Ric(X,Y) obtained from ∇ is no longer symmetric in general, due to the fact that the induced Riemann curvature R on the normalized null hypersurface (M,g,N) fails to have the usual algebraic curvature symmetries in general. Precisely, the induced Ricci tensor Ric is given by Ric X Y denotes the Ricci curvature of the ambient manifold. We define the (0,2)-symmetrized Ricci tensor Ric 0 on the null hypersurface by is induced Ricci tensor curvature on a null hypersurface. But By substituting previous terms in the above expression of Ric η (X,Y) we get the desired formula.

Corollary 4.2
Let (M,g,N) be a closed normalized of a null hypersurface with rigged vector field and τ N (ξ)=0 and (0,2)-symmetrized Ricci tensor Ric 0 on null hypersurface in a ( 2)â n + pseudo-Riemannian manifold with constant curvature k. Then,

Theorem 4.3
Let (M,g,N) be a closed normalized null hypersurface with rigged vector field ξ and τ N (ξ)=0 in a ( 2)â n + pseudo-Riemannian manifold. Then Proof. To get eqn. (45), take Y= in the (4.1). Recall that from a geometric point of view and in practice, one gets the scalar curvature by contracting with a (non-degenerate) metric the (symmetric) Ricci curvature. It turns out that in the null geometry setting, such a scalar quantity cannot be calculated by the usual way (degenerancy of the induced metric and the failure of symmetry in the induced Ricci curvature) [12]. This justify introduction of a symmetrized Ricci curvature tensor Ric 0 and our use of the associated non-degenerate metric g η in calculating this scalar quantity. More precisely, the extrinsic scalar curvature r 0 on the rigged null hypersurface (M,g,N) is given by g η -trace of the symmetrized Ricci curvature Ric 0 . With respect to a local quasiorthonormal frame Now let r n denote the scalar curvature of the non-degenerate metric g η on M that is the contraction of Ric η with respect to g. In the following, we state a formula relating the extrinsic scalar curvature r 0 to the associated scalar curvature r n Theorem 4.4 Let (M,g,N) be a closed normalized null hypersurface with rigged vector field ξ and τ N (ξ)=0 in a pseudo-Riemannian manifold. Then Proof. We have  Then, we have the following [1].

Corollary 4.3
Let (M,g,N) be a closed normalized null hypersurface of a (n+2)dimensional Lorentzian space and τ(ξ)=0. Then Where X and Y are ortogonal in  (N) and π=span{X,Y}

Proof.
Observe that a plane π⊂(N) is both non-degenerate with respect to g and g (simultaneously) or not. Now, eqn. (51) is a direct use of eqn. (37) in the eqn. (49), taking into account the fact without loss of generality, we have assumed X and Y g η -unit and orthogonal in (N) (and hence also for g).
, we infer that ( , ) = ( , ) = 1 Here, for a vector field X, we brief X s and X 0 for PX and η(X)ξ respectively, where P is the morphism projection of TM onto (N) and then, X=X 0 +X 5 Using eqn. (38), we get Now, we recall that the ambient is Lorentzian space form with curvature (=k). Then, by Gauss-Codazzi equations an being in mind 0 , (1), , it is easy to chek that In particular, for if the screen distribution is conformal with conformal factor , then Hence, set Y=ξ to get: i.e τ N (X)=0, which shows that τ N vanishes on M. Then we get by the closed assumption