KYUNGPOOK Math. J. 2019; 59(4): 783-795  
Some Results on Null Hypersurfaces in (LCS)-manifolds
Samuel Ssekajja
School of Mathematics, University of Witwatersrand, Private Bag 3, Wits 2050, South Africa
e-mail : samuel.ssekajja@wits.ac.za or ssekajja.samuel.buwaga@aims-senegal.org
Received: January 7, 2019; Revised: June 25, 2019; Accepted: October 21, 2019; Published online: December 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

We prove that a Lorentzian concircular structure (LCS)-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there are no totally geodesic ascreen null hypersurfaces of a conformally flat (LCS)-manifold.

Keywords: null hypersurfaces, Ascreen null hypersurfaces, Lorentzian concircular structure manifolds.
1. Introduction

Since the middle of the twentieth century, Riemannian geometry has had a substantial influence on several main areas of mathematical sciences. Primarily, semi-Riemannian (in particular, Lorentzian [3]) geometry has its roots in Riemannian geometry. On the other hand, the situation is quite different for null manifolds with a degenerate metric, since one fails to use, in the usual way, the theory of non-null geometry. Null submanifolds have numerous applications in mathematical physics and General relativity (see [16] for details). This prompted Duggal-Bejancu [6] and Duggal-Sahin [8] to introduce the geometry of null submanifolds. They introduced a non-degenerate screen distribution to construct a null transversal vector bundle which is non-intersecting to its null tangent bundle and developed local geometry of null curves, hypersurfaces, and submanifolds. Their approach is extrinsic contrary to the intrinsic approach of Kupeli [12]. Based on the above books, many authors picked interest in null geometry, for instance, see [1, 2, 5, 9, 10, 11, 13, 15] and many more references cited therein.

In [17, 18], the geometry of Lorentzian concircular structure (LCS)-manifolds is extensively studied. As these spaces are semi-Riemannian, they naturally admit null subspaces. The aim of this paper is to introduce the geometry of null subspaces of (LCS)-manifolds. In particular, we study the geometry of null hypersurfaces of these spaces. Unlike the null hypersurfaces of Sasakian manifolds, null hypersurfaces (LCS)-manifolds are never tangent to the characteristic vector field of (LCS)-manifolds. Moreover, an ascreen null hypersurface admits a symmetric Ricci tensor. In case of a conformally flat (LCS)-manifolds, we prove that no totally geodesic ascreen null hypersurfaces can be admitted. Moreover, a screen conformal or screen totally umbilic null hypersurface is proper totally umbilic. The paper is arranged as follows; In Section 2, we quote some basic notions on (LCS)-manifolds and null hypersurfaces of semi-Riemannian manifolds. Section 3, we prove several new characterization theorems on null hypersurfaces of (LCS)-manifolds.

2. Preliminaries

An (n+2)-dimensional Lorentzian manifold is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric , that is, admits a smooth symmetric tensor field of type (0, 2) such that for each point p, the tensor : Tp×Tp –→ ℝ is a non-degenerate inner product of signature (−,+, . . . , +), where Tp denotes the tangent vector space of at p and ℝ is the real number space. A non-zero vector vTp is said to be timelike (resp., non-spacelike, null, spacelike) if it satisfies p(v, v) < 0 (resp., ≤ 0,= 0,> 0) [6, 8, 16]. Furthermore, the category to which a given vector falls is called its causal character. From now on, we denote by Γ(E) the module of smooth sections of a vector bundle E over .

Let (M̄, ḡ) be a Lorentzian manifold. A vector field V defined by (X, V) = A(X), for any X ∈ Γ(TM̄), is said to be a concircular [17, 18] vector field if, for any X, Y ∈ Γ(TM̄), we have (∇̄X A)Y = α[(X, Y) − ω(X) A(Y)], where α is a non-vanishing smooth function and ω is a closed 1-form. Here, ∇̄ denotes the Levi-Civita connection of with respect to . Suppose that admits a unit timelike concircular vector field ζ, called the characteristic vector field of the manifold. Then, we have

g¯(ζ,ζ)=-1.

Since ζ is a unit concircular vector field, it follows that there exists a non-zero 1-form θ such that for

g¯(X,ζ)=θ(X),XΓ(TM¯),

and the following relation holds

(¯Xθ)Y=α[g¯(X,Y)+θ(X)θ(Y)],

for all X, Y ∈ Γ(TM̄), and α is a non-vanishing smooth function satisfying

Xα=dα(X)=ρθ(X).

Here, ρ is a smooth function given by ρ = −ζα.

Let us put

φ¯X=(1/α)¯Xζ,XΓ(TM¯).

Then, by (2.3) and (2.5), we have

φ¯X=X+θ(X)ζ,

which follows that φ̄ is a symmetric (1, 1) tensor field called the structure tensor field of the manifold. Thus, the Lorentzian manifold together with the unit timelike concircular vector field ζ, its associated 1-form θ and a (1, 1) tensor field φ̄ is said to be a Lorentzian concircular structure manifold (briefly, an (LCS)-manifold) [17, 18]. In particular, if α = 1, then we obtain the LP-Sasakian structure of Matsumoto [14]. In an (LCS)-manifold, the following relations hold for all vector fields X, Y ∈ Γ(TM̄):

φ¯2X=X+θ(X)ζ,φ¯ζ=0,θφ¯=0,θ(ζ)=-1,g¯(φ¯X,φ¯Y)=g¯(X,Y)+θ(X)θ(Y),(¯Xφ¯)Y=α[g¯(X,Y)ζ+2θ(X)θ(Y)ζ+θ(Y)X],¯Xζ=αφ¯X.

Let (M, g) be a null hypersurface of . This means that, at each xM the restriction g = |TxM is degenerate. That is; there exists a non-zero UTxM such that g(U, X) = 0, for all XTxM. Therefore, in null setting, the normal bundle TM of the null hypersurface M 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 and plays an important role in the introduction of the main induced geometric objects on M. The approach of [6, 8], which we adopt here, consist of fixing, on the null hypersurface, a geometric data formed by a null section and a screen distribution. By screen distribution on M, we mean a complimentary bundle of TM in TM. It is then a rank n non-degenerate distribution over M. 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 [12]. We denote by S(TM) the screen distribution over M. Then we have the decomposition

TM=S(TM)TM,

where ⊥ denotes the orthogonal direct sum. From [6] or [8], it is known that for a null hypersurface equipped with a screen distribution, there exists a unique rank 1 vector subbundle tr(TM) of TM̄ over M, such that for any non-zero section ξ of TM on a coordinate neighborhood , there exists a unique section N of tr(TM) on satisfying

g¯(N,ξ)=1,g¯(N,N)=g¯(N,W)=0,WΓ(S(TM)U).

It then follows that

TM¯M=S(TM){TMtr(TM)}=TMtr(TM),

where ⊕ denote the direct (non-orthogal) sum. We call tr(TM) a (null) transversal vector bundle along M. In fact, from (2.11) and (2.12) one shows that, conversely, a choice of a transversal bundle tr(TM) determines uniquely the screen distribution S(TM). A vector field N as in (2.11) is called a null transversal vector field of M. It is then noteworthy that the choice of a null transversal vector field N along M determines both the null transversal vector bundle, the screen distribution S(TM) and a unique radical vector field, say ξ, satisfying (2.11). The name screen distribution is justified as follows; in the case M is a null cone of a 4-dimensional Lorentzian manifold, the integral curves of vector fields in TM are null (lightlike) rays and fibers of S(TM) can be visualized as screen that are transversal to these rays.

Let ∇ and ∇* denote the induced connections on M and S(TM), respectively, and P be the projection of TM onto S(TM), then the local Gauss-Weingarten equations of M and S(TM) are the following [6]

¯XY=XY+B(X,Y)N,¯XN=-ANX+τ(X)N,XPY=X*PY+C(X,PY)ξ,Xξ=-Aξ*X-τ(X)ξ,Aξ*ξ=0,

for all X, Y ∈ Γ(TM), ξ ∈ Γ(TM) and N ∈ Γ(tr(TM)). In the above setting, B is the local second fundamental form of M and C is the local second fundamental form on S(TM). AN and Aξ* are the shape operators on TM and S(TM) respectively, while τ is a 1-form on TM. The above shape operators are related to their local fundamental forms by

g(Aξ*X,Y)=B(X,Y),g(ANX,PY)=C(X,PY),

for any X, Y ∈ Γ(TM). Moreover, g¯(Aξ*X,N)=0, and (AN X, N) = 0, for all X ∈ Γ(TM). From these relations, we notice that Aξ* and AN are both screen-valued operators. Let ϑ = (N, ·) be a 1-form metrically equivalent to N defined on . Take η = i*ϑ to be its restriction on M, where i : M is the inclusion map. Then it is easy to show that

(Xg)(Y,Z)=B(X,Y)η(Z)+B(X,Z)η(Y),

for all X, Y, Z ∈ Γ(TM). Consequently, ∇ is generally not a metric connection with respect to g. However, the induced connection ∇* on S(TM) is a metric connection.

Denote by the curvature tensor of the connection ∇̄ on . Using the Gauss-Weingarten formulae (2.13)–(2.16), we obtain the following curvature relations (see details in [6, 8]).

g¯(R¯(X,Y)Z,ξ)=(XB)(Y,Z)-(YB)(X,Z)+τ(X)B(Y,Z)-τ(Y)B(X,Z),g¯(R¯(X,Y)PZ,N)=(XC)(Y,PZ)-(YC)(X,PZ)-τ(X)C(Y,PZ)+τ(Y)C(X,PZ),

for all X, Y, Z ∈ Γ(TM), ξ ∈ Γ(TM) and N ∈ Γ(tr(TM)).

3. Classification Results

Let us start off with the following important observation. Let (M, g) be a null hypersurface of a (LCS)-manifold (M̄, ḡ). As is Lorenztian, any screen distribution over M is Riemannian. Next, since ζ is a global vector field of , we decompose it as follows;

ζ=W+aξ+bN,

where a and b are smooth functions given by

a=θ(N)and b=θ(ξ),

and W a smooth section tangent to S(TM). Using (3.1) and (2.1), we have g(W, W)+2ab = −1. Suppose that a or b vanishes, then it follows that g(W, W) = −1. Consequently, W is a unit timelike vector field of S(TM). This is a contradiction as S(TM) is Riemannian. Thus, we have the following.

Lemma 3.1

There exist no null hypersurface of a (LCS)-manifold (M̄, ḡ) such that ζ is tangent or transversal to M.

In the theory of null hypersurfaces of Sasakian manifolds, it is possible to select a screen distribution S(TM) containing φ̄TM and φ̄tr(TM) as subbundles. See [8, 11, 10] for details. This does not hold for null hypersurfaces of a (LCS)-manifold (M̄, ḡ). More precisely, we have the following result.

Theorem 3.2

Let (M, g) be a null hypersurface of a (LCS)-manifold (M̄, ḡ). Then φ̄TMS(TM) and φ̄tr(TM) ⊄ S(TM). Moreover, φ̄TMTM = {0} and φ̄tr(TM) ∩ tr(TM) = {0}.

Proof

First, we not that φ̄ξ or φ̄N are non-zero sections. In fact, suppose that φ̄ξ = 0 then using (2.8) and (3.2), we have 0 = (φ̄ξ, φ̄ξ) = b2. This, together with (3.1), suggests that ζ is tangent to M. This is a contraction in view of Lemma 3.1. In the same way, one can show that φ̄N ≠ 0. Next, assume that φ̄ξ belongs to S(TM). Then, by (2.6) and (2.10), we have 0 = (φ̄ξ, ξ) = b2, from which ζ turns out to be tangent to M which is a contradiction by Lemma 3.1. In the same way, if φ̄N belongs to S(TM) we see from (2.11) and (2.6) that 0 = (φ̄N, N) = a2. In this case (3.1) gives ζ = W + bN. As ζ is timelike, we obtain g(W, W) = −1. This is a contradiction as S(TM) is Riemannian. Therefore, φ̄TMS(TM) and φ̄tr(TM) ⊄ S(TM). Next, suppose that φ̄TMTM ≠ {0}. Then there exist a non-vanishing smooth function β such that φ̄ξ = βξ. Applying φ̄ to this equation and using (2.7), we get (β2 − 1)φ̄ξ = . Applying φ̄ to this last relation and considering (2.7), one gets β(β2 − 1)ξ = 0, which implies that β = ±1 since β ≠ 0. Now let β = −1 such that φ̄ξ + ξ = 0. Taking the inner product of this relation with ξ and using (2.6), we get b2 = 0. This is a contradiction since by Lemma 3.1, ζ can not be tangent to M. On the other hand, if β = 1 we get φ̄ξ = ξ. But (2.6) implies φ̄ξ = ξ + . From these two relations we deduce that = 0. Since ζ is a unit timelike vector field, the last relation gives b = 0, which is a contradiction as before. Thus, φ̄TMTM = {0}. In the same way, one can show that φ̄tr(TM) ∩ tr(TM) = {0}, which completes the proof.

Using the language of [11], we will say that a null hypersurface (M, g) of a (LCS)-manifold (M̄, ḡ) is ascreen if the characteristic vector field ζ belongs to S(TM)[= TMtr(TM)]. Equivalently, M is ascreen if W = 0. On an ascreen null hypersurface, the following holds.

Theorem 3.3

Let (M, g) be a null hypersurface of a (LCS)-manifold M̄. Then M is an ascreen null hypersurface of M̄ if and only if φ̄TM = φ̄tr(TM).

Proof

Following the method of [11], suppose that M is ascreen null hypersurface, then (2.1) reduces to ζ = + bN, where a = θ(N) and b = θ(ξ) are non-vanishing smooth functions. Applying φ̄ to this relation and using the fact that φ̄ζ = 0, we get φ̄ξ + bφ̄N = 0. Thus, one gets φ̄ξ = ωφ̄N, where ω=-ba0, a non vanishing smooth function. This implies that φ̄TMφ̄tr(TM) ≠ {0}. Since rank φ̄TM = rank φ̄tr(TM) = 1, it follows that φ̄TM = φ̄tr(TM). Conversely, suppose that φ̄TM = φ̄tr(TM). Then, there exists a non-vanishing smooth function ω such that φ̄ξ = ωφ̄N. Taking the inner product of this relation with respect to φ̄ξ and φ̄N in turn, we get b2 = ω(ab + 1) and ωa2 = ab + 1, respectively. Since ω ≠ 0, we have a ≠ 0, b ≠ 0 and b2 = (ωa)2. The latter gives b = ±ωa. The case b = ωa implies that ab = ωa2 = ab + 1, which is a contradiction. Thus b = −ωa, from which 2ab = −1. Since ω=-ba, a ≠ 0 and φ̄ξ = ωφ̄N, it is easy to see that aφ̄ξ + bφ̄N = 0. Applying φ̄ to this equation, and using b2 = ω(ab + 1) together with 2ab = −1, we get ζ = + bN. Therefore, M is ascreen null hypersurface of , which completes the proof.

Now, we will construct an example of this class of hypersurface in (LCS)-manifold.

Example 3.4

Consider a 3-dimensional manifold = {(x, y, z) ∈ ℝ3}, where (x, y, z) are the standard coordinates in ℝ3. Let {E1, E2, E3} be linearly independent global frame on given by E1 = ez(x+y∂y), E2 = ezy and E3 = e−2zz. Let be the Lorentzian metric defined by (E1, E2) = (E1, E3) = (E2, E3) = 0, (E1, E1) = (E2, E2) = 1 and (E3, E3) = −1. Let θ be the 1-form defined by θ(X) = (X, E3), for any X ∈ Γ(TM̄). Let φ̄ be a (1,1) tensor field defined by φ̄E1 = E1, φ̄E2 = E2 and φ̄E3 = E3. Then using the linearity of φ̄ and we have θ(E3) = −1, φ̄2X = X + θ(X)E3 and (φ̄X, φ̄Y) = (X, Y) + θ(X)θ(Y), for any X, Y ∈ Γ(TM̄). Thus, for ζ = E3, (φ̄, ζ, θ, ḡ) defines a Lorentzian paracontact structure on , which is also a (LCS)-structure. Consequently, (M̄,φ̄, ζ, θ, ḡ) is a (LCS)-manifold with α = e−2z such that = ρθ(X), where ρ = 2e−4z (see [18]). Now, we define a null hypersurface M of (M̄,φ̄, ζ, θ, ḡ) as M = {(x, y, z) ∈ ℝ3 : y = z}. Thus, TM is spanned by U1 = E1 and U2 = y + z, in which TM is spanned by ξ = U2. The transversal bundle tr(TM) is spanned by N, where N = (α/2)(yz). By using the definition of φ̄, we have φ̄N = (α/2)φ̄ξ. Hence, by Theorem 3.3 we have φ̄TM = φ̄tr(TM). Next, applying φ̄ to ζ = + bN and considering the previous relations, we get 2a + αb = 0. Since 2ab + 1 = 0, we obtain ζ=(α/2)ξ-(1/α)N. Therefore, M is an ascreen null hypersurface of .

A null hypersurface (M, g) of a semi-Riemannian manifold (M̄, ḡ) is called totally umbilic [6, p. 106] if there exist a smooth function μ on a coordinate neighborhood such that AE*X=μPX, or equivalently,

B(X,PY)=μg(X,Y),X,YΓ(TM).

In case μ = 0 on , we say that M is totally geodesic otherwise it is proper totally umbilic in . Furthermore, M is screen totally umbilic [6, p. 109] if there exist a smooth function ϱ on a coordinate neighborhood such that ANX = ϱPX, or equivalently,

C(X,PY)=ϱg(X,Y),X,YΓ(TM).

In case ϱ = 0 on , we say that M is screen totally geodesic otherwise it is proper screen totally umbilic in .

Let (M, g) be a null hypersurface. We say that M is screen conformal [8, p. 60] if there exist a non-vanishing smooth function ψ on a coordinate neighborhood such that ANX=ψAE*X, or equivalently,

C(X,PY)=ψB(X,Y),X,YΓ(TM).

In case ψ is a constant function, we say that M is screen homothetic.

Suppose that M is an ascreen null hypersurface of . In view of (3.1), we have ζ = + bN, where a and b are given by (3.2). Differentiating this relation and using (2.14) and (2.16), we get

αφ¯X=-aAξ*X-bANX+[Xa-aτ(X)]ξ+[Xb+bτ(X)]N,

for any X ∈ Γ(TM). Taking the inner product of (3.6) with N and ξ, in turn, we get

Xa-aτ(X)=αη(X)+αaθ(X),         Xb+bτ(X)=αbθ(X),

in which we have used the fact X = PX + η(X)ξ, for any X ∈ Γ(TM). Replacing X with ξ in (3.7), gives

ξa-aτ(ξ)=(1/2)αand ξb+bτ(ξ)=αb2.

On the other hand, taking the inner product of (3.6) with PY, where Y ∈ Γ(TM), we have

aB(X,PY)+bC(X,PY)=-αg(PX,PY).

Notice from (3.9) that C is symmetric on S(TM). Setting X = ξ in (3.9) and using (2.16) together with the fact b ≠ 0, we get

C(ξ,PY)=0,YΓ(TM).

In view of (3.3), (3.4), (3.5), (3.7), (3.9) and (3.10), we have the following result.

Theorem 3.5

Let (M, g) be an ascreen null hypersurface of a (LCS)-manifold M̄. Then the following holds;

(1) zero is an eigenvalue of AN with respect to ξ,

(2) M is a screen integrable null hypersurface and locally isometric to, whereis a null curve tangent to TMand M′ is a leaf of S(TM),

(3) a screen conformal or screen totally umbilic M is proper totally umbilic,

(4) a screen totally geodesic M is never totally geodesic,

(5) there exist no M with a or b constant and τ = 0.

Proof

Parts (1), (3), (4) and (5) are obvious. Part (2) follows from (3.9) and similar arguments as in [7], and the proof is completed.

Remark 3.6

In [11], it was proved (see Theorem 3.4 therein) that a Sasakian manifold does not admit any ascreen null hypersurfaces which are screen totally umbilic or screen conformal. We remark, based on Theorem 3.5, that this is not the case with ascreen null hypersurfaces of (LCS)-manifolds. Futhermore, it was shown, by the same author in Theorem 3.5, that S(TM) is never parallel, and if dim M̄ >3 then S(TM) is not integrable. We note once again that this is not the same with ascreen null hypersurfaces of (LCS)-manifolds.

It is well-known that the Ricci tensor of null hypersurface (and generally of null submanifold) is not symmetric. This is because the induced connection is not a metric connection (see relation (2.18)). In line with the above, the authors in [6, p. 99] (also see [8]) proves the following result.

Theorem 3.7.([6])

Let (M, g) be a null hypersurface of a semi-Riemannian manifold (M̄, ḡ). Then the Ricci tensor of the induced connectionis symmetric, if and only if, each 1-form τ induced by S(TM) is closed, i.e., dτ = 0, on.

With reference to Theorem 3.7, we have the following result.

Theorem 3.8

The Ricci tensor of an ascreen null hypersurface (M, g) of a (LCS)- manifold (M̄, ḡ) is symmetric.

Proof

By the second relation in (3.7) and the fact that b ≠ 0, we have

τ(X)=αθ(X)-Xbb,XΓ(TM).

Differentiating (3.11) and using (2.9) and (2.13), we get

Yτ(X)=(Yα)θ(X)+αθ(YX)+aαB(X,Y)+α2g¯(X,φ¯Y)-Y(Xb)b+(Xb)(Yb)b2,

for any X, Y ∈ Γ(TM). Interchanging X and Y in (3.12) and subtracting (3.12) from the new relation, we get

Xτ(Y)-Yτ(X)=(Xα)θ(Y)-(Yα)θ(X)+αθ([X,Y])+Y(Xb)b-X(Yb)b,

in which we have used the symmetries of B and φ̄. On the other hand, (3.11) gives

τ([X,Y])=αθ([X,Y])-[X,Y]bb.

Then, by (3.13), (3.14) and the definition of , we get

2dτ(X,Y)=[Xτ(Y)-Yτ(X)-τ([X,Y])]=(Xα)θ(Y)-(Yα)θ(X),

in which we have considered the fact Y (Xb) − X(Y b) + [X, Y]b = 0. In view of (2.4), we have = ρθ(X), and hence (3.15) reduces to

2dτ(X,Y)=ρ[θ(X)θ(Y)-θ(Y)θ(X)]=0.

Finally, considering (3.16) and Theorem 3.7, we see that the Ricci tensor of M is symmetric. This completes the proof.

Remark 3.9

A similar conclusion, as in Theorem 3.8, can be arrived at if the first relation of (3.7), i.e., τ (X) = [Xaαη(X) − aαθ(X)]/a is used. In fact, differentiating this relation with respect to Y ∈ Γ(TM) and considering (2.9), (2.13), (2.14) and (2.17), we derive

Yτ(X)=Y(Xa)a-(Xa)(Ya)a2-η(X)Y(αa)-αaη(YX)-αaτ(Y)η(X)+αaC(Y,PX)-aαB(X,Y)-αθ(YX)-α2g¯(X,φ¯Y)-(Yα)θ(X).

Interchanging X and Y in (3.17) and subtracting, while considering the facts B, φ̄ are symmetric and = ρθ(X) (see (2.4)), we get

Xτ(Y)-Yτ(X)=X(Ya)a-Y(Xa)a+Y(αa)η(X)-X(αa)η(Y)-αaη([X,Y])+αaτ(X)η(Y)-αaτ(Y)η(X)+αaC(X,PY)-αaC(Y,PX)-αθ([X,Y]).

On the other hand, using the expression for Xa in (3.7), we derive

Y(αa)η(X)-X(αa)η(Y)=[Yαa-α2aθ(Y)]η(X)-[Xαa-α2aθ(X)]η(Y)+αaτ(X)η(Y)-αaτ(Y)η(X).

Also, by (3.9), we have

τ([X,Y])=[X,Y]aa-αaη([X,Y])-αθ([X,Y])

Thus, putting (3.18), (3.19), (3.20) and (2.4) together, we derive

2dτ(X,Y)=[Yαa-α2aθ(Y)]η(X)-[Xαa-α2aθ(X)]η(Y)+αa[C(X,PY)-C(Y,PX)].=α2-ρa[θ(X)η(Y)-θ(Y)η(X)]+αa[C(X,PY)-C(Y,PX)].

Observe that the right hand side of (3.21) vanishes on M since S(TM) is integrable (see Theorem 3.5) and that C(ξ, PX) = 0, for all X ∈ Γ(TM), by (3.10). That is = 0 and thus, by Theorem 3.8, M admits a symmetric Ricci tensor.

Let = λξ, then it follows that Ñ = (1)N. Moreover, = B and

τ(X)=τ˜(X)+Xln(λ),XΓ(TMU).

This shows that B and τ depend on the section ξ on . By Theorem 3.8 and Poincaré’s lemma we obtain τ (X) = Xf, where f is a smooth function on . Let λ = exp(f) in (3.22), then, τ̃ = 0 on . Thus, we have

Corollary 3.10

Let (M, g) be an ascreen null hypersurface of a (LCS)-manifold (M̄, ḡ). There exist a pair {ξ, N} onsuch that the corresponding 1-form τ from the Weingarten equation vanishes.

A semi-Riemannian manifold (M̄, ḡ) of constant sectional curvature c is called a semi-Riemannian space form (see [16, p. 80]) and denoted by (c). The curvature tensor field of (c) is given by

R¯(X,Y)Z=c{g¯(Y,Z)X-g¯(X,Z)Y},X,Y,ZΓ(TM¯).

Let (c) be a (LCS)-manifold of constant curvature c. Then by Proposition 3.1 of [18, p. 451], c satisfies c = (α2ρ). Using this fact together with (3.9) and (3.23), we get the following result.

Theorem 3.11

Let (M, g) be a screen totally geodesic ascreen null hypersurface of a (LCS)-space form M̄(α2ρ) then α2 = ρ. Moreover, M is a proper totally umbilic null hypersurface whose immersion into M̄ is affinely equivalent to the graph immersion of a certain function F : M −→ ℝ.

Proof

A proof follows easily as in [6].

The notion of quasi-constant curvature was introduced by Chen and Yano [4]. Moreover, it was shown in [18] that a conformally flat (LCS)-manifold (M̄, ḡ) is of quasi-constant curvature and its curvature tensor satisfies

R¯(X,Y,Z,W)=c1[g¯(Y,Z)g¯(X,W)-g¯(X,Z)g¯(Y,W)]+c2[g¯(X,W)θ(Y)θ(Z)-g¯(Y,W)θ(X)θ(Z)+g¯(Y,Z)θ(X)θ(W)-g¯(X,Z)θ(Y)θ(W)],X,Y,Z,WΓ(TM¯),

where c1 and c2(≠ 0) are smooth functions. We denote such a manifold by (c1, c2). The following result holds.

Theorem 3.12

Let (M, g) be an ascreen null hypersurface of a conformally flat (LCS)-manifold M̄(c1, c2) of dimension > 3. Then, M is totally umbilic if and only if the umbiliciy factor μ in (3.3) is a solution of the following differential equations

ξμ+μτ(ξ)-μ2-c2b2=0andPXμ+μτ(PX)=0,XΓ(TM).

Moreover, if M is totally umbilic then it is a screen totally umbilic null hypersurface with screen umbilicity factor ϱ := −(α+)/b in (3.4) satisfying the following pair of differential equations

ξϱ-ϱτ(ξ)-μϱ-c1+c22=0andPXϱ-ϱτ(PX)=0,XΓ(TM).
Proof

Replacing W, Z by ξ, PZ, respectively, in (3.24) and using the fact b = θ(ξ), we get

R¯(X,Y,PZ,ξ)=c2bθ(X)g(Y,PZ)-c2bθ(Y)g(X,PZ),

for all X, Y, Z ∈ Γ(TM). On the other hand, using (2.19), (2.15), (2.16), (2.18) and (3.3), we get

R¯(X,Y,PZ,ξ)=[Xμ+μτ(X)-μ2η(X)]g(Y,PZ)-[Yμ+μτ(Y)-μ2η(Y)]g(X,PZ).

In view of (3.25) and (3.26), we have

[Xμ+μτ(X)-μ2η(X)-c2bθ(X)]g(Y,PZ)=[Yμ+μτ(Y)-μ2η(Y)-c2bθ(Y)]g(X,PZ),

for any X, Y, Z ∈ Γ(TM). Replacing X (or Y) by ξ, we get ξμ+μτ(ξ)−μ2c2b2 = 0. Again replacing X, Y by PX, PY, respectively, in (3.27) and using the fact that S(TM) is non-degenerate, we have

[PXμ+μτ(PX)]PY=[PYμ+μτ(PY)]PX.

Next, suppose that there exist such that PX′μ + μτ(PX′) ≠ 0 at xM. It then follows from (3.28) that all vectors from the fiber S(TM)x are collinear with (PX′)x. This is a contradiction as dim(S(TM))x > 1, which proves the first part of the theorem.

Observe, from (3.9), that M is also screen totally umbilic with ϱ := −(α+)/b. Following the same steps as above, this time with (2.20), (2.15), (2.18), (3.4), (3.10) and (3.24), we get

[Xϱ-ϱτ(X)-ϱμη(X)-c1η(X)-c2aθ(X)]g(Y,PZ)=[Yϱ-ϱτ(Y)-ϱμη(Y)-c1η(Y)-c2aθ(Y)]g(X,PZ),

for all X, Y, Z ∈ Γ(TM). Replacing X (or Y) by ξ in (3.29) and using the fact b = θ(ξ), we get ξϱϱτ(ξ) − ϱμc1c2ab = 0. As M is ascreen, we have 2ab+1 = 0. Considering these two relations, we have ξϱ-ϱτ(ξ)-μϱ-c1+c22=0. The last differential equation follows as in (3.28) and the following arguments. This completes the proof.

It then follows from Theorem 3.12 that the following result holds.

Corollary 3.13

(1) There exist no totally geodesic ascreen null hypersurfaces of a conformally flat (LCS)-manifold M̄(c1, c2).

(2) There exist no screen totally geodesic ascreen null hypersurfaces of a conformally flat (LCS)-manifold M̄(c1, c2) with c1 = 0.

Acknowledgements

The author is very grateful to the reviewer and editor for providing valuable suggestions to improve the quality of this paper. Part of this work was done while the author was still at the University of KwaZulu-Natal.

References
  1. C. Atindogbé. Scalar curvature on lightlike hypersurfaces. Appl Sci., 11(2009), 9-18.
  2. C. Atindogbé, MM. Harouna, and J. Tossa. Lightlike hypersurfaces in Lorentzian manifolds with constant screen principal curvatures. Afr Diaspora J Math., 16(2)(2014), 31-45.
  3. JK. Beem, PE. Ehrlich, and KL. Easley. Global Lorentzian geometry. Second Edition, , Marcel Dekker, New York, 1996.
  4. BY. Chen, and K. Yano. Hypersurfaces of a conformally flat space. Tensor (NS)., 26(1972), 318-322.
  5. J. Dong, and X. Liu. Totally umbilical lightlike hypersurfaces in Robertson-Walker spacetimes. ISRN Geom., (2014):Art. ID 974695, 10 pp.
    CrossRef
  6. KL. Duggal, and A. Bejancu. Lightlike submanifolds of semi-Riemannian manifolds and applications. Mathematics and Its Applications, 364, Kluwer Academic Publishers, Dordrecht, 1996.
  7. KL. Duggal, and DH. Jin. Null curves and hypersurfaces of semi-Riemannian manifolds, , World Scientific, Hackensack, NJ, 2007.
    CrossRef
  8. KL. Duggal, and B. Sahin. Differential geometry of lightlike submanifolds. Frontiers in Mathematics, , Birkhäuser Verlag, Basel, 2010.
  9. M. Hassirou. Kaehler lightlike submanifolds. J Math Sci Adv Appl., 10(1–2)(2011), 1-21.
  10. DH. Jin. Geometry of lightlike hypersurfaces of an indefinite sasakian manifold. Indian J Pure Appl Math., 41(4)(2010), 569-581.
    CrossRef
  11. DH. Jin. Ascreen lightlike hypersurfaces of an indefinite Sasakian manifold. J Korean Soc Math Educ Ser B: Pure Appl Math., 20(1)(2013), 25-35.
    CrossRef
  12. DN. Kupeli. Singular semi-Riemannian geometry. Mathematics and Its Applications, 366, Kluwer Academic Publishers, Dordrecht, 1996.
  13. F. Massamba, and S. Ssekajja. Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds. Arab J Math., 5(2016), 87-101.
    CrossRef
  14. K. Matsumoto. On Lorentzian paracontact manifolds. Bull Yamagata Univ Natur Sci., 12(2)(1989), 151-156.
  15. M. Navarro, O. Palmas, and DA. Solis. Null screen isoparametric hypersurfaces in Lorentzian space forms. Mediterr J Math., 15(2018) Art. 215, 14 pp.
    CrossRef
  16. B. O’Neill. Semi-Riemannian geometry with applications to relativity, , Academic Press, New York, 1983.
  17. AA. Shaikh. On Lorentzian almost paracontact manifolds with a structure of the concircular type. Kyungpook Math J., 43(2003), 305-314.
  18. AA. Shaikh. Some results on (LCS)n-manifolds. J Korean Math Soc., 46(2009), 449-461.
    CrossRef


This Article

Social Network Service

e-submission

Archives

Indexed/Covered by