### Article

KYUNGPOOK Math. J. 2019; 59(4): 783-795

**Published online** December 23, 2019

Copyright © Kyungpook Mathematical Journal.

### 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

### Abstract

We prove that a Lorentzian concircular structure (

**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 (

### 2. Preliminaries

An (_{p}_{p}_{p}_{p}_{p}

Let (_{X}

Since

and the following relation holds

for all

Here,

Let us put

Then, by (

which follows that

Let (_{TxM} is _{x}_{x}^{⊥} of the null hypersurface ^{⊥} in ^{⊥} [12]. We denote by

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 ^{⊥} on a coordinate neighborhood , there exists a unique section

It then follows that

where ⊕ denote the direct (non-orthogal) sum. We call ^{⊥} are null (lightlike) rays and fibers of

Let ∇ and ∇^{*} denote the induced connections on

for all ^{⊥}) and _{N}

for any _{N}_{N}^{*}

for all ^{*} on

Denote by

for all ^{⊥}) and

### 3. Classification Results

Let us start off with the following important observation. Let (

where

and

### Lemma 3.1

In the theory of null hypersurfaces of Sasakian manifolds, it is possible to select a screen distribution ^{⊥} and

### Theorem 3.2

^{⊥} ⊄ ^{⊥} ∩ ^{⊥} = {0}

**Proof**

First, we not that ^{2}. This, together with (^{2}, from which ^{2}. In this case (^{⊥} ⊄ ^{⊥} ∩ ^{⊥} ≠ {0}. Then there exist a non-vanishing smooth function ^{2} − 1)^{2} − 1)^{2} = 0. This is a contradiction since by Lemma 3.1, ^{⊥} ∩ ^{⊥} = {0}. In the same way, one can show that

Using the language of [11], we will say that a null hypersurface (^{⊥}[= ^{⊥} ⊕

### Theorem 3.3

^{⊥} =

**Proof**

Following the method of [11], suppose that ^{⊥} ∩ ^{⊥} = rank ^{⊥} = ^{⊥} = ^{2} = ^{2} = ^{2} = (^{2}. The latter gives ^{2} = ^{2} =

Now, we will construct an example of this class of hypersurface in (

### Example 3.4

Consider a 3-dimensional manifold ^{3}}, where (^{3}. Let {_{1}, _{2}, _{3}} be linearly independent global frame on _{1} = ^{−}^{z}_{x}_{y}_{2} = ^{−}^{z}_{y}_{3} = ^{−2}^{z}_{z}_{1}, _{2}) = _{1}, _{3}) = _{2}, _{3}) = 0, _{1}, _{1}) = _{2}, _{2}) = 1 and _{3}, _{3}) = −1. Let _{3}), for any _{1} = _{1}, _{2} = _{2} and _{3} = _{3}. Then using the linearity of _{3}) = −1, ^{2}_{3} and _{3}, (^{−2}^{z}^{−4}^{z}^{3} : _{1} = _{1} and _{2} = _{y}_{z}^{⊥} is spanned by _{2}. The transversal bundle _{y}_{z}^{⊥} =

A null hypersurface (

In case _{N}

In case

Let (

In case

Suppose that

for any

in which we have used the fact

On the other hand, taking the inner product of (

Notice from (

In view of (

### Theorem 3.5

(1) _{N}

(2) ^{⊥}

(3)

(4)

(5)

**Proof**

Parts (1), (3), (4) and (5) are obvious. Part (2) follows from (

### 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 (

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 (

### Theorem 3.7.([6])

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

### Theorem 3.8

**Proof**

By the second relation in (

Differentiating (

for any

in which we have used the symmetries of

Then, by (

in which we have considered the fact

Finally, considering (

### Remark 3.9

A similar conclusion, as in Theorem 3.8, can be arrived at if the first relation of (

Interchanging

On the other hand, using the expression for

Also, by (

Thus, putting (

Observe that the right hand side of (

Let

This shows that

### Corollary 3.10

A semi-Riemannian manifold (

Let ^{2} −

### Theorem 3.11

^{2} − ^{2} =

**Proof**

A proof follows easily as in [6].

The notion of

where _{1} and _{2}(≠ 0) are smooth functions. We denote such a manifold by _{1}, _{2}). The following result holds.

### Theorem 3.12

_{1}, _{2})

**Proof**

Replacing

for all

In view of (

for any ^{2}−_{2}^{2} = 0. Again replacing

Next, suppose that there exist such that _{x}_{x}_{x}

Observe, from (

for all _{1} − _{2}

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

### Corollary 3.13

(1) _{1}, _{2}).

(2) _{1}, _{2}) _{1} = 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

- C. Atindogbé.
Scalar curvature on lightlike hypersurfaces . Appl Sci.,11 (2009), 9-18. - 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. - JK. Beem, PE. Ehrlich, and KL. Easley. Global Lorentzian geometry. Second Edition,
, Marcel Dekker, New York, 1996. - BY. Chen, and K. Yano.
Hypersurfaces of a conformally flat space . Tensor (NS).,26 (1972), 318-322. - J. Dong, and X. Liu.
Totally umbilical lightlike hypersurfaces in Robertson-Walker spacetimes . ISRN Geom.,(2014):Art. ID 974695, 10 pp. - KL. Duggal, and A. Bejancu.
Lightlike submanifolds of semi-Riemannian manifolds and applications . Mathematics and Its Applications,364 , Kluwer Academic Publishers, Dordrecht, 1996. - KL. Duggal, and DH. Jin. Null curves and hypersurfaces of semi-Riemannian manifolds,
, World Scientific, Hackensack, NJ, 2007. - KL. Duggal, and B. Sahin.
Differential geometry of lightlike submanifolds . Frontiers in Mathematics,, Birkhäuser Verlag, Basel, 2010. - M. Hassirou.
Kaehler lightlike submanifolds . J Math Sci Adv Appl.,10 (1–2)(2011), 1-21. - DH. Jin.
Geometry of lightlike hypersurfaces of an indefinite sasakian manifold . Indian J Pure Appl Math.,41 (4)(2010), 569-581. - 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. - DN. Kupeli.
Singular semi-Riemannian geometry . Mathematics and Its Applications,366 , Kluwer Academic Publishers, Dordrecht, 1996. - F. Massamba, and S. Ssekajja.
Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds . Arab J Math.,5 (2016), 87-101. - K. Matsumoto.
On Lorentzian paracontact manifolds . Bull Yamagata Univ Natur Sci.,12 (2)(1989), 151-156. - M. Navarro, O. Palmas, and DA. Solis.
Null screen isoparametric hypersurfaces in Lorentzian space forms . Mediterr J Math.,15 (2018) Art. 215, 14 pp. - B. O’Neill. Semi-Riemannian geometry with applications to relativity,
, Academic Press, New York, 1983. - AA. Shaikh.
On Lorentzian almost paracontact manifolds with a structure of the concircular type . Kyungpook Math J.,43 (2003), 305-314. - AA. Shaikh.
Some results on (LCS) . J Korean Math Soc.,_{n}-manifolds46 (2009), 449-461.