Hypersurfaces of Lorentzian para-Sasakian manifolds

TL;DR

This paper explores the geometric properties of invariant and noninvariant hypersurfaces within Lorentzian para-Sasakian manifolds, establishing conditions for their structures and providing illustrative examples.

Contribution

It introduces new conditions for hypersurfaces to be locally product manifolds and establishes a Lorentzian para-Sasakian structure on invariant hypersurfaces.

Findings

Noninvariant hypersurfaces admit almost product structures.

Conditions for hypersurfaces to be locally product manifolds.

Examples illustrating invariant and noninvariant hypersurfaces.

Abstract

In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We investigate hypersurfaces of affinely cosymplectic and normal (1,1,1) almost contact manifolds. It is proved that a noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold. Some necessary and sufficient conditions have been given for a noninvariant hypersurface of a Lorentzian para-Sasakian manifold to be locally product manifold. We establish a Lorentzian para-Sasakian structure for an invariant hypersurface of a Lorentzian para-Sasakian manifold. Finally we give some examples for invariant and noninvariant hypersurfaces of a Lorentzian para-Sasakian manifold.

Equations

$$\phi ({\xi }_{\alpha })$$

$${\phi }^2$$

$${\eta }^{\alpha }\circ \phi$$

$${\eta }^{\alpha }({\xi }_{\beta })$$

$$rank(\phi )$$

$$\eta (\overline{X})$$

$$\overline{g}(\phi \overline{X},\phi \overline{Y})$$

$$\Phi (\overline{X},\overline{Y})$$

$$({\overline{\nabla }}_{\overline{X}}\Phi )(\overline{Y},\overline{Z})$$

$$\Phi (\overline{X},\overline{Y})=\frac{1}{2}(({\overline{\nabla }}_{\overline{X}}\eta )\overline{Y}+({\overline{\nabla }}_{\overline{Y}}\eta )\overline{X}).$$

$$({\overline{\nabla }}_{\overline{X}}\phi )\overline{Y}=\eta (\overline{Y})\overline{X}+\overline{g}(\overline{X},\overline{Y})\xi +2\eta (\overline{X})\eta (\overline{Y})\xi .$$

$$J^{\prime }(\overline{X},f\frac{d}{dt})=(\phi \overline{X}-f\xi ,\eta (\overline{X})\frac{d}{dt}),$$

$$[\phi ,\phi ]+d\eta \otimes \xi =0,$$

$$[\phi ,\phi ](\overline{X},\overline{Y})=[\phi \overline{X},\phi \overline{Y}]-\phi [\phi \overline{X},\overline{Y}]-\phi [\overline{X},\phi \overline{Y}]+{\phi }^2[\overline{X},\overline{Y}],$$

$$i:M\to \overline{M},$$

$$\phi i_{\ast }X=i_{\ast }JX+\alpha (X)\xi ,$$

$$i_{\ast }X+\eta (i_{\ast }X)\xi =i_{\ast }(J^{2}X)+\alpha (JX)\xi .$$

$$J^{2}X=X$$

$$\alpha (JX)=\eta (i_{\ast }X)=i^{\ast }(\eta X),$$

$$C\alpha =i^{\ast }\eta .$$

$${\overline{\nabla }}_{i_{\ast }X}{i}_{\ast }Y=i_{\ast }{\nabla }_{X}Y+h(X,Y)\xi ,$$

$$S(\overline{X},\overline{Y})=[\phi \overline{X},\phi \overline{Y}]-\phi [\phi \overline{X},\overline{Y}]-\phi [\overline{X},\phi \overline{Y}]+{\phi }^2[\overline{X},\overline{Y}]+d\eta (\overline{X},\overline{Y})\xi ,$$

$$L_{\xi }\phi =0\text{ and }{L}_{\xi }\eta =0,$$

$$S(\overline{X},\overline{Y})$$

$$S(i_{\ast }X,i_{\ast }Y)$$

$$d\alpha (JX,Y)+d\alpha (X,JY)=2C\alpha ([X,Y]).$$

$$({\nabla }_X{i}_{\ast })Y={\overline{\nabla }}_{i_{\ast }X}{i}_{\ast }Y-i_{\ast }({\nabla }_{X}Y),$$

$$({\nabla }_X{i}_{\ast })Y=h(X,Y)\xi ,h(X,Y)=h(Y,X),$$

$${\overline{\nabla }}_{i_{\ast }X}\xi =-i_{\ast }AX+w(X)\xi ,$$

$$({\overline{\nabla }}_{i_{\ast }X}\phi )i_{\ast }Y$$

$$\overline{\nabla }\phi =0\text{, }\overline{\nabla }\eta =0.$$

$$\overline{\nabla }\xi =0.$$

Full text

Hypersurfaces of Lorentzian para-Sasakian Manifolds

Selcen Yüksel Perktaş, Erol Kılıç, Sadık Keleş

Abstract. In this paper, we study the invariant and noninvariant hypersurfaces of $(1,1,1)$ almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an $(1,1,1)$ almost contact manifold admits an almost product structure. We investigate hypersurfaces of affinely cosymplectic and normal $(1,1,1)$ almost contact manifolds. It is proved that a noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold. Some necessary and sufficient conditions have been given for a noninvariant hypersurface of a Lorentzian para-Sasakian manifold to be locally product manifold. We establish a Lorentzian para-Sasakian structure for an invariant hypersurface of a Lorentzian para-Sasakian manifold. Finally we give some examples for invariant and noninvariant hypersurfaces of a Lorentzian para-Sasakian manifold.

M.S.C. 2000. 53C25, 53C42, 53C50.

Keywords. Invariant Hypersurfaces, Non-Invariant Hypersurfaces, $(1,1,1)$ Almost Contact Manifolds, Lorentzian Almost Paracontact manifolds, Lorentzian para-Sasakian Manifolds.

1 Introduction

Hypersurfaces of an almost contact manifold have been studied by Blair [2], Eum [5], Goldberg and Yano [7], Ludden [8] and the others. In 1970, Goldberg and Yano [7] defined noninvariant hypersurfaces of almost contact manifolds. A hypersurface such that the transform of a tangent vector of the hypersurface by the tensor $\varphi$ defining the almost contact structure is never tangent to the hypersurface is called a noninvariant hypersurface of almost contact manifold[7]. The authors [7] showed that a noninvariant hypersurface of an almost contact manifolds admits an almost complex structure and a distinguished 1-form induced by the contact form of the manifold. They also investigated the noninvariant hypersurface of an almost contact metric manifold.

In 1976, Sato [13] studied a structure similiar to the almost contact structure, namely almost paracontact structure. In [1], T. Adati studied hypersurfaces of an almost paracontact manifold. A. Bucki [3] considered hypersurfaces of an almost r-paracontact Riemannian manifold. Some properties of invariant hypersurfaces of an almost r-paracontact Riemannian manifold were investigated in [4] by A. Bucki and A. Miernowski. Moreover, in [10], I. Mihai and K. Matsumoto studied submanifolds of an almost r-paracontact Riemannian manifold of P-Sasakian type.In [6] the authors studied invariant and noninvariant hypersurfaces of almost R-paracontact manifolds. Singh [14] defined $(e_{1},e_{2},r)$ almost contact structure as a generalization of many known structures, which are obtained by taking particular values of $(e_{1},e_{2})$ and $r$ (see also [15]). The study of Lorentzian almost paracontact manifolds was initiated by Matsumoto in 1989 [9]. Also he introduced the notion of Lorentzian para-Sasakian ( for short, LP-Sasakian ) manifold. I. Mihai and R. Rosca [11] defined the same notion independently and thereafter many authors [18, 12, 16, 17] studied LP-Sasakian manifolds and their submanifolds.

In the present paper, we study invariant and noninvariant hypersurfaces of $(1,1,1)$ almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We investigate the invariant hypersurfaces with two different conditions: when the characteristic vector field $\xi$ is everywhere tangent to the hypersurfaces and when the characteristic vector field $\xi$ does not belong to the tangent hyperplane of the hypersurfaces. Section 2 is devoted to preliminaries. In section 3 we show that a noninvariant hypersurface of a $(1,1,1)$ almost contact manifold with the characteristic vector field $\xi$ nowhere tangent to the hypersurface admits an almost product structure. In section 4 we study hypersurfaces of affinely cosymplectic and normal $(1,1,1)$ almost contact manifolds. In section 5 it is proved that a noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold. We also find a necessary and sufficient condition for a noninvariant hypersurface of a Lorentzian para-Sasakian manifold to be locally product manifold. In section 5 we establish a Lorentzian para-Sasakian structure for an invariant hypersurface of a Lorentzian para-Sasakian manifold with the characteristic vector field $\xi$ tangent to the hypersurface. In the last section we give some examples for invariant and noninvariant hypersurfaces of an $(1,1,1)$ almost contact manifold, a Lorentzian almost paracontact manifold and a Lorentzian para-Sasakian manifold.

2 Preliminaries

Let $\overline{M}$ be an n-dimensional differentiable manifold. If there exist a tensor field $\varphi$ of type $(1,1)$, $r$-linearly independent vector fields $\xi_{\alpha}$ and $r$ 1-forms $\eta^{\alpha}$ on $\overline{M}$ such that [15]

[TABLE]

where $e_{1},$ $e_{2}$ take values $\pm 1$ independently, $I$ denotes the identity map of $\Gamma(T\overline{M})$ and $\otimes$ is the tensor product, then the structure $(\varphi,\xi_{\alpha},\eta^{\alpha})$ is said to be an almost $(e_{1},e_{2})$-$r$-contact structure or in short $(e_{1},e_{2},r)$ ac structure and the manifold $\overline{M}$ with the $(e_{1},e_{2},r)$ ac structure is called an $(e_{1},e_{2},r)$ ac manifold.

Let $\overline{M}$ be an $(e_{1},e_{2},r)$ ac manifold. Then the following relations hold on $\overline{M}$ [14]:

[TABLE]

Now, consider that $\overline{M}$ is a $(1,1,1)$ ac manifold. Then $\overline{M}$ admits a Lorentzian metric $\overline{g}$, such that

[TABLE]

for all $\overline{X},\,\overline{Y}\in\Gamma(T\overline{M})$. In this case $\overline{M}$ is said to admit a Lorentzian almost paracontact structure $(\varphi,\xi,\eta,\overline{g}).$ Then we get

[TABLE]

where $\overline{\nabla}$ is the Levi-Civita connection with respect to $\overline{g}$. It is clear that Lorentzian metric $\overline{g}$ makes $\xi$ a timelike unit vector field, i.e, $\overline{g}(\xi,\xi)=-1.$ The manifold $\overline{M}$ equipped with a Lorentzian almost paracontact structure $(\phi,\xi,\eta,\overline{g})$ is called a Lorentzian almost  paracontact manifold (for short $LAP$-manifold) [9, 19].

A Lorentzian almost  paracontact manifold $\overline{M}$ endowed with the structure $(\varphi,\xi,\eta,\overline{g})$ is called a Lorentzian paracontact manifold ( for short LP-manifold) [9] if

[TABLE]

A Lorentzian almost  paracontact manifold $\overline{M}$ endowed with the structure $(\varphi,\xi,\eta,\overline{g})$ is called a Lorentzian para Sasakian manifold ( for short LP-Sasakian) [9] if

[TABLE]

We note that in a LP-Sasakian manifold the $1$-form $\eta$ is closed.

Let $\overline{M}\times R$ be a product manifold, where $\overline{M}$ is an $(1,1,1)$ ac manifold. The tensor field $J^{\prime}$ of type $(1,1)$ on $\overline{M}\times R$ defined by

[TABLE]

where $f$ is a $C^{\infty}$ real-valued function  and $\overline{X}\in\Gamma(T\overline{M}),$ satisfies $J^{\prime 2}=I$ and thus provides an almost product structure on $\overline{M}\times R.$ If the induced almost product structure on $\overline{M}\times R$ is integrable then the $(1,1,1)$ ac structure on  $\overline{M}$ is said to be normal [15]. Since the vanishing of the Nijenhius tensor $[J^{\prime},J^{\prime}]$ is a necessary and sufficient condition for integrability, the condition of the normality in terms of Nijenhius tensor $[\varphi,\varphi]$ of $\varphi$ is (see [15])

[TABLE]

where

[TABLE]

for all $\overline{X},\overline{Y}\in\Gamma(T\overline{M})$.

3 Non-Invariant Hypersurfaces of $(1,1,1)$ ac

Manifolds

Let $\overline{M}$ is a $(1,1,1)$ ac manifold. Consider an $(n-1)$-dimensional manifold $M$ imbedded in $\overline{M}$ with the imbedding map

[TABLE]

and assume that for each $p\in M$ the vector $\xi_{i(p)}$ does not belong to the tangent hyperplane of the hypersurface. Then we have

[TABLE]

where $J$ is a tensor field of type $(1,1)$ , $\alpha$ is a $1$-form on $M$ and  $i_{\ast}$ is the differential of the immersion $i$ of $M$ into $\overline{M}.$ If $\alpha\neq 0$, then the submanifold $i(M)$ is called a noninvariant hypersurface of $\overline{M}$. On the other hand, if the $1$-form $\alpha$ vanishes, $i(M)$ is called an invariant hypersurface of $\overline{M}$ (see [7]). A hypersurface may, of course, be neither invariant nor noninvariant. Throughout this section, unless specified otherwise $i(M)$ will be a noninvariant hypersurface of the $(1,1,1)$ ac manifold $\overline{M}.$

Theorem 3.1

If $M$ is a noninvariant hypersurface of a $(1,1,1)$ ac manifold $\overline{M}$ with $\xi$ nowhere tangent to $M$, then $M$ admits an almost product structure.

Proof. By applying $\varphi$ to (3.1) and using (2.1)-(2.4), we have

[TABLE]

Then from (3.1), we get

[TABLE]

and

[TABLE]

where $X\in\Gamma(TM)$ and $i^{\ast}$ is the dual map of $i_{\ast}.$ Thus $J$ acts as an almost product structure on $M$.

If we define a $1$-form $C\alpha$ on $M$ by $C\alpha(X)=\alpha(JX)$ then from (3.3) we can write

[TABLE]

Thus, the hypersurface $M$ admits a $1$-form $\alpha$ whose vanishing means that the tangent hyperplane of the hypersurface is invariant under $\varphi$.

Now, let $\overline{\nabla}$ be a symmetric affine connection on $\overline{M}$ and define an affine connection $\nabla$ on $M$ with respect to the affine normal $\xi$ by

[TABLE]

where $h$ is a symmetric tensor field of type (0,2) on $M$ which is called the second fundamental form of $M$ with respect to $\xi$.

Suppose that the $(1,1,1)$ ac structure is normal. Then, the torsion field $S$ of type $(1,2)$ on $M$ which is defined by

[TABLE]

for all $\overline{X},\overline{Y}\in\Gamma(T\overline{M}),$ vanishes. By taking $\overline{Y}=\xi$ in (3.5), we get

[TABLE]

where $L_{\xi}$ is the Lie derivative operator with respect to $\xi.$ From (3.5) the tensor field $S$ is also expressed by

[TABLE]

or

[TABLE]

By using (3.1) and (3.4), we obtain

[TABLE]

Therefore, we have

Theorem 3.2

A noninvariant hypersurface of a normal $(1,1,1)$ ac manifold $\overline{M}$ is a locally product manifold which has a $1$-form $\alpha=C^{-1}i^{\ast}\eta$ such that its differential satisfies

[TABLE]

Corollary 3.3

An invariant hypersurface of a $(1,1,1)$ ac manifold is an almost product manifold. If the $(1,1,1)$ ac manifold is normal, then the almost product structure is integrable.

Theorem 3.4

Let $\xi$ be an infinitesimal automorphism of the $(1,1,1)$ ac manifold $\overline{M}$. If, for every noninvariant hypersurface, the induced almost product structure $J$ is integrable and the differential of the induced $1$-form $\alpha=C^{-1}i^{\ast}\eta$ satisfies (3.9) then $\overline{M}$ is normal.

4 Hypersurfaces of affinely cosymplectic and

normal $(1,1,1)$ ac manifolds

Let $\overline{M}$ be a $(1,1,1)$ ac manifold with a symmetric affine connection $\overline{\nabla}$ and $\nabla$ denotes the induced connection on the noninvariant hypersurface $M$. If we write

[TABLE]

then the Gauss and Weingarten equations are

[TABLE]

and

[TABLE]

where $h$ and $A$ are the second fundamental tensors of type $(0,2)$ and $(1,1)$, respectively of $M$ with respect to $\xi,$ and $w$ is a 1-form on $M$ defining the connection on the affine normal bundle.

By using (3.1), (4.2) and (4.3) we get

[TABLE]

Then we will investigate the following two cases:

**Case I : ** Let $\overline{M}$ be an affinely cosymplectic $(1,1,1)$ ac manifold, that is, $\overline{M}$ be a $(1,1,1)$ ac manifold with a symmetric affine connection $\overline{\nabla}$ such that

[TABLE]

From (3) we can easily see that an affinely cosymplectic $(1,1,1)$ ac manifold is normal. Also by using (2.1) and (2.2), we can show that (4.5) implies that

[TABLE]

Therefore, by (4.3), we have

[TABLE]

Moreover, since $\overline{\nabla}\varphi=0$ then from (4) we have

[TABLE]

and

[TABLE]

**Case II : ** Let $\overline{M}$ be a normal $(1,1,1)$ ac manifold such that $\varphi=\overline{\nabla}\xi.$ Then by using (3.1) and (4.3), we have

[TABLE]

that  is

[TABLE]

and

[TABLE]

If $AX=0,$ for all $X\in\Gamma(TM)$, then from (4.3) it is obvious that $\overline{\nabla}_{i_{\ast}X}\xi$ and $\xi$ are propotional. So affine normals are parallel along the hypersurface. In this case, the hypersurface $M$ is said to be totally flat.

Proposition 4.1

Let $M$ be a noninvariant hypersurface of an affinely cosymplectic $(1,1,1)$ ac manifold. Then $M$ is totally flat and

[TABLE]

Corollary 4.2

Let $M$ be an invariant hypersurface of an affinely cosymplectic $(1,1,1)$ ac manifold. Then

[TABLE]

Proposition 4.3

Let $M$ be a noninvariant hypersurface of a normal $(1,1,1)$ ac manifold such that $\varphi=\overline{\nabla}\xi.$ Then

[TABLE]

and

[TABLE]

5 Hypersurfaces of Lorentzian almost

paracontact manifolds

A $(1,1,1)$ ac manifold $\overline{M}$ admitting a Lorentzian metric $\overline{g}$ such that

[TABLE]

where $\overline{X},\overline{Y}\in\Gamma(T\overline{M})$, is called Lorentzian almost paracontact manifold and denoted by $(\overline{M},\varphi,\eta,\overline{g}).$

Proposition 5.1

Let $(M,J,\alpha,g)$ be a noninvariant hypersurface of $(\overline{M},\varphi,\eta,\overline{g})$ where $g$ is the induced metric on $M$, that is, $i^{\ast}\overline{g}=g.$ Then the hypersurface $(M,J,\alpha,g)$ admits an almost product metric

[TABLE]

Proof. From (5.2), we can write

[TABLE]

By using (2.1) in (5.4), we obtain

[TABLE]

The induced metric $g$ on $(M,J,\alpha)$ can be defined by

[TABLE]

So if we use (3.3) and (5.4) in (5.5), then we have

[TABLE]

that is,

[TABLE]

If we denote $g+\alpha\otimes\alpha$ by $G$, the proof is completed.

Corollary 5.2

A noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold.

Now, let define 2-forms

[TABLE]

and

[TABLE]

$\Phi$   and $\ \Omega$ are called the fundamental forms of the Lorentzian almost paracontact manifold $(\overline{M},\varphi,\eta,\overline{g})$ and the submanifold $(M,J,G)$ of $\overline{M}$, respectively. Then we have

Lemma 5.3

Let $\Phi$   and $\ \Omega$ are the fundamental forms of  $(\overline{M},\varphi,\eta,\overline{g})$ and $(M,J,\alpha,G)$, respectively. Then

[TABLE]

Proof. For $X,Y\in\Gamma(TM)$, by using definitions of the fundamental forms, (3.1) and (5.3), we get

[TABLE]

Hence, we obtain

[TABLE]

Theorem 5.4

Let $(M,J,\alpha,G)$ be a noninvariant hypersurface of the Lorentzian para-Sasakian manifold $(\overline{M},\varphi,\eta,\overline{g})$. Then

(a)

$J=-A$,

(b)

$\alpha=w$.

Proof. Since $(\overline{M},\varphi,\eta,\overline{g})$ is a Lorentzian para-Sasakian manifold, we have

[TABLE]

By using (4.3) and (3.1), we get

[TABLE]

which gives

[TABLE]

Theorem 5.5

If $M$ is a noninvariant hypersurface of a Lorentzian para-Sasakian manifold $(\overline{M},\varphi,\eta,\overline{g})$, then

(a)

$(\nabla_{X}J)(Y)=\alpha(Y)JX-C\alpha(Y)X,$**

(b)

$\overline{g}(i_{\ast}X,i_{\ast}Y)+2C\alpha(X)C\alpha(Y)=h(X,JY)+(\nabla_{X}\alpha)(Y)+\alpha(X)\alpha(Y).$**

Proof. By using (3.1) and (4.1) we obatin

[TABLE]

On the other hand, since $(\overline{M},\varphi,\eta,\overline{g})$ is a Lorentzian para-Sasakian manifold, from (2.11) we also have

[TABLE]

By considering $C\alpha=i^{\ast}\eta$ and equating the components of (5) and (5.8), we get (a) and (b) in the theorem.

As an immediate consequence we have the following:

Corollary 5.6

Let $M$ be a noninvariant hypersurface of the Lorentzian para-Sasakian manifold $(\overline{M},\varphi,\eta,\overline{g})$ with the induced almost product structure $J.$ Then $M$ is a locally product manifold if and only if

[TABLE]

Now, let $\overline{M}$ be a (1,1,1) ac manifold and $M$ be an invariant hypersurface of $\overline{M}.$ Assume that for each $p\in M$ the vector $\xi_{i(p)}$ belongs to the tangent hyperplane of the hypersurface. For an invariant hypersurface of a $(1,1,1)$ ac manifold we can write

[TABLE]

where $\psi$ is a tensor of type (1,1) on the hypersurface $M$ and $X\in\Gamma(TM).$ Applying $\varphi$ to the both sides of the equation (5.10), we get

[TABLE]

If we denote

[TABLE]

and

[TABLE]

then we have

[TABLE]

Furthermore,

[TABLE]

and

[TABLE]

that is

[TABLE]

Thus we have,

Theorem 5.7

Let $M$ be an invariant hypersurface of a $(1,1,1)$ ac manifold $(\overline{M},\varphi,\eta,\xi)$ and $\xi\in\Gamma(TM)$. Then $M$ is a $(1,1,1)$ ac manifold with the structure $(\psi,\xi^{\ast},\eta^{\ast})$ where $i_{\ast}\xi^{\ast}=\xi$ and $\eta^{\ast}(X)=\eta(i_{\ast}X)$, for all $X\in\Gamma(TM)$.

Theorem 5.8

Let $M$ be an invariant hypersurface of a $(1,1,1)$ ac manifold $(\overline{M},\varphi,\eta,\xi)$ with $\xi\in\Gamma(TM)$. If $\overline{M}$ is normal, then $M$ is also normal.

Proof. By using (3.5), we can write

[TABLE]

for all $X,Y\in\Gamma(TM)$. If we use (5.10), (5.12) and (5.13) in (5), we get

[TABLE]

Hence, we have the assertion of the theorem.

Theorem 5.9

Let $M$ be an invariant hypersurface of a Lorentzian almost paracontact manifold $(\overline{M},\varphi,\eta,\overline{g})$ where $\xi\in\Gamma(TM)$. Then $M$ is also a Lorentzian almost paracontact manifold.

Proof. From Theorem 5.7. it follows that an invariant hypersurface $M$ of $\overline{M}$ is a $(1,1,1)$ ac manifold with the structure $(\psi,\xi^{\ast},\eta^{\ast}).$ Let $g^{\ast}$ be the induced metric on $M$. Then we have

[TABLE]

Since $\overline{M}$ is a Lorentzian almost paracontact manifold, then by using (5.13) in (5.19) we get

[TABLE]

Moreover,

[TABLE]

which completes the proof.

Theorem 5.10

Let $(\overline{M},\varphi,\eta,\overline{g})$ be a Lorentzian para Sasakian manifold.Then an invariant hypersurface with $\xi\in\Gamma(TM)$ of $\overline{M}$ is also a Lorentzian para Sasakian manifold.

Proof. Let $\overline{M}$ be a Lorentzian para-Sasakian manifold. Then we have

[TABLE]

where $\overline{\nabla}$ is a Levi-Civita connection with respect to $\overline{g}.$ From (5.10) and (5.12), we can write

[TABLE]

By using (3.4) in the last equation, we obtain

[TABLE]

where $\nabla$ is the induced connection on $M$ and $N$ is normal to $M.$ If we consider normal and tangent components of above equation we get

[TABLE]

Since $\overline{M}$ be a Lorentzian para Sasakian manifold from (2.11), we have

[TABLE]

for all $X,Y\in\Gamma(TM).$ By using (5.10), (5.12) and (5.13) in (5.22), we obtain

[TABLE]

On the other hand, from (3.4) and (5.10), one can get

[TABLE]

where $\nabla$ is the induced connection on $M$ and $N$ is normal to $M.$By equating right hand sides of equations (5.23) and (5.24), we have

[TABLE]

This completes the proof.

6 Examples

Example 6.1

Let $\overline{M}$, be the 5-dimensional real number space with a coordinate system $(x,y,z,t,s)$. Defining

[TABLE]

[TABLE]

[TABLE]

the set $(\varphi,\xi,\eta)$ becomes a $(1,1,1)$ ac structure in $\overline{M}$.

Let $M_{1}$ be a hypersurface of $\overline{M}$ which is given by $s=x$ with the imbedding map $i:M_{1}\rightarrow\overline{M}$ . Then

[TABLE]

is a local basis for the tangent hyperplane of $M_{1}$ and $N_{1}=(1,0,0,0,-1)$ is the normal vector field of the hypersurface. It is obvious that the characteristic vector field $\xi_{i(p)}$, $p\in M_{1}$, does not belong to the tangent hyperplane of $M_{1}$. A tangent vector field of the hypersurface can be written by $X\equiv i_{*}X=f_{1}u_{1}+f_{2}u_{2}+f_{3}u_{3}+f_{4}u_{4}$ for some smooth functions $f_{i}$, $1\leq i\leq 4$, on $M$. Then we have

[TABLE]

which shows that $M_{1}$ is a noninvariant hypersurface of $\overline{M}$.

Now let us consider the hypersurface $M_{2}$ of the $(1,1,1)$ ac manifold $\overline{M}$ defining by $x=y$ and let $i:M_{2}\rightarrow\overline{M}$ be the imbedding map of $M_{2}$ into $M$. In this case the set

[TABLE]

is a local basis for the tangent hyperplane and $N_{2}=(1,-1,0,0,0)$ is the normal vector field of $M_{2}$. The characteristic vector field belongs to the tangent hyperplane of the hypersurface. For any tangent vector field $X\equiv i_{*}X=h_{1}v_{1}+h_{2}v_{2}+h_{3}v_{3}+h_{4}v_{4}$ of the hypersurface we have

[TABLE]

where $h_{i}$, $1\leq i\leq 4$, are some smooth functions on $M_{2}$. From (6.1) we see that that $M_{2}$ is an invariant hypersurface of $\overline{M}$.

Example 6.2

Let $\overline{M}$ be the 5-dimensional real number space with a coordinate system $(x,y,z,t,s)$. In $\overline{M}$ we define

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then $(\varphi,\xi,\eta,g)$ is a Lorentzian almost paracontact structure in $\overline{M}$.

Let $M$ be a hypersurface of $\overline{M}$ which is defined by $s=x$ with the imbedding map $i:M\rightarrow\overline{M}$ . Then the set

[TABLE]

is a local basis for the tangent hyperplane of $M$ and $N=(1,0,0,0,-1)$ is the normal vector field of the hypersurface. Since $\xi_{i(p)}=\frac{1}{2}(u_{1}-N)_{i(p)}$, it can be easily seen that the characteristic vector field $\xi_{i(p)}$, $p\in M$, does not belong to the tangent hyperplane of $M$. Moreover, since $\varphi u_{1}=u_{1},\,\varphi u_{2}=u_{2},\,\varphi u_{3}=u_{3},\,\varphi u_{4}=u_{4}$, then $M$ is an invariant hypersurface of $\overline{M}$ with the characteristic vector field $\xi_{i(p)}$, $p\in M$, which does not belong to the tangent hyperplane of the hypersurface.

Example 6.3

Let $\overline{M}$ be the 3-dimensional real number space with a coordinate system $(x,y,z)$. If we define

[TABLE]

[TABLE]

[TABLE]

on $\overline{M}$, then $(\varphi,\xi,\eta,g)$ is a Lorentzian almost paracontact structure in $\overline{M}$.

Assume that $M$ be a surface of $\overline{M}$ given by $x=\arcsin y$ with the imbedding map $i:M\rightarrow\overline{M}$ . Then

[TABLE]

forms a local basis for the tangent plane of $M$ and $N=(\sqrt{1-y^{2}},-1,0)$ is the normal vector field of the surface. For any tangent vector field $X$ of the surface we have

[TABLE]

where $X\equiv i_{*}X=f_{1}u_{1}+f_{2}u_{2}$ for some smooth functions $f_{1},f_{2}$ on $M$. From (6.2) we obtain that $M$ is an invariant surface of $\overline{M}$ with the characteristic vector field $\xi_{i(p)}$, $p\in M$, belonging to the tangent plane of the surface.

Example 6.4

Let $\overline{M}=R^{3}$ be the 3-dimensional real number space with a coordinate system $(x,y,z)$. We define

[TABLE]

Then $(\varphi,\xi,\eta,g)$ is a Lorentzian para-Sasakian structure on $\overline{M}$.

Let $M_{1}$ be a surface of $\overline{M}$ with the imbedding map $i:M_{1}\rightarrow\overline{M}$ which is given by

[TABLE]

Then ${u_{1}=(1,0,1),u_{2}=(0,1,1)}$ is a local basis for the tangent plane of the surface. The vector field

[TABLE]

is a normal vector field of $M_{1}$. Since

[TABLE]

then for each $p\in M_{1}$ the characteristic vector field $\xi_{i(p)}$ does not belong to the tangent plane of the surface. A tangent vector field of the surface can be written by $X\equiv i_{*}X=f_{1}u_{1}+f_{2}u_{2}$ for some smooth functions $f_{1},\,f_{2}$ on $M$. By using (6.3) we have

[TABLE]

From (3.1) and (6.4) we get

[TABLE]

and

[TABLE]

where $J$ acts an almost product structure on $M_{1}$. Consequently, $M_{1}$ is a noninvariant surface of the Lorentzian para-Sasakian manifold $\overline{M}$ with $\xi$ nowhere tangent to $M_{1}$.

Let $M_{2}$ be another surface of $\overline{M}$ which is given by

[TABLE]

Then ${v_{1}=(\frac{1}{1+y^{2}},1,0),v_{2}=(0,0,1)}$ forms a local orthogonal basis for the tangent plane of the surface and

[TABLE]

is a normal vector field of $M_{2}$. It is obvious that the characteristic vector field of the manifold belongs to the tangent plane of $M_{2}$. For any tangent vector field $i_{*}Y\equiv Y$ of the surface where $i:M_{2}\rightarrow\overline{M}$ is an imbedding map into the Lorentzian para-Sasakian manifold $\overline{M}$ we can write $i_{*}Y=\gamma_{1}v_{1}+\gamma_{2}v_{2}$ for some smooth functions $\gamma_{1},\,\gamma_{2}$ on $M_{2}$. By using (6.3) we have

[TABLE]

which shows that $M_{2}$ is a noninvariant surface of the Lorentzian para-Sasakian manifold $\overline{M}$ with $\xi$ tangent to the surface.