## 1 Introduction

Contact structure has most important applications in physics. Many authors gave their valuable and essential results on differential geometry [2], [7], [8]. Firstly contact manifolds were defined by Boothby and Wang [6]. In 1959, Gray defined almost contact manifold by the condition that the structural group of the tangent bundle is reducible to *U*(*n*) *×* 1 [8]. Sasakian introduced Sasaki manifold, which is an almost contact manifold with a special kind a Riemannian metric [15]. Compared to that Sasakian manifolds have only recently become subject of deeper research in mathematics, mechanics and physics [3, 18]. To study manifolds with negative curvature, Bishop and O’Neill introduced the notion of warped product as a generalization of Riemannian product [4]. In 1960’s and 1970’s, when almost contact manifolds were studied as an odd dimensional counterpart of almost complex manifolds, the warped product was used to make examples of almost contact manifolds [18]. In addition, S. Tanno classified the connected (2*n* + 1) dimensional almost contact manifold *M* whose automorphism group has maximum dimension (*n* + 1)^{2} in [18]. For such a manifold, the sectional curvature of plane sections containing *ξ* is a constant, say *c*. Then there are three classes:

*i)**c >*0, M is homogeneous Sasakian manifold of constant holomorphic sectional curvature.*ii)**c*= 0, M is the global Riemannian product of a line or a circle with a K*ä*hler manifold of constant holomorphic sectional curvature.*iii)**c <*0, M is warped product space ℝ ×ℂ_{f}^{n}.

Kenmotsu obtained some tensorial equations to characterize manifolds of the third case.

In 1972, Kenmotsu abstracted the differential geometric properties of the third case. In [9], Kenmotsu studied a class of almost contact Riemannian manifold which satisfy the following two condition,

He showed normal an almost contact Riemannian manifold with (1.1) but not quasi Sasakian hence not Sasakian. He characterized warped product space *L* ×* _{f}* ℂ

*E*by an almost contact Riemannian manifold with (1.1). Moreover, he showed that every point of an almost contact Riemannian manifold with (1.1) has a neighborhood which is a warped (−

^{n}*ɛ,ɛ*) ×

*where*

_{f}V*f*(

*t*) =

*ce*and

^{t}*V*is Kähler.

In 1981, Janssens and Vanhecke [10], an almost contact metric manifold satisfiying this (1.1) is called a Kenmotsu manifold. Some authors studied Kenmotsu manifold [1], [12], [13], [16], [19].

At the same time, in the year 1969, Takahashi [17] has introduced the Sasakian manifolds with Pseudo-Riemannian metric and prove that one can study the Lorentzian Sasakian structure with an indefinite metric. Furthermore, in 1990, K. L. Duggal [7] has initiated the space time manifolds with contact structure and analyzed the paper of Takahashi. In 1991, Roşça introduced Lorentzian Kenmotsu manifold [14].

Our aim in the present note is to extend the study of some properties curvature to the setting of a Lorentzian Kenmotsu manifod. We first rewiev, in section 2, basic formula and definition of aLorentzian Kenmotsu manifold. In section 3, we introduced *ℒ* - sectional curvature of Lorentzian Kenmotsu manifold. In section 4, we call semi invariant submanifold of Lorentzian Kenmotsu manifold. In section 5, we study semi invariant submani-fold of Lorentzian Kenmotsu space form, In last section, we investigate semi invariant products of a Lorentzian Kenmotsu manifold.

## 2 Lorentzian Kenmotsu Manifolds

Let *M* be a real (2*n* + 1)− dimensional differentiable manifold endowed with an almost contact structure (*ϕ*,*η*,*ξ*), where *ϕ* is a tensor field of type (1,1), *η* is a 1−form, and *ξ* is a vector field on *M* satisfying

*M*is called an

*almost contact manifold*. It follows that

*ϕ*(

*ξ*) = 0,

*η ○ ϕ*= 0 ,

*rankϕ*= 2

*n*. If there exists a semi-Riemannian metric

*g*satisfying

*ϕ,η,ξ,g*) is called a

*Lorentzian almost contact structure*and

*M*is said to be a

*Lorentzian almost contact manifold*.

For a Lorentzian almost contact manifold we also have *η*(*X*) = *ɛg*(*X*,*ξ*). We note that *ξ* is neither a light-like nor a spacelike vector fields on *M*. We note that *ξ* is the time-like vector field. We consider a local basis {*e*_{1},...,*e*_{2n},ξ } in *T M* i.e.

*e*

_{1},...,

*e*

_{2n}are spacelike vector fields.

Then a 2 −form Φ is defined by Φ(*X*,*Y*) = *g*(*X*,*ϕY*), for any *X*,*Y* ∈ Γ(*T M*), called the *fundamental* 2−*form*. Moreover, a Lorentzian almost contact manifold is *normal* if

*ϕ*,

*ϕ*] is denoting the Nijenhuis tensor field associated to

*ϕ*.

*Let M be a Lorentzian almost contact manifold of dimension* (2*n* + 1), *with* (*ϕ,ξ ,η,g*). *M is said to be a Lorentzian almost Kenmotsu manifold if* 1−*form η is closed and d*Φ = −2*η* ∧ *Φ. A normal Lorentzian almost Kenmotsu manifold M is called a Lorentzian Kenmotsu manifold [14].*

*Let* (*M*,*ϕ*,*ξ*,*η*,*g*) *be a Lorentzian almost contact manifold. M is a Lorentzian Kenmotsu manifold if and only if*

*for all X,Y*∈ Γ(

*T M*),

*where*∇

*is Levi-Civita connection on M [14].*

*Let M be* (2*n*+1)-*dimensional a Lorentzian Kenmotsu manifold with structure* (*ϕ,ξ,η,g*). *Then we have*

*for all X ∈*Γ(

*T M*)

*[14].*

Let *K*(*X _{p}*,

*Y*) be the sectional curvature for 2−plane spanned by

_{p}*X*and

_{p}*Y*,

_{p}*p*∈

*M. M*is said to have constant

*ϕ−*holomorphic sectional curvature if

*K*(

*X*,

_{p}*ϕX*) is constant for any point

_{p}*p*and for any unit vector

*X*≠ 0 such that

_{p}*η*(

*X*) = 0.

_{p}A Lorentzian Kenmotsu manifold is said to be a Lorentzian Kenmotsu space form if it has constant *ϕ−*holomorphic section curvature *c* and then, it is denoted by *M*(*c*). The curvature tensor field *R* of *M*(*c*) is given by,

*X*,

*Y*,

*Z*,

*W*∈ Γ(

*T M*).

By virtue of (2.5), we have the following proposition.

*A Lorentzian Kenmotsu manifold of constant ϕ−holomorphic sectional curvature cannot be flat manifold.*

Also, the Ricci curvature of M is given by

*X*,

*Y*∈ Γ(

*T M*). Then from (2.5) on

*M*(

*c*), we have,

*X*,

*Y*∈ Γ(

*T M*).

*A Lorentzian Kenmotsu manifold of constant ϕ−holomorphic sectional curvature cannot be η-Einstain manifold.*

## 3 *ℒ*-sectional curvature of Lorentzian Kenmotsu manifold

Let *M* be Lorentzian Kenmotsu manifold. Therefore, *T M* splits into two complementary subbundles *Imϕ* (whose differentiable distribution is usually denoted by *ℒ*) and ker*ϕ* (whose differentiable distribution is usually denoted by *ℳ*) The sectional curvature of planar sections spanned by vector fields of *ℒ* called *ℒ −*sectional curvature.

In what follows, we denote by *ℳ* the distribution spanned by the structure vector field *ξ* and by *ℒ* its orthogonal complementary distribution. Then we have,

If *X ∈ ℳ* we have *ϕX* = 0 and if *X ∈ ℒ* we have *η*(*X*) = 0, that is, *ϕ*^{2}*X* = −*X*.

From (2.5) the *ℒ −sectional curvature* of Lorentzian Kenmotsu space form is given by

*Let M be Lorentzian Kenmotsu space form. If ℒ −sectional curvature K _{ℒ} is constant equal to c, then c* = −1.

We can chose *X* and *Y* such that *g*(*X,ϕY* ) = 0. Thus, from (3.1) we deduce

*Let M be Lorentzian Kenmotsu maifold and X,Y ∈ ℒ. In this case, the scalar curvature of M is*

*Let M be Lorentzian Kenmotsu manifold and X,Y ∈ ℒ. Then M is Einstein manifold.*

For all *X,Y ∈ ℒ*, using (2.6), we can proof that *M* is Einstein manifold.

## 4 Semi Invariant Submanifolds of a Lorentzian Kenmotsu Space Form

*An* (2*m* + 1)−*dimensional Riemannian submanifold M of Lorentzian Kenmotsu space form
*

^{⊥}

*on M satisfying:*

*(i)**T M*=*D*⊕*D*^{⊥}⊕*sp*{*ξ*};*(ii)**The distribution D is invariant under ϕ, that is ϕD*=_{x}*D*∈_{x}for any x*M*;*(iii)**The distribution D*^{⊥}*is anti-invariant under ϕ, that is,*∈ M for any x$\phi {D}_{x}^{\perp}\subseteq {T}_{x}^{\perp}M$ *M, where T*_{x}M and T_{x}M^{ ⊥}*are the tangent space of M at x.*

Now, we choose a local field of orthonormal frames {*E*_{1},...,*E*_{2}_{p},*E*_{2}_{p}_{+1},...,*E*_{2}_{m},ξ } on *M*. Then we have,

*dimD*= 2

*p*and

*dimD*

^{⊥}=

*q*.

Then if *p* = 0 we have an anti-invariant submanifold tangent to *ξ* and if *q* = 0, we have an invariant submanifold. Now, we give the following example.

*In what follows*, (ℝ^{2n+1},*ϕ*,*η*,*ξ*, *g*) *will denote the manifold* ℝ^{2n+1}*with its usual Lorentzian Kenmotsu structure given by*

*(x*

_{1},...,

*x*,

_{n}*y*

_{1},...,

*y*ℝ

_{n},z) denoting the Cartesian coordinates on^{2n+1}.

*The consider a submanifold of*ℝ

^{7}

*defined by*

*Then local frame of T M is given by*

*and we have*

*which are the a basis of T*

^{⊥}

*M. We determine D*

_{1}=

*sp*{

*e*

_{1},

*e*

_{2}

*} and D*

_{2}=

*sp*{

*e*

_{3},

*e*

_{4}}.

*Then D*

_{1},

*D*

_{2}

*are invariant and anti-invariant distribution, respectively. Thus T M*=

*D*

_{1}⊕

*D*

_{2}⊕

*sp*{

*ξ*}

*is a semi invariant submanifold of*ℝ

^{7}.

Let
*g*. Then Gauss and Weingarten formulas are given by

*X*,

*Y*∈ Γ(

*T M*) and

*N*∈ Γ(

*T*

^{⊥}

*M*). ∇

^{⊥}is the connection in the normal bundle,

*h*is the second fundamental from of

*A*is the Weingarten endomorphism associated with

_{N}*N*. The second fundamental form

*h*and the shape operator

*A*are related with by

Let *M* be semi invariant submanifold of
*M* is said to be totally geodesic if *h*(*X*,*Y*) = 0, for any *X*,*Y* ∈ Γ(*T M*).

We denote by
*R* the curvature tensor fields associated with

*X*,

*Y*,

*Z*,

*W*∈ Γ(

*T M*).

On the other hand, let *M* be a semi invaiant submanifold of a Lorentzian Kenmotsu space form
*M* has constant *ϕ*-sectional curvature *c* if and only if the Riemannian curvature tensor

*Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from
*,

*for all X,Y*∈ Γ(

*T M*).

Let Γ(*T M*) = *sp*{*e*_{1},...,*e _{p}*,

*e*

_{p}_{+1},...,

*e*,

_{q}*e*

_{p}_{+}

_{q}

_{+1}} such that {

*e*

_{1},...,

*e*} are tangent to

_{p}*D*

_{1}and {

*e*

_{p}_{+1},...,

*e*} are tangent to

_{q}*D*

_{2}. Then we have,

Now, using (4.6), we get

*Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from**If M is totally geodesic, then M is an η-Einstein manifold.*

*Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from**Then we have scalar curvature*

From (4.7) by using *X* = *Y* = *e _{k}* we get

The proof is completed.

*Let M be semi invariant submanifold of Lorentzian Kenmotsu space form**Then*

*for all X*,

*Y*,

*Z*,

*W*∈ Γ(

*D*

^{⊥}).

Using (4.6). For all *X*,*Y*,*Z*,*W* ∈ Γ(*D*^{⊥}), since *ϕX*,*ϕY*,*ϕZ*,*ϕW* ∈ *ϕD*^{⊥}⊂ *T M*^{⊥} we have (4.8).

*Let M be semi invariant submanifold of Lorentzian Kenmotsu space form**If D*^{⊥}*is totally geodesic, then D*^{⊥}*is flat if and only if c* = −3.

*Let M be semi invariant submanifold of Lorentzian Kenmotsu space form**Then*

*for all X*,

*Y*∈ Γ(

*D*

^{⊥}),

*where S is Ricci tensor.*

Using (4.8). From
*X*,*Y* ∈ Γ(*D*^{⊥}), we have equation (4.9).

*Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from
*

^{⊥}

*is totally geodesic, then distribution D*

^{⊥}

*is Einstein.*

*Let M be semi inavariant submanifold of Lorentzian Kenmotsu space form**If D*^{⊥}*is totally geodesic, then*

*where τ is the scalar curvature.*

*Let M be semi invariant submanifold of Lorentzian Kenmotsu space form
*(

*c*).

*Then the Ricci curvature determined by D*

*for all X*,

*Y*∈ Γ(

*D*).

For all *X*,*Y* ∈ Γ(*D*), from (4.8) we have

Then, from

*Let M be semi invariant submanifold of Lorentzian Kenmotsu space form**Then the scalar curvature determined by D is given*

*Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from
*

*Let M be semi invariant submanifold of Lorentzian Kenmotsu space form**Then, ϕ-sectional curvature of D is −c if and only if D is totally geodesic.*

Using (4.6). For all *X* ∈ Γ(*D*),

Then,

## 5 Semi Invarinat Product in a Lorentzian Kenmotsu Space Form

Let *M* be a semi invariant submanifold of a Lorentzian Kenmotsu space form
*M* is a semi invariant product if the distribution *D* ⊕ *sp*{*ξ* } is integrable and locally *M* is a Riemannian product *M*_{1}*× M*_{2}, where *M*_{1} (resp. *M*_{2}) is leaf of *D* ⊕ *sp*{*ξ* }
(resp. *D*^{
⊥}
). If we have *pq* ≠ 0, we say that *M* is a proper semi invariant product.

*Let M be a proper semi invariant product of a Lorentzian Kenmotsu space form**Then*

*for any unit vector fields X*∈

*D and Z*∈

*D*

^{⊥}.

Using (4.6) and *ϕZ ∈* Γ(*ϕD*^{⊥}) *⊂ T M*^{⊥} this complates the proof.

*Let M be a proper semi invariant product of a Lorentzian Kenmotsu space form**Then*,

Since *h* is fundamental form, we have

*Let M be a proper semi invariant product of an a Lorentzian Kenmotsu space form**Then*,

*for all X*,

*Y*∈ Γ(

*D ⊕ sp*{ξ })

*and Z,W ∈*Γ(

*D*

^{ ⊥}).

Let *M* be semi invariant submanifold of Lorentzian Kenmotsu manifold
*Z,W ∈* Γ(*D*^{⊥}),

Using (4.6), which complates the proof.

*Let M be a proper semi invariant product of a Lorentzian Kenmotsu
*

*for all X,Y*∈ Γ(

*D*)

*and Z*,

*W ∈*Γ(

*D*

^{⊥}

*⊕ sp*{ξ }).

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