Some Properties Curvture of Lorentzian Kenmotsu Manifolds

In this paper different curvature tensors on Lorentzian Kenmotsu manifod are studied. We investigate constant φ−holomorphic sectional curvature and L -sectional curvature of Lorentzian Kenmotsu manifolds, obtaining conditions for them to be constant of Lorentzian Kenmotsu manifolds in such condition. We calculate the Ricci tensor and scalar curvature for all the cases. Moreover we investigate some properties of semi invariant submanifolds of a Lorentzian Kenmotsu space form. We show that if a semi-invariant submanifold of a Lorentzian Kenmotsu space form M is totally geodesic, then M is an η−Einstein manifold. We consider sectional curvature of semi invariant product of a Lorentzian Kenmotsu manifolds.


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 (2n + 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 Kähler manifold of constant holomorphic sectional curvature.
iii) c < 0, M is warped product space R × f C 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 CE n 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 (−ε, ε) × f V where f (t) = ce t and V is Kähler.
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 submanifold of Lorentzian Kenmotsu space form, In last section, we investigate semi invariant products of a Lorentzian Kenmotsu manifold.

Lorentzian Kenmotsu Manifolds
Let M be a real (2n + 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 then M is called an almost contact manifold. It follows that ϕ(ξ ) = 0, η • ϕ = 0, rankϕ = 2n. If there exists a semi-Riemannian metric g satisfying then (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) = εg(X, ξ ).We note that ξ is neither a lightlike nor a spacelike vector fields on M. We note that ξ is the time-like vector field. We consider a local basis g(e i , e j ) = δ i j and g(ξ , ξ ) = −1 that is e 1 , ..., e 2n are spacelike vector fields.
Let K(X p ,Y p ) be the sectional curvature for 2−plane spanned by X p and Y p , p ∈ M. M is said to have constant ϕ−holomorphic sectional curvature if K(X p , ϕX p ) is constant for any point p and for any unit vector X p = 0 such that η(X p ) = 0.
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, By virtue of (2.5), we have the following proposition. Also, the Ricci curvature of M is given by for all X,Y ∈ Γ(T M).

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 Proof. We can chose X and Y such that g(X, ϕY ) = 0. Thus , from (3.1) we deduce Proof. For all X,Y ∈ L , using (2.6), we can proof that M is Einstein manifold. where dimD = 2p 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 n , y 1 , ..., y n , z) denoting the Cartesian coordinates on R 2n+1 . The consider a submanifold of R 7 defined by M = X(u, v, k, l,t) = (u, k, 0, v, 0, l,t).
Then local frame of T M is given by Let ∇ be the Levi-Civita connection of M with respect to the g. Then Gauss and Weingarten formulas are given by for any X,Y ∈ Γ(T M) and N ∈ Γ(T ⊥ M). ∇ ⊥ is the connection in the normal bundle, h is the second fundamental from of M and A N is the Weingarten endomorphism associated with N. The second fundamental form h and the shape operator A are related with by g(h(X,Y ), N) = g(A N X,Y ).  Z)).
for all X,Y ∈ Γ(T M).
Proof. From (4.7) by using X = Y = e k we get The proof is completed.
for all X,Y ∈ Γ(D ⊥ ), where S is Ricci tensor.
where τ is the scalar curvature.

Semi Invarinat Product in a Lorentzian Kenmotsu Space Form
Let M be a semi invariant submanifold of a Lorentzian Kenmotsu space form M. We say that 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. for any unit vector fields X ∈ D and Z ∈ D ⊥ .
Proof. Using (4.6) and ϕZ ∈ Γ(ϕD ⊥ ) ⊂ T M ⊥ this complates the proof. Proof. Since h is fundamental form, we have h(E k , ξ ) 2 from (4.1) h(E k , E l ) 2 which gives (5.2). Using (4.6), which complates the proof. T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k