Some Properties Curvture of Lorentzian Kenmotsu Manifolds

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 ℂⁿ.

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, (1.1) $\begin{array}{l} (\nabla_{X} φ) Y & = - η (Y) φ X - g (X, φ Y) ξ \\ \nabla_{X} ξ & = X - η (X) ξ \end{array}$ \begin{array}{*{20}{l}}{({\nabla _X}\varphi )Y}&{ = - \eta (Y)\varphi X - g(X,\varphi Y)\xi }\\{{\nabla _X}\xi }&{ = X - \eta (X)\xi }\end{array}

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 ℂEⁿ 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 (−ɛ,ɛ) ×_f V where f(t) = ce^t and 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.

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 (2.1) $φ^{2} = - I + η \otimes ξ, η (ξ) = 1 .$ {\varphi ^2} = - I + \eta \otimes \xi ,\,\,\eta (\xi ) = 1. 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 (2.2) $g (φ X, φ Y) = g (X, Y) - ɛ η (X) η (Y), g (ξ, ξ) = ɛ = - 1$ g(\varphi X,\varphi Y) = g(X,Y) - \varepsilon \eta (X)\eta (Y),\,\,\,\,g(\xi ,\xi ) = \varepsilon = - 1 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) = 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 {e₁,...,e_2n,ξ } in T M i.e. $g (e_{i}, e_{j}) = δ_{i j} and g (ξ, ξ) = - 1$ g({e_i},{e_j}) = {\delta _{ij}}\,{\rm{and}}\,g(\xi ,\xi ) = - 1 that is e₁,...,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 $N = [φ, φ] + 2 d η \otimes ξ = 0$ N = [\varphi ,\varphi ] + 2d\eta \otimes \xi = 0 where [ϕ,ϕ] is denoting the Nijenhuis tensor field associated to ϕ.

Definition 2.1

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

Theorem 2.1

Let (M,ϕ,ξ,η,g) be a Lorentzian almost contact manifold. M is a Lorentzian Kenmotsu manifold if and only if(2.3) $(\nabla_{X} φ) Y = - g (φ X, Y) ξ + η (Y) φ X$ \left( {{\nabla _X}\varphi } \right)Y = - g(\varphi X,Y)\xi + \eta (Y)\varphi Xfor all X,Y ∈ Γ(T M), where ∇ is Levi-Civita connection on M [14].

Corollary 2.1

Let M be (2n+1)-dimensional a Lorentzian Kenmotsu manifold with structure (ϕ,ξ,η,g). Then we have(2.4) $\nabla_{X} ξ = - φ^{2} X$ {\nabla _X}\xi = - {\varphi ^2}Xfor all X ∈ Γ(T M) [14].

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, (2.5) $\begin{array}{l} R (X, Y, Z, W) & = \frac{c + 3}{4} {g (X, W) g (Y, Z) - g (X, Z) g (Y, W)} \\ + \frac{c - 1}{4} {g (φ X, W) g (φ Y, Z) - g (φ X, Z) g (φ Y, W) - 2 g (φ X, Y) g (φ Z, W) \\ + g (X, Z) η (Y) η (W) - g (Y, Z) η (X) η (W) + g (Y, W) η (X) η (Z)} . \end{array}$ \begin{array}{*{20}{l}}{R(X,Y,Z,W)}&{ = \frac{{c + 3}}{4}\{ g(X,W)g(Y,Z) - g(X,Z)g(Y,W)\} }\\{}&{ + \frac{{c - 1}}{4}\{ g(\varphi X,W)g(\varphi Y,Z) - g(\varphi X,Z)g(\varphi Y,W) - 2g(\varphi X,Y)g(\varphi Z,W)}\\{}&{ + g(X,Z)\eta (Y)\eta (W) - g(Y,Z)\eta (X)\eta (W) + g(Y,W)\eta (X)\eta (Z)\} .}\end{array} where X,Y,Z,W ∈ Γ(T M).

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

Proposition 2.2

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

Also, the Ricci curvature of M is given by $S (X, Y) = \underset{i = 1}{\sum^{2 n + 1}} R (E_{i}, X, Y, E_{i}),$ S(X,Y) = \mathop {\mathop {\sum }\limits^{2n + 1} }\limits_{i = 1} R({E_i},X,Y,{E_i}), for X,Y ∈ Γ(T M). Then from (2.5) on M(c), we have, (2.6) $S (X, Y) = \frac{(c - 3) n + (c + 1)}{2} g (φ X, φ Y) - 2 n η (X) η (Y)$ S(X,Y) = \frac{{(c - 3)n + (c + 1)}}{2}g(\varphi X,\varphi Y) - 2n\eta (X)\eta (Y) for all X,Y ∈ Γ(T M).

Proposition 2.3

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

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 kerϕ (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, $T M = ℒ \oplus ℳ .$ TM = \mathcal{L} \oplus \mathcal{M}.

If X ∈ ℳ we have ϕX = 0 and if X ∈ ℒ we have η(X) = 0, that is, ϕ²X = −X.

From (2.5) the ℒ −sectional curvature of Lorentzian Kenmotsu space form is given by (3.1) $K_{ℒ} (X, Y) = \frac{c - 3}{4} + 3 \frac{c + 1}{4} g {(X, φ Y)}^{2}$ {K_\mathcal{L}}(X,Y) = \frac{{c - 3}}{4} + 3\frac{{c + 1}}{4}g{(X,\varphi Y)^2}

Corollary 3.1

Let M be Lorentzian Kenmotsu space form. If ℒ −sectional curvature K_ℒ is constant equal to c, then c = −1.

Proof

We can chose X and Y such that g(X,ϕY ) = 0. Thus, from (3.1) we deduce $c = \frac{c - 3}{4} \Rightarrow c = - 1 .$ c = \frac{{c - 3}}{4} \Rightarrow c = - 1.

Corollary 3.2

Let M be Lorentzian Kenmotsu maifold and X,Y ∈ ℒ. In this case, the scalar curvature of M is $τ = - n (2 n + 1) .$ \tau = - n(2n + 1).

Proposition 3.1

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

Proof

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

Semi Invariant Submanifolds of a Lorentzian Kenmotsu Space Form

Definition 4.1

An (2m + 1)−dimensional Riemannian submanifold M of Lorentzian Kenmotsu space form $\bar{M}$ \overline M is called a semi invariant submanifold if ξ is tangent to M and there exists on M two differentiable distributions D and D^⊥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, $φ D_{x}^{⊥} \subseteq T_{x}^{⊥} M$ \varphi D_x^ \bot \subseteq T_x^ \bot M M for any x ∈ M, where T_xM and T_xM^⊥are the tangent space of M at x.

Now, we choose a local field of orthonormal frames {E₁,...,E₂_p,E₂_p₊₁,...,E₂_m,ξ } on M. Then we have, (4.1) $D = sp {E_{1}, ..., E_{2 p}}, D^{⊥} = sp {E_{2 p + 1}, ..., E_{2 m}}$ D = sp\{ {E_1},...,{E_{2p}}\} ,\,\,\,\,{D^ \bot } = sp\{ {E_{2p + 1}},...,{E_{2m}}\} 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.

Example 4.1

In what follows, (ℝ²ⁿ⁺¹,ϕ,η,ξ, g) will denote the manifold ℝ²ⁿ⁺¹with its usual Lorentzian Kenmotsu structure given by $\begin{matrix} η = dz, ξ = \frac{\partial}{\partial z} \\ φ (\underset{i = 1}{\sum^{n}} (X_{i} \frac{\partial}{\partial x_{i}} + Y_{i} \frac{\partial}{\partial y_{i}}) + Z \frac{\partial}{\partial z}) = \underset{i = 1}{\sum^{n}} (Y_{i} \frac{\partial}{\partial x_{i}} - X_{i} \frac{\partial}{\partial y_{i}}) + \underset{i = 1}{\sum^{n}} Y_{i} y_{i} \frac{\partial}{\partial z} \\ g = e^{- 2 z} (\underset{i = 1}{\sum^{n}} d x_{i} \otimes d x_{i} + d y_{i} \otimes d y_{i}) - ɛ dz \otimes dz \end{matrix}$ \begin{array}{*{20}{c}}{\eta = dz,\,\,\,\xi = \frac{\partial }{{\partial z}}}\\{\varphi (\mathop {\mathop {\sum }\limits^n }\limits_{i = 1} ({X_i}\frac{\partial }{{\partial {x_i}}} + {Y_i}\frac{\partial }{{\partial {y_i}}}) + Z\frac{\partial }{{\partial z}}) = \mathop {\mathop {\sum }\limits^n }\limits_{i = 1} ({Y_i}\frac{\partial }{{\partial {x_i}}} - {X_i}\frac{\partial }{{\partial {y_i}}}) + \mathop {\mathop {\sum }\limits^n }\limits_{i = 1} {Y_i}{y_i}\frac{\partial }{{\partial z}}}\\{g = {e^{ - 2z}}(\mathop {\mathop {\sum }\limits^n }\limits_{i = 1} d{x_i} \otimes d{x_i} + d{y_i} \otimes d{y_i}) - \varepsilon dz \otimes dz}\end{array}(x₁,...,x_n,y₁,...,y_n,z) denoting the Cartesian coordinates on ℝ²ⁿ⁺¹. The consider a submanifold of ℝ⁷defined by $M = X (u, v, k, l, t) = (u, k, 0, v, 0, l, t) .$ M = X(u,v,k,l,t) = (u,k,0,v,0,l,t).

Then local frame of T M is given by $\begin{array}{l} e_{1} = \frac{\partial}{\partial x_{1}}, e_{2} = \frac{\partial}{\partial y_{1}}, \\ e_{3} = \frac{\partial}{\partial x_{2}}, e_{4} = \frac{\partial}{\partial y_{3}}, \\ e_{5} = \frac{\partial}{\partial z} = ξ \end{array}$ \begin{array}{l}{e_1} = \frac{\partial }{{\partial {x_1}}},{e_2} = \frac{\partial }{{\partial {y_1}}},\\{e_3} = \frac{\partial }{{\partial {x_2}}},{e_4} = \frac{\partial }{{\partial {y_3}}},\\{e_5} = \frac{\partial }{{\partial z}} = \xi \\\end{array}and we have $e_{1}^{*} = \frac{\partial}{\partial x_{3}}, e_{2}^{*} = \frac{\partial}{\partial y_{2}}$ e_1^ * = \frac{\partial }{{\partial {x_3}}},\,\,\,e_2^ * = \frac{\partial }{{\partial {y_2}}}which are the a basis of T^⊥M. We determine D₁ = sp{e₁,e₂} and D₂ = sp{e₃,e₄}. Then D₁, D₂are invariant and anti-invariant distribution, respectively. Thus T M = D₁ ⊕ D₂ ⊕ sp{ξ } is a semi invariant submanifold of ℝ⁷.

Let $\bar{\nabla}$ \overline \nabla be the Levi-Civita connection of $\bar{M}$ \overline M with respect to the g. Then Gauss and Weingarten formulas are given by (4.2) ${\bar{\nabla}}_{X} Y = \nabla_{X} Y + h (X, Y)$ {\overline \nabla _X}Y = {\nabla _X}Y + h(X,Y)(4.3) ${\bar{\nabla}}_{X} N = \nabla_{X}^{⊥} N - A_{N} X$ {\overline \nabla _X}N = \nabla _X^ \bot N - {A_N}X 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 $\bar{M}$ \overline 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 (4.4) $g (h (X, Y), N) = g (A_{N} X, Y) .$ g(h(X,Y),N) = g({A_N}X,Y).

Let M be semi invariant submanifold of $\bar{M}$ \overline M . M is said to be totally geodesic if h(X,Y) = 0, for any X,Y ∈ Γ(T M).

We denote by $\bar{R}$ \overline R and R the curvature tensor fields associated with $\bar{\nabla}$ \overline \nabla and ∇ respectively. The Gauss equation is given by (4.5) $\bar{R} (X, Y, Z, W) = R (X, Y, Z, W) + g (h (X, Z), h (Y, W)) - g (h (X, W), h (Y, Z))$ \overline R (X,Y,Z,W) = R(X,Y,Z,W) + g(h(X,Z),h(Y,W)) - g(h(X,W),h(Y,Z)) for all X,Y,Z,W ∈ Γ(T M).

On the other hand, let M be a semi invaiant submanifold of a Lorentzian Kenmotsu space form $\bar{M}$ \overline M . Then using (2.5) and (4.5), a semi invariant submanifold M has constant ϕ-sectional curvature c if and only if the Riemannian curvature tensor $\bar{R}$ \overline R satisfied (4.6) $\begin{array}{l} R (X, Y, Z, W) & = \frac{c + 3}{4} {g (X, W) g (Y, Z) - g (X, Z) g (Y, W)} \\ + \frac{c - 1}{4} {g (φ X, W) g (φ Y, Z) - g (φ X, Z) g (φ Y, W) - 2 g (φ X, Y) g (φ Z, W) \\ + g (X, Z) η (Y) η (W) - g (Y, Z) η (X) η (W) + g (Y, W) η (X) η (Z)} \\ + g (h (X, W), h (Y, Z)) - g (h (Y, W), g (X, Z)) . \end{array}$ \begin{array}{*{20}{l}}{R(X,Y,Z,W)}&{ = \frac{{c + 3}}{4}\{ g(X,W)g(Y,Z) - g(X,Z)g(Y,W)\} }\\{}&{ + \frac{{c - 1}}{4}\{ g(\varphi X,W)g(\varphi Y,Z) - g(\varphi X,Z)g(\varphi Y,W) - 2g(\varphi X,Y)g(\varphi Z,W)}\\{}&{ + g(X,Z)\eta (Y)\eta (W) - g(Y,Z)\eta (X)\eta (W) + g(Y,W)\eta (X)\eta (Z)\} }\\{}&{ + g(h(X,W),h(Y,Z)) - g(h(Y,W),g(X,Z)).}\end{array}

Theorem 4.1

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from $\bar{M} (c)$ \overline M (c) . Then we get Ricci tensor of M, (4.7) $\begin{array}{l} S (X, Y) & = {\frac{c + 3}{4} (p + q - 3) + 3 \frac{c - 1}{2}} g (X, Y) \\ - {\frac{c - 1}{4} (p + q - 6) + \frac{c + 1}{2}} η (X) η (Y) \\ + \overset{p + q}{\sum_{i = 1}} {g (h (X, Y), h (E_{i}, E_{i})) - g (h (E_{i}, Y), h (X, E_{i}))} \end{array}$ \begin{array}{*{20}{l}}{S(X,Y)}&{ = \{ \frac{{c + 3}}{4}(p + q - 3) + 3\frac{{c - 1}}{2}\} g(X,Y)}\\{}&{ - \{ \frac{{c - 1}}{4}(p + q - 6) + \frac{{c + 1}}{2}\} \eta (X)\eta (Y)}\\{}&{ + \mathop {\mathop {\sum }\limits_{i = 1} }\limits^{p + q} \{ g(h(X,Y),h({E_i},{E_i})) - g(h({E_i},Y),h(X,{E_i}))\} }\end{array}for all X,Y ∈ Γ(T M).

Proof

Let Γ(T M) = sp{e₁,...,e_p,e_p₊₁,...,e_q,e_p₊_q₊₁} such that {e₁,...,e_p} are tangent to D₁ and {e_p₊₁,...,e_q} are tangent to D₂. Then we have, $S (X, Y) = \overset{q}{\sum_{i = 1}} R (X, E_{i}, E_{i}, Y) .$ S(X,Y) = \mathop {\mathop {\sum }\limits_{i = 1} }\limits^p R(X,{E_i},{E_i},Y) + \mathop {\mathop {\sum }\limits_{i = p + 1} }\limits^q R(X,{E_i},{E_i},Y) + R(X,\xi ,\xi ,Y).

Now, using (4.6), we get $\begin{array}{l} S (X, Y) & = {\frac{c + 3}{4} (p - 1) + 3 \frac{c - 1}{4}} g (X, Y) + \frac{c - 1}{4} (3 - p) η (X) η (Y) \\ + \overset{p}{\sum_{i = 1}} {g (h (X, Y), h (E_{i}, E_{i})) - g (h (E_{i}, Y), h (X, E_{i}))} \\ + {\frac{c + 3}{4} (q - 1) + 3 \frac{c - 1}{4}} g (X, Y) + \frac{c - 1}{4} (3 - q) η (X) η (Y) \\ + \overset{q}{\sum_{i = p + 1}} g (h (X, Y), h (E_{i}, E_{i})) - g (h (E_{i}, Y), h (X, E_{i})) \\ - \frac{c + 3}{4} g (X, Y) - {\frac{c + 3}{4} + \frac{c - 1}{4}} η (X) η (Y) \end{array}$ \begin{array}{*{20}{l}}{S(X,Y)}&{ = \{ \frac{{c + 3}}{4}(p - 1) + 3\frac{{c - 1}}{4}\} g(X,Y) + \frac{{c - 1}}{4}(3 - p)\eta (X)\eta (Y)}\\{}&{ + \mathop {\mathop {\sum }\limits_{i = 1} }\limits^p \{ g(h(X,Y),h({E_i},{E_i})) - g(h({E_i},Y),h(X,{E_i}))\} }\\{}&{ + \{ \frac{{c + 3}}{4}(q - 1) + 3\frac{{c - 1}}{4}\} g(X,Y) + \frac{{c - 1}}{4}(3 - q)\eta (X)\eta (Y)}\\{}&{ + \mathop {\mathop {\sum }\limits_{i = p + 1} }\limits^q g(h(X,Y),h({E_i},{E_i})) - g(h({E_i},Y),h(X,{E_i}))}\\{}&{ - \frac{{c + 3}}{4}g(X,Y) - \{ \frac{{c + 3}}{4} + \frac{{c - 1}}{4}\} \eta (X)\eta (Y)}\end{array} which gives proof.

Corollary 4.1

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from $\bar{M} (c)$ \overline M (c). If M is totally geodesic, then M is an η-Einstein manifold.

Proposition 4.2

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from $\bar{M} (c)$ \overline M (c) . Then we have scalar curvature $\begin{array}{l} τ & = {\frac{c + 3}{4} (p + q - 3) + 3 \frac{c - 1}{2}} (p + q - 1) \\ + \frac{c - 1}{4} (p + q - 6) + \frac{c + 1}{2} + \frac{1}{{(p + q + 1)}^{2}} {‖ H ‖}^{2} + {‖ h ‖}^{2} . \end{array}$ \begin{array}{*{20}{l}}\tau &{ = \{ \frac{{c + 3}}{4}(p + q - 3) + 3\frac{{c - 1}}{2}\} (p + q - 1)}\\{}&{ + \frac{{c - 1}}{4}(p + q - 6) + \frac{{c + 1}}{2} + \frac{1}{{{{(p + q + 1)}^2}}}{{\left\| H \right\|}^2} + {{\left\| h \right\|}^2}.}\end{array}

Proof

From (4.7) by using X = Y = e_k we get $τ = \overset{p + q + 1}{\sum_{k = 1}} S (e_{k}, e_{k}) .$ \tau = \mathop {\mathop {\sum }\limits_{k = 1} }\limits^{p + q + 1} S({e_k},{e_k}).

The proof is completed.

Proposition 4.3

Let M be semi invariant submanifold of Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then(4.8) $\begin{array}{l} R (X, Y, Z, W) & = \frac{c + 3}{4} {g (X, W) g (Y, Z) - g (X, Z) g (Y, W)} \\ + g (h (X, W), h (Y, Z)) - g (h (Y, W), g (X, Z)) \end{array}$ \begin{array}{*{20}{l}}{R(X,Y,Z,W)}&{ = \frac{{c + 3}}{4}\{ g(X,W)g(Y,Z) - g(X,Z)g(Y,W)\} }\\{}&{ + g(h(X,W),h(Y,Z)) - g(h(Y,W),g(X,Z))}\end{array}for all X,Y,Z,W ∈ Γ(D^⊥).

Proof

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

Corollary 4.2

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

Proposition 4.4

Let M be semi invariant submanifold of Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then(4.9) $S (X, Y) = \frac{c + 3}{4} (q - 1) g (X, Y) + \overset{q}{\sum_{i = 1}} {g (h (X, Y), h (E_{i}, E_{i})) - g (h (E_{i}, Y), h (X, E_{i}))}$ S(X,Y) = \frac{{c + 3}}{4}(q - 1)g(X,Y) + \mathop {\mathop {\sum }\limits_{i = 1} }\limits^q \{ g(h(X,Y),h({E_i},{E_i})) - g(h({E_i},Y),h(X,{E_i}))\} for all X,Y ∈ Γ(D^⊥), where S is Ricci tensor.

Proof

Using (4.8). From $S (X, Y) = \overset{q}{\sum_{i = 1}} R (X, E_{i}, E_{i}, Y)$ S(X,Y) = \mathop {\mathop {\sum }\limits_{i = 1} }\limits^q R(X,{E_i},{E_i},Y) , for all X,Y ∈ Γ(D^⊥), we have equation (4.9).

Corollary 4.3

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from $\bar{M}$ \overline M . If D^⊥is totally geodesic, then distribution D^⊥is Einstein.

Corollary 4.4

Let M be semi inavariant submanifold of Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). If D^⊥is totally geodesic, then $τ_{D^{⊥}} = \frac{c + 3}{4} q (q - 1)$ {\tau _{{D^ \bot }}} = \frac{{c + 3}}{4}q(q - 1)where τ is the scalar curvature.

Proposition 4.5

Let M be semi invariant submanifold of Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c)(c). Then the Ricci curvature determined by D $S (X, Y) = {\frac{c + 3}{4} (p - 1) + 3 \frac{c - 1}{4}} g (X, Y)$ S(X,Y) = \{ \frac{{c + 3}}{4}(p - 1) + 3\frac{{c - 1}}{4}\} g(X,Y)for all X,Y ∈ Γ(D).

Proof

For all X,Y ∈ Γ(D), from (4.8) we have $\begin{array}{l} R (X, Y, Z, W) & = \frac{c + 3}{4} {g (X, W) g (Y, Z) - g (X, Z) g (Y, W)} \\ + \frac{c - 1}{4} {g (X, φ W) g (Y, φ Z) - g (X, φ Z) g (Y, φ W) \end{array}$ \begin{array}{*{20}{l}}{R(X,Y,Z,W)}&{ = \frac{{c + 3}}{4}\{ g(X,W)g(Y,Z) - g(X,Z)g(Y,W)\} }\\{}&{ + \frac{{c - 1}}{4}\{ g(X,\varphi W)g(Y,\varphi Z) - g(X,\varphi Z)g(Y,\varphi W)}\end{array}

Then, from $S (X, Y) = \overset{p}{\sum_{i = 1}} R (X, E_{i}, E_{i}, Y)$ S(X,Y) = \mathop {\mathop {\sum }\limits_{i = 1} }\limits^p R(X,{E_i},{E_i},Y) , using last equation, this complates the proof.

Corollary 4.5

Let M be semi invariant submanifold of Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then the scalar curvature determined by D is given $τ_{D} = p \frac{(c + 3) 2 p - 1) + 3 (c - 1)}{4} .$ {\tau _D} = p\frac{{(c + 3)2p - 1) + 3(c - 1)}}{4}.

Corollary 4.6

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from $\bar{M}$ \overline M . If D is totally geodesic, then distribution D is Einstein.

Theorem 4.6

Let M be semi invariant submanifold of Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then, ϕ-sectional curvature of D is −c if and only if D is totally geodesic.

Proof

Using (4.6). For all X ∈ Γ(D), $\begin{array}{l} R (X, φ X, X, φ X) & = \frac{c + 3}{4} {g (X, φ X) g (φ X, X) - g (X, X) g (φ X, φ X)} \\ + \frac{c - 1}{4} {g (φ X, φ X) g (φ^{2} X, X) - g (φ X, X) g (φ^{2} X, φ X) \\ - 2 g (φ X, φ X) g (φ X, φ X) \\ + g (h (X, φ X), h (φ X, X)) - g (h (φ X, φ X), h (X, X)) . \end{array}$ \begin{array}{*{20}{l}}{R(X,\varphi X,X,\varphi X)}&{ = \frac{{c + 3}}{4}\{ g(X,\varphi X)g(\varphi X,X) - g(X,X)g(\varphi X,\varphi X)\} }\\{}&{ + \frac{{c - 1}}{4}\{ g(\varphi X,\varphi X)g({\varphi ^2}X,X) - g(\varphi X,X)g({\varphi ^2}X,\varphi X)}\\{}&{ - 2g(\varphi X,\varphi X)g(\varphi X,\varphi X)}\\{}&{ + g(h(X,\varphi X),h(\varphi X,X)) - g(h(\varphi X,\varphi X),h(X,X)).}\end{array}

Then, $R (X, φ X, X, φ X) = - c - 2 {‖ h (X, X) ‖}^{2} .$ R(X,\varphi X,X,\varphi X) = - c - 2{\left\| {h(X,X)} \right\|^2}.

Semi Invarinat Product in a Lorentzian Kenmotsu Space Form

Let M be a semi invariant submanifold of a Lorentzian Kenmotsu space form $\bar{M}$ \overline 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 M₁× M₂, where M₁ (resp. M₂) is leaf of D ⊕ sp{ξ } (resp. D^⊥ ). If we have pq ≠ 0, we say that M is a proper semi invariant product.

Theorem 5.1

Let M be a proper semi invariant product of a Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then(5.1) $R (X, φ X, Z, φ Z) = 2 ({‖ h (X, Z) ‖}^{2} - \frac{c - 1}{4})$ R(X,\varphi X,Z,\varphi Z) = 2({\left\| {h(X,Z)} \right\|^2} - \frac{{c - 1}}{4})for any unit vector fields X ∈ D and Z ∈ D^⊥.

Proof

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

Theorem 5.2

Let M be a proper semi invariant product of a Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then, (5.2) ${‖ h ‖}^{2} \geq pq (1 - c) + 2 qp$ {\left\| h \right\|^2} \ge pq(1 - c) + 2qp

Proof

Since h is fundamental form, we have $\begin{array}{l} {‖ h ‖}^{2} & = \overset{2 p}{\sum_{i, j = 1}} {‖ h (E_{i}, E_{j}) ‖}^{2} + \overset{2 m}{\sum_{k, l = 2 p + 1}} {‖ h (E_{k}, E_{l}) ‖}^{2} \\ + 2 \overset{2 p}{\sum_{i = 1}} \overset{2 m}{\sum_{k = 2 p + 1}} {‖ h (E_{i}, E_{k}) ‖}^{2} + 2 \overset{2 m}{\sum_{k = 2 p + 1}} {‖ h (E_{k}, ξ) ‖}^{2} \end{array}$ \begin{array}{*{20}{l}}{{{\left\| h \right\|}^2}}&{ = \mathop {\mathop {\sum }\limits_{i,j = 1} }\limits^{2p} {{\left\| {h({E_i},{E_j})} \right\|}^2} + \mathop {\mathop {\sum }\limits_{k,l = 2p + 1} }\limits^{2m} {{\left\| {h({E_k},{E_l})} \right\|}^2}}\\{}&{ + 2\mathop {\mathop {\sum }\limits_{i = 1} }\limits^{2p} \mathop {\mathop {\sum }\limits_{k = 2p + 1} }\limits^{2m} {{\left\| {h({E_i},{E_k})} \right\|}^2} + 2\mathop {\mathop {\sum }\limits_{k = 2p + 1} }\limits^{2m} {{\left\| {h({E_k},\xi )} \right\|}^2}}\end{array} from (4.1) ${‖ h ‖}^{2} = pq (1 - c) + 2 qp + \overset{2 p}{\sum_{i, j = 1}} {‖ h (E_{i}, E_{j}) ‖}^{2} + \overset{2 m}{\sum_{k, l = 2 p + 1}} {‖ h (E_{k}, E_{l}) ‖}^{2}$ {\left\| h \right\|^2} = pq(1 - c) + 2qp + \mathop {\mathop {\sum }\limits_{i,j = 1} }\limits^{2p} {\left\| {h({E_i},{E_j})} \right\|^2} + \mathop {\mathop {\sum }\limits_{k,l = 2p + 1} }\limits^{2m} {\left\| {h({E_k},{E_l})} \right\|^2} which gives (5.2).

Proposition 5.3

Let M be a proper semi invariant product of an a Lorentzian Kenmotsu space form $\bar{M} (c)$ \overline M (c). Then, $R (X, Y, Z, W) = 0$ R(X,Y,Z,W) = 0for all X,Y ∈ Γ(D ⊕ sp{ξ }) and Z,W ∈ Γ(D^⊥ ).

Proof

Let M be semi invariant submanifold of Lorentzian Kenmotsu manifold $\bar{M}$ \overline M . Then for all Z,W ∈ Γ(D^⊥), $φ Z, φ W \in φ D^{⊥} \subset T M^{⊥} .$ \varphi Z,\varphi W \in \varphi {D^ \bot } \subset T{M^ \bot }.

Using (4.6), which complates the proof.

Proposition 5.4

Let M be a proper semi invariant product of a Lorentzian Kenmotsu $\bar{M}$ \overline M . Then $R (X, Y, Z, W) = 0$ R(X,Y,Z,W) = 0for all X,Y ∈ Γ(D) and Z,W ∈ Γ(D^⊥⊕ sp{ξ }).

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Some Properties Curvture of Lorentzian Kenmotsu Manifolds

Published Online: Mar 31, 2020

Page range: 283 - 292

Received: Jun 24, 2019

Accepted: Aug 20, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00026

KeywordsLorentz Kenmotsu manifold, projective curvature tensor, ℒ-sectional curvature, semi invariant submanifold

© 2020 Ramazan Sari, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Lorentz Kenmotsu manifold, projective curvature tensor, ℒ-sectional curvature, semi invariant submanifold