Some Properties Curvture of Lorentzian Kenmotsu Manifolds

Ramazan Sari 1
  • 1 Gümüşhacıköy Hasan Duman Vocational Schools, 05300, Amasya, Turkey
Ramazan Sari
  • Gümüşhacıköy Hasan Duman Vocational Schools, Amasya University, 05300, Amasya, Turkey
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Abstract

In this paper different curvature tensors on Lorentzian Kenmotsu manifod are studied. We investigate constant ϕ–holomorphic sectional curvature and -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.

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 (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:

  1. i)c > 0, M is homogeneous Sasakian manifold of constant holomorphic sectional curvature.
  2. ii)c = 0, M is the global Riemannian product of a line or a circle with a Kähler manifold of constant holomorphic sectional curvature.
  3. iii)c < 0, M is warped product space ℝ ×fn.

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,

(Xφ)Y=η(Y)φXg(X,φY)ξXξ=Xη(X)ξ

He showed normal an almost contact Riemannian manifold with (1.1) but not quasi Sasakian hence not Sasakian. He characterized warped product space L ×fEn 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) = cet 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 - 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 (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

φ2=I+ηξ,η(ξ)=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
g(φX,φY)=g(X,Y)ɛη(X)η(Y),g(ξ,ξ)=ɛ=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,...,e2n,ξ } in T M i.e.

g(ei,ej)=δijandg(ξ,ξ)=1
that is e1,...,e2n 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]+2dηξ=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 = −2ηΦ. 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

(Xφ)Y=g(φX,Y)ξ+η(Y)φX
for all X,Y ∈ Γ(T M), whereis 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

Xξ=φ2X
for all X ∈ Γ(T M) [14].

Let K(Xp,Yp) be the sectional curvature for 2−plane spanned by Xp and Yp, pM. M is said to have constant ϕ−holomorphic sectional curvature if K(Xp,ϕXp) is constant for any point p and for any unit vector Xp≠ 0 such that η(Xp) = 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,

R(X,Y,Z,W)=c+34{g(X,W)g(Y,Z)g(X,Z)g(Y,W)}+c14{g(φX,W)g(φY,Z)g(φX,Z)g(φY,W)2g(φX,Y)g(φZ,W)+g(X,Z)η(Y)η(W)g(Y,Z)η(X)η(W)+g(Y,W)η(X)η(Z)}.
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)=2n+1i=1R(Ei,X,Y,Ei),
for X,Y ∈ Γ(T M). Then from (2.5) on M(c), we have,
S(X,Y)=(c3)n+(c+1)2g(φX,φY)2nη(X)η(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.

3 -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 ) and kerϕ (whose differentiable distribution is usually denoted by ) The sectional curvature of planar sections spanned by vector fields of called ℒ −sectional curvature.

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

TM=.

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

K(X,Y)=c34+3c+14g(X,φY)2

Corollary 3.1

Let M be Lorentzian Kenmotsu space form. If ℒ −sectional curvature K 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=c34c=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(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.

4 Semi Invariant Submanifolds of a Lorentzian Kenmotsu Space Form

Definition 4.1

An (2m + 1)−dimensional Riemannian submanifold M of Lorentzian Kenmotsu space form M¯ is called a semi invariant submanifold if ξ is tangent to M and there exists on M two differentiable distributions D and Don M satisfying:

  1. (i)T M = DDsp{ξ };
  2. (ii)The distribution D is invariant under ϕ, that is ϕDx = Dx for any xM;
  3. (iii)The distribution Dis anti-invariant under ϕ, that is, φDxTxM M for any xM, where TxM and TxMare the tangent space of M at x.

Now, we choose a local field of orthonormal frames {E1,...,E2p,E2p+1,...,E2m,ξ } on M. Then we have,

D=sp{E1,...,E2p},D=sp{E2p+1,...,E2m}
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 manifold2n+1with its usual Lorentzian Kenmotsu structure given by

η=dz,ξ=zφ(ni=1(Xixi+Yiyi)+Zz)=ni=1(YixiXiyi)+ni=1Yiyizg=e2z(ni=1dxidxi+dyidyi)ɛdzdz
(x1,...,xn,y1,...,yn,z) denoting the Cartesian coordinates on2n+1. The consider a submanifold of7defined by
M=X(u,v,k,l,t)=(u,k,0,v,0,l,t).

Then local frame of T M is given by

e1=x1,e2=y1,e3=x2,e4=y3,e5=z=ξ
and we have
e1*=x3,e2*=y2
which are the a basis of TM. We determine D1 = sp{e1,e2} and D2 = sp{e3,e4}. Then D1, D2are invariant and anti-invariant distribution, respectively. Thus T M = D1D2sp{ξ } is a semi invariant submanifold of7.

Let ¯ be the Levi-Civita connection of M¯ with respect to the g. Then Gauss and Weingarten formulas are given by

¯XY=XY+h(X,Y)
¯XN=XNANX
for any X,Y ∈ Γ(T M) and N ∈ Γ(TM). ∇ is the connection in the normal bundle, h is the second fundamental from of M¯ and AN 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(ANX,Y).

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

We denote by R¯ and R the curvature tensor fields associated with ¯ and ∇ respectively. The Gauss equation is given by

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 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 R¯ satisfied

R(X,Y,Z,W)=c+34{g(X,W)g(Y,Z)g(X,Z)g(Y,W)}+c14{g(φX,W)g(φY,Z)g(φX,Z)g(φY,W)2g(φX,Y)g(φZ,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)).

Theorem 4.1

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from M¯(c) . Then we get Ricci tensor of M,

S(X,Y)={c+34(p+q3)+3c12}g(X,Y){c14(p+q6)+c+12}η(X)η(Y)+i=1p+q{g(h(X,Y),h(Ei,Ei))g(h(Ei,Y),h(X,Ei))}
for all X,Y ∈ Γ(T M).

Proof

Let Γ(T M) = sp{e1,...,ep,ep+1,...,eq,ep+q+1} such that {e1,...,ep} are tangent to D1 and {ep+1,...,eq} are tangent to D2. Then we have,

S(X,Y)=i=1qR(X,Ei,Ei,Y).

Now, using (4.6), we get

S(X,Y)={c+34(p1)+3c14}g(X,Y)+c14(3p)η(X)η(Y)+i=1p{g(h(X,Y),h(Ei,Ei))g(h(Ei,Y),h(X,Ei))}+{c+34(q1)+3c14}g(X,Y)+c14(3q)η(X)η(Y)+i=p+1qg(h(X,Y),h(Ei,Ei))g(h(Ei,Y),h(X,Ei))c+34g(X,Y){c+34+c14}η(X)η(Y)
which gives proof.

Corollary 4.1

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space fromM¯(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 fromM¯(c) . Then we have scalar curvature

τ={c+34(p+q3)+3c12}(p+q1)+c14(p+q6)+c+12+1(p+q+1)2H2+h2.

Proof

From (4.7) by using X = Y = ek we get

τ=k=1p+q+1S(ek,ek).

The proof is completed.

Proposition 4.3

Let M be semi invariant submanifold of Lorentzian Kenmotsu space formM¯(c). Then

R(X,Y,Z,W)=c+34{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))
for all X,Y,Z,W ∈ Γ(D).

Proof

Using (4.6). For all X,Y,Z,W ∈ Γ(D), since ϕX,ϕY,ϕZ,ϕWϕDT M we have (4.8).

Corollary 4.2

Let M be semi invariant submanifold of Lorentzian Kenmotsu space formM¯(c). If Dis totally geodesic, then Dis flat if and only if c = −3.

Proposition 4.4

Let M be semi invariant submanifold of Lorentzian Kenmotsu space formM¯(c). Then

S(X,Y)=c+34(q1)g(X,Y)+i=1q{g(h(X,Y),h(Ei,Ei))g(h(Ei,Y),h(X,Ei))}
for all X,Y ∈ Γ(D), where S is Ricci tensor.

Proof

Using (4.8). From S(X,Y)=i=1qR(X,Ei,Ei,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 M¯ . If Dis totally geodesic, then distribution Dis Einstein.

Corollary 4.4

Let M be semi inavariant submanifold of Lorentzian Kenmotsu space formM¯(c). If Dis totally geodesic, then

τD=c+34q(q1)
where τ is the scalar curvature.

Proposition 4.5

Let M be semi invariant submanifold of Lorentzian Kenmotsu space form M¯(c)(c). Then the Ricci curvature determined by D

S(X,Y)={c+34(p1)+3c14}g(X,Y)
for all X,Y ∈ Γ(D).

Proof

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

R(X,Y,Z,W)=c+34{g(X,W)g(Y,Z)g(X,Z)g(Y,W)}+c14{g(X,φW)g(Y,φZ)g(X,φZ)g(Y,φW)

Then, from S(X,Y)=i=1pR(X,Ei,Ei,Y) , using last equation, this complates the proof.

Corollary 4.5

Let M be semi invariant submanifold of Lorentzian Kenmotsu space formM¯(c). Then the scalar curvature determined by D is given

τD=p(c+3)2p1)+3(c1)4.

Corollary 4.6

Let M be a semi-invariant submanifold of a Lorentzian Kenmotsu space from M¯ . If D is totally geodesic, then distribution D is Einstein.

Theorem 4.6

Let M be semi invariant submanifold of Lorentzian Kenmotsu space formM¯(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),

R(X,φX,X,φX)=c+34{g(X,φX)g(φX,X)g(X,X)g(φX,φX)}+c14{g(φX,φX)g(φ2X,X)g(φX,X)g(φ2X,φX)2g(φX,φX)g(φX,φX)+g(h(X,φX),h(φX,X))g(h(φX,φX),h(X,X)).

Then,

R(X,φX,X,φX)=c2h(X,X)2.

5 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 Dsp{ξ } is integrable and locally M is a Riemannian product M1× M2, where M1 (resp. M2) is leaf of Dsp{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 formM¯(c). Then

R(X,φX,Z,φZ)=2(h(X,Z)2c14)
for any unit vector fields XD and ZD.

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 formM¯(c). Then,

h2pq(1c)+2qp

Proof

Since h is fundamental form, we have

h2=i,j=12ph(Ei,Ej)2+k,l=2p+12mh(Ek,El)2+2i=12pk=2p+12mh(Ei,Ek)2+2k=2p+12mh(Ek,ξ)2
from (4.1)
h2=pq(1c)+2qp+i,j=12ph(Ei,Ej)2+k,l=2p+12mh(Ek,El)2
which gives (5.2).

Proposition 5.3

Let M be a proper semi invariant product of an a Lorentzian Kenmotsu space formM¯(c). Then,

R(X,Y,Z,W)=0
for all X,Y ∈ Γ(D ⊕ sp{ξ }) and Z,W ∈ Γ(D ).

Proof

Let M be semi invariant submanifold of Lorentzian Kenmotsu manifold M¯ . Then for all Z,W ∈ Γ(D),

φZ,φWφDTM.

Using (4.6), which complates the proof.

Proposition 5.4

Let M be a proper semi invariant product of a Lorentzian Kenmotsu M¯ . Then

R(X,Y,Z,W)=0
for all X,Y ∈ Γ(D) and Z,W ∈ Γ(D⊕ sp{ξ }).

References

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    M. Atçeken, S. Dirik and Ü. Yıldırım, (2017), An inequality for warped product semi-inavriant submanifolds of a normal paracontact metric manifold, Filomat, 31(19), 6233–6240

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    Y.J. Suh, K. Mandal and U.C. De, (2017), A note on invariant submanifolds of CR-Integrable almost Kenmotsu manifolds, Filomat, 31(19), 6211–6218.

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    T. Takahashi, (1969), Sasakian manifold with pseudo-Riemannian metric, Tohoku Math. J., 21(2), 271–290.

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  • [1]

    M. Atçeken, S. Dirik and Ü. Yıldırım, (2017), An inequality for warped product semi-inavriant submanifolds of a normal paracontact metric manifold, Filomat, 31(19), 6233–6240

  • [2]

    Benn, I.M.; Tucker, R.W., (1987), An introduction to Spinors and Geometry with Applications in Physics, Adam Hilger, Bristol.

  • [3]

    D.E. Blair, (2002), Riemannian geometry of contact and symplectic manifolds, Progr. Math., 203, Birkhäuser Boston, Boston, MA.

  • [4]

    R.L. Bishop and B. O’Neill, (1969), Manifolds of negative curvature, Trans. Amer. Math. Soc.145, 1–50.

  • [5]

    Bishop, Richard L.; Goldberg, Samuel I., (1980), Tensor Analysis on Manifolds, New York, Dover.

  • [6]

    W. M. Boothby and H. C. Wang, (1958), On contact manifolds, Ann. of Math. 68(2), 721–734.

  • [7]

    Duggal. K. L., (1990), Speace time manifold and contact Manifolds, Int. J. of math. and mathematical science,13,545–554.

  • [8]

    J. W. Gray, (1959), Some global properties of contact structures, Ann. of Math., 69(2), 421–450.

  • [9]

    K. Kenmotsu, (1972), A class of almost contact Riemannian manifolds, TohokuMath. J. II Ser., 24, 93–103.

  • [10]

    D. Janssens and L. Vanhecke, (1981), Almost contact structures and curvature tensors, Kodai Math. j., 4, 1–27.

  • [11]

    Mihai,I., Oiaga, A. and Roşca,R., (1999), Lorentzian Kenmotsu Manifolds Having Two Skew Symmetric Conformai Vector, Bull. Math. Soc. Sc. Math. Roumanie, 42(90), 237–251.

  • [12]

    G. Pitiş, (2007), Geometry of Kenmotsu manifolds, Publishing House of Transilvania University of Braşov, Braşov.

  • [13]

    D.G. Prakasha,C.S. Bagewadi and N.S. Basavarajappa, (2008), On Lorentzian β-Kenmotsu manifolds, Int. J. Math. Anal, 17–20, 919–927.

  • [14]

    R. Roşca, (1991), On Lorentzian Kenmotsu manifolds, Atti Accad. Peloritana Pericolanti, Cl. Sci., 69, 15–30.

  • [15]

    S. Sasaki, (1960), On differentiate manifolds with certain structures which are closely related to almost contact structure I, Tohoku Math. J., 12, 459–476

  • [16]

    Y.J. Suh, K. Mandal and U.C. De, (2017), A note on invariant submanifolds of CR-Integrable almost Kenmotsu manifolds, Filomat, 31(19), 6211–6218.

  • [17]

    T. Takahashi, (1969), Sasakian manifold with pseudo-Riemannian metric, Tohoku Math. J., 21(2), 271–290.

  • [18]

    S. Tanno, (1969), Sasakian manifolds with constant ϕ-holomorphic sectional curvature, Tohoku Math. J., 21(2), 501–507.

  • [19]

    S. Uddin, C. Ozel and V.A. Khan, (2012), Warped product CR-submanifolds of Lorentzian β -Kenmotsu manifolds, Pub. De L. Mat., 92(106), 157–163.