# Hemi-Slant Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

Authors: S. M. Khursheed Haider; Mamta Thakur; Advin
Journal: ISRN Geometry
 (2012)

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## Abstract

We introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We give an example of hemi-slant lightlike submanifold and establish two characterization theorems for the existence of such submanifolds. We prove some theorems which ensure the existence of minimal hemi-slant lightlike submanifolds and obtain a condition under which the induced connection∇ on M is a metric connection. An example of proper minimal hemi-slant lightlike submanifolds is also given.


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## Body

## 1. Introduction

Given a semi-Riemannian manifold, one can consider its lightlike submanifold whose study is important from application point of view and difficult in the sense that the intersection of normal vector bundle and tangent bundle of these submanifolds is nonempty. This unique feature makes the study of lightlike submanifolds different from the study of nondegenerate submanifolds. The general theory of lightlike submanifolds was developed by Kupeli [1]; Duggal and Bejancu [2]. On the other hand, the concepts of transversal and screen transversal lightlike submanifolds of an indefinite Kaehler manifold were given by Sahin [3, 4]. In Sasakian setting, these submanifolds were investigated by Yildirim and Sahin [5–7].In the present paper, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. The paper is arranged as follows. In Section2, we recall definitions for indefinite Kenmotsu manifolds and give basic information on the lightlike geometry needed for this paper. In Section 3, after defining hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold, we give an example which ensures the existence of hemi-slant lightlike submanifolds. We obtain integrability conditions of distributions involved, investigate the geometry of leaves of distributions, and establish two characterization theorems for the existence of hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold. We prove that there does not exist curvature invariant hemi-slant lightlike submanifold in an indefinite Kenmotsu space form with some condition on c and obtain a geometric condition under which the induced connection on M is a metric connection. In Section 4, we study minimal hemi-slant lightlike submanifolds and give an example of proper minimal hemi-slant lightlike submanifolds immersed in R29.


## 2. Preliminaries

An odd dimensional semi-Riemannian manifold(M¯,g¯) is said to be an indefinite contact metric manifold [8] if there exists a (1,1) tensor field ϕ, a vector field V, called the characteristic vector field, and its 1-form η satisfying
(2.1)g¯(ϕX,ϕY)=g¯(X,Y)-ϵη(X)η(Y),g¯(V,V)=ϵϕ2X=-X+η(X)V,g¯(X,V)=ϵη(X),∀X,Y∈Γ(TM),
where ϵ=±1. It follows that ϕV=0,ηoϕ=0,η(V)=ϵ.An indefinite almost contact metric manifoldM¯ is said to be an indefinite Kenmotsu manifold [9] if
(2.2)∇¯XV=-X+η(X)V,(2.3)(∇¯Xϕ)Y=-g¯(ϕX,Y)V+ϵη(Y)ϕX.
for any X,Y∈Γ(TM¯).We follow [2] for the notation and formulae used in this paper. A submanifold (Mm,g) immersed in a semi-Riemannian manifold (M¯m+n,g¯) is called a lightlike submanifold if the metric g induced from g¯ is degenerate and the radical distribution RadTM is of rank r, where 1≤r≤m. Let S(TM) be a screen distribution which is a semi-Riemannian complementary distribution of RadTM in TM, that is,
(2.4)TM=RadTM⊥S(TM).
Consider a screen transversal vector bundle S(TM⊥), which is a semi-Riemannian complementry vector bundle of RadTM in TM⊥. Since for any local basis {ξi} of RadTM, there exist a local null frame {Ni} of sections with values in the orthogonal complement of S(TM⊥) in [S(TM)]⊥ such that g¯(ξi,Nj)=δij, it follows that there exist a lightlike transversal vector bundle ltr(TM) locally spanned by {Ni} [2, page 144]. Let tr(TM) be complementary (but not orthogonal) vector bundle to TM in TM¯|M. Then
(2.5)tr(TM)=ltr(TM)⊥S(TM⊥).TM¯|M=S(TM)⊥[(RadTM)⊕ltr(TM)]⊥S(TM⊥).
The Gauss and Weingarten equations are as follows:
(2.6)∇¯XY=∇XY+h(X,Y),∀X,Y∈Γ(TM),∇¯XU=-AUX+∇XtU,∀X∈Γ(TM),U∈Γ(tr(TM)),
where {∇XY,AUX} and {h(X,Y),∇XtU} belong to Γ(TM) and Γ(tr(TM)), respectively, ∇ and ∇t are linear connections on M and on the vector bundle tr(TM), respectively. Moreover, we have
(2.7)∇¯XY=∇XY+hl(X,Y)+hs(X,Y),(2.8)∇¯XN=-ANX+∇XlN+Ds(X,N),(2.9)∇¯XW=-AWX+∇XsW+Dl(X,W),
for each X,Y∈Γ(TM), N∈Γ(ltr(TM)) and W∈Γ(S(TM⊥)). If we denote the projection of TM on S(TM) by P, then by using (2.7)–(2.9) and the fact that ∇¯ is a metric connection, we obtain
(2.10)g¯(hs(X,Y),W)+g¯(Y,Dl(X,W))=g(AWX,Y),(2.11)g¯(Ds(X,N),W)=g¯(N,AWX).
From the decomposition of the tangent bundle of a lightlike submanifold, we have
(2.12)∇XPY=∇X*PY+h*(X,PY),(2.13)∇Xξ=-Aξ*X+∇X*tξ,
for X,Y∈Γ(TM) and ξ∈Γ(RadTM). By using the above equation, we get
(2.14)g¯(hl(X,PY),ξ)=g(Aξ*X,PY).
It is important to note that the induced connection ∇ on M is not a metric connection whereas ∇* is a metric connection on S(TM).We denote curvature tensors ofM¯ and M by R¯ and R, respectively. The Gauss equation for M is given by
(2.15)R¯(X,Y)Z=R(X,Y)Z+Ahl(X,Z)Y-Ahl(Y,Z)X+Ahs(X,Z)Y-Ahs(Y,Z)X+(∇Xhl)(Y,Z)-(∇Yhl)(X,Z)+Dl(X,hs(Y,Z))-Dl(Y,hs(X,Z))+(∇Xhs)(Y,Z)-(∇Yhs)(X,Z)+Ds(X,hl(Y,Z))-Ds(Y,hl(X,Z)),
for all X,Y,Z∈Γ(TM).The curvature tensorR¯ of an indefinite Kenmotsu space form M¯(c) is given by [9] as follows:
(2.16)R¯(X,Y)Z=c-34{g¯(Y,Z)X-g¯(X,Z)Y}+c+14{η(X)η(Z)Y-η(Y)η(Z)X+g¯(X,Z)η(Y)V-g¯(Y,Z)η(X)V+g¯(ϕY,Z)ϕX-g¯(ϕX,Z)ϕY-2g¯(ϕX,Y)ϕZ},
for any X,Y∈Γ(TM¯).From now on, we denote(M,g,S(TM),S(TM⊥)) by M in this paper.


## 3. Hemi-Slant Lightlike Submanifolds

The concept of pseudoslant (antislant) submanifolds with definite metric is given by Carriazo [10] which was renamed by Sahin as hemi-slant submanifolds and studied their warped product in Kaehler manifolds [11]. In this section, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold for which we need the following lemma from [3].Lemma 3.1.
LetM be a 2q-lightlike submanifold of an indefinite Kaehler manifold M¯ with constant index 2q such that 2q<dim(M). Then the screen distribution S(TM) of lightlike submanifold M is Riemannian.In view of the above lemma, we can define hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold as follows.Definition 3.2.
LetM be a 2q-lightlike submanifold of an indefinite Kenmotsu manifold M¯ of index 2q with the structure vector field V tangent to M such that 2q<dimM. Then, the submanifold M is said to be hemi-slant lightlike submanifold of M¯ if the following conditions are satisfied: (i)
RadTM is a distribution on M such that
(3.1)ϕ(RadTM)=ltr(TM);(ii)
for allx∈U⊂M and for each non zero vector field X tangent to S(TM)=Dθ⊥{V} if X and V are linearly independent, then the angle θ(X) between ϕX and the vector space S(TM) is constant, where Dθ is the complementary distribution to V in the screen distribution S(TM).The angleθ(X) is called the slant angle of the distribution Dθ. A hemi-slant lightlike submanifold is said to be proper if Dθ≠0 and θ≠0,π/2.From Definition3.2, we have the following decomposition:
(3.2)TM=RadTM⊥Dθ⊥{V},tr(TM)=ltr(TM)⊥FDθ⊥μ,TM¯=(RadTM⊕ltr(TM))⊥Dθ⊥FDθ⊥μ⊥{V}.Denoting(Rq2m+1,ϕo,V,η,g) as a manifold Rq2m+1 with its usual Kenmotsu structure given by
(3.3)η=dz,V=∂Z,g¯=η⊗η+e2z{-(Σi=1q/2dxi⊗dxi+dyi⊗dyi)+Σi=q+1mdxi⊗dxi+dyi⊗dyi},ϕo(Σi=1m(Xi∂xi+Yi∂yi)+Z∂z)=Σi=1m(Yi∂xi-Xi∂yi),
where (xi,yi,z) are the cartesian coordinates.The existence of proper hemi-slant lightlike submanifolds in indefinite Kenmotsu manifolds is ensured by the following example.Example 3.3.
LetM be a submanifold of M¯=(R29,g¯) defined by
(3.4)x1=s,x2=t,x3=usinw,x4=sinu,y1=t,y2=s,y3=ucosw,y4=cosu,
where u,w∈(0,π/2) and R29 is a semi-Euclidean space of signature (-,-,+,+,+,+,+,+,+).
Then, a local frame ofTM is given by
(3.5)ξ1=e-z(∂x1+∂y2),ξ2=e-z(∂x2+∂y1),X1=e-z(sinw∂x3+cosu∂x4+cosw∂y3-sinu∂y4),X2=e-z(ucosw∂x3-usinw∂y3),V=∂Z.
We see that M is a 2-lightlike submanifold with RadTM=span{ξ1,ξ2} and S(TM)=span{X1,X2}⊥{V} which is Riemannian and can be easily proved that S(TM) is a slant distribution with slant angle θ=π/4. Moreover, the screen transversal bundle S(TM⊥) is spanned by
(3.6)W1=e-z(sinu∂x4+cosu∂y4),W2=e-z(sinw∂x3-cosu∂x4+cosw∂y3+sinu∂y4),
and the transversal lightlike bundle ltr(TM) is spanned by
(3.7)N1=-e-z2(∂x1-∂y2),N2=e-z2(∂x2-∂y1).
It is not difficult to see that
(3.8)ϕξ1=N2,ϕξ2=N1.
Hence, M is a hemi-slant lightlike submanifold of R29.For anyX∈Γ(TM), we write
(3.9)ϕX=TX+FX,
where TX and FX are the tangential and transversal components of ϕX, respectively. Similarly, for W∈Γ(tr(TM)), we write
(3.10)ϕW=BW+CW,
where BW and CW are the tangential and transversal components of ϕW, respectively. If the projections on the distributions S(TM)=Dθ⊥{V} and RadTM are denoted by Q¯1 and Q2, respectively, then any X tangent to M can be written as
(3.11)X=Q¯1X+Q2X.
We observe that Q¯1X=Q1X+η(X)V. By applying ϕ to (3.11) and using (3.9), we conclude that
(3.12)ϕX=TQ1X+FQ1X+FQ2X,(3.13)ϕQ2X=FQ2X,TQ2X=0,TQ1X∈Γ(S(TM)).The following two theorems are the characterizations of hemi-slant lightlike submanifolds which are similar to the characterization of slant submanifolds in indefinite Hermitian manifolds given by Sahin [12].Theorem 3.4.
The necessary and sufficient condition for a2q-lightlike submanifold M of an indefinite Kenmotsu manifold M¯ of index 2q with structure vector field V tangent to M to be hemi-slant lightlike submanifold is that (i)
ϕ(ltr(TM)) is a distribution on M;(ii)
for any vector fieldX tangent to M linearly independent of structure vector field V, there exists a constant λ∈[-1,0] such that
(3.14)(Q1T)2Q1X=λ(Q1¯X-η(Q1¯X)V),where λ=-cos2θ and θ is the slant angle of M.Proof.
Assume thatM is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then ϕ(RadTM)=ltr(TM), from which we have ϕ(ltr(TM))=Rad(TM). Hence ϕ(ltr(TM)) is a distribution on M. In order to prove (ii), we observe that, for any X∈Γ(TM) linearly independent of structure vector field V and Q1X∈Γ(Dθ), one has
(3.15)cosθ(Q1X)=-g(Q1X,(Q1T)2Q1X)|Q1X||TQ1X|.
But cosθ(Q1X) can also be computed as
(3.16)cosθ(Q1X)=|TQ1X||ϕQ1X|.
From (3.15) and (3.16), we arrive at
(3.17)cos2θ(Q1X)=-g(Q1X,(Q1T)2Q1X)|Q1X|2.
Taking into account that θ(Q1X) is constant on Dθ, from (3.17) we infer that
(3.18)(Q1T)2Q1X=λQ1X=λ(Q¯1X-η(Q¯1X)V),
where we have used λ=-cos2θ(Q1X). Clearly λ∈[-1,0]. Converse part directly follows from (i) and (ii).Theorem 3.5.
The necessary and sufficient condition for a2q-lightlike submanifold M of an indefinite Kenmotsu manifold M¯ of index 2q with structure vector field V tangent to M to be hemi-slant lightlike submanifold is that (i)
ϕ(ltr(TM)) is a distribution on M,(ii)
for any vector field X tangent toM linearly independent of structure vector field V, there exists a constant μ∈[-1,0] such that
(3.19)BFQ¯1X=μ(Q¯1X-η(Q¯1X)V),
whereλ=-cos2θ and θ is the slant angle of M.Proof.
IfM is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯ of index 2q, then ϕ(ltr(TM)) is a distribution on M. For the proof of (ii), applying ϕ to (3.12) and using (3.9) and (3.10), we arrive at
(3.20)-X+η(X)V=(Q1T)2Q1X+FQ1TQ1X+BFQ1X+CFQ1X+BFQ2X.
Comparing the components of screen distribution on both sides of the above equation, we get
(3.21)-Q1¯X+η(Q1¯X)V=(Q1T)2Q1X+BFQ1X.
In view of Theorem 3.4 and the fact that M is a hemi-slant lightlike submanifold, we conclude that
(3.22)(Q1T)2Q1X=-cos2θ(Q¯1X-η(Q¯1X)V).
Hence (ii) follows from (3.21) and (3.22) together with FV=0. Conversely, assume that (i) and (ii) hold. From (ii) and (3.21), we observe that
(3.23)(Q1T)2Q1X=-(1+μ)Q1X.
If we take -(1+μ)=λ, then λ∈[-1,0]. Hence the proof of Theorem follows from Theorem 3.4.Corollary 3.6.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then
(3.24)g(TQ1¯X,TQ1¯Y)=cos2θ|S(TM)[g(Q¯1X,Q¯1Y)-η(Q¯1X)η(Q¯1Y)],(3.25)g(FQ1¯X,FQ1¯Y)=sin2θ|S(TM)[g(Q¯1X,Q¯1Y)-η(Q¯1X)η(Q¯1Y)]
for any X,Y∈Γ(TM).Differentiating (3.12), taking into account that M¯ is an indefinite Kenmotsu manifold and then comparing tangential, lightlike transversal and screen transversal parts, we obtain
(3.26)(∇XTQ1)Y=AFQ1YX+AFQ2YX+Bhl(X,Y)+Bhs(X,Y)-g(ϕX,Y)V+η(Y)TQ1X(3.27)(∇XF)Q2Y=-hl(X,TQ1Y)-Dl(X,FQ1Y)+η(Y)FQ2X(3.28)(∇XF)Q1Y=Chs(X,Y)-hs(X,TQ1Y)-Ds(X,FQ2Y)+η(Y)FQ1X.The integrability conditions of the distributions involved in the definition of hemi-slant lightlike submanifolds are given by the following theorems.Theorem 3.7.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯ of index 2q. Then the distribution RadTM is integrable if and only if(i)
AFQ2YX-AFQ2XY=0;(ii)
Ds(X,FQ2Y)=Ds(Y,FQ2X)for any X,Y∈Γ(RadTM).Proof.
For anyX,Y∈Γ(RadTM), from (3.26) we have
(3.29)-TQ1∇XY=AFQ2YX+Bhl(X,Y)+Bhs(X,Y)-g(ϕX,Y)V.
Interchanging X and Y in (3.29) and subtracting (3.29) from the resulting equation, we get
(3.30)TQ1[X,Y]=AFQ2XY-AFQ2YX+2g(ϕX,Y)V.
Taking inner product of the above equation with ϕZ for Z∈Γ(Dθ), after simplification, we obtain
(3.31)g([X,Y],Z)=sec2θ{g(AFQ2XY-AFQ2YX),ϕZ}.
Also, from (3.28) we have
(3.32)-FQ1∇XY=Chs(X,Y)-Ds(X,FQ2Y).
Interchanging the role of X and Y in (3.32) and subtracting (3.32) from the resulting equation, we get
(3.33)FQ1[X,Y]=Ds(X,FQ2Y)-Ds(Y,FQ2X).
Taking inner product of above equation with ϕZ, we arrive at
(3.34)g([X,Y],Z)=cosec2θg(Ds(X,FQ2Y)-Ds(Y,FQ2X),ϕZ).
Hence our assertion follows from (3.31) and (3.34).Theorem 3.8.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯ of index 2q. Then the distribution Dθ⊥{V} is integrable if and only if
(3.35)hl(X,TQ1Y)-hl(Y,TQ1X)+Dl(X,FQ1Y)-Dl(Y,FQ1X)=0,
for any X,Y∈Γ(S(TM)).Proof.
For anyX,Y∈Γ(Dθ⊥{V}), from (3.27) we have
(3.36)-FQ2∇XY=-hl(X,TQ1Y)-Dl(X,FQ1Y).
Interchanging X and Y in (3.36) and subtracting (3.36) from the resulting equation, we obtain
(3.37)FQ2[X,Y]=hl(X,TQ1Y)-hl(Y,TQ1X)+Dl(X,FQ1Y)-Dl(Y,FQ1X),
which proves our assertion.In view of the above argument, if we restrict the vector fieldsX and Y to Dθ, then we see that the distribution Dθ is not integrable.The following two theorems deals with the totally geodesic foliations of the leaves of the distributionsS(TM)=Dθ⊥{V} and RadTM.Theorem 3.9.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the distribution Dθ⊥{V} defines totally geodesic foliation in M if and only if AFQ1TQ1YX-BDl(X,FQ1Y) has no component in RadTM for any X,Y∈Γ(S(TM)).Proof.
Using (2.7), (2.9), (3.12), and (3.14), for any X,Y∈Γ(S(TM)) and N∈Γ(ltr(TM)), we obtain
(3.38)g¯(∇¯XY,N)=cos2g¯(∇¯XY,N)+g¯(AFQ1TQ1YX-BDl(X,FQ1Y),N),
which can be written as
(3.39)sin2θg¯(∇¯XY,N)=g¯(AFQ1TQ1YX-BDl(X,FQ1Y),N).
Thus, our assertion follows from (3.39).Theorem 3.10.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯ of index 2q. Then the distribution RadTM defines totally geodesic foliation in M if and only if h*(ξ1,TQ1X)-AFQ1Xξ1 has no component in RadTM for any ξ1∈Γ(RadTM) and X∈Γ(Dθ).Proof.
For anyξ1,ξ2∈Γ(RadTM) and X∈Γ(Dθ), from (2.7), (2.9), (2.12), and (3.12), we obtain
(3.40)g(∇ξ1ξ2,X)=-g¯(ϕξ2,h*(ξ1,TQ1X)-AFQ1Xξ1).
Hence our assertion follows from (3.40).The necessary and sufficient conditions under which the induced connection∇ on a hemi-slant lightlike submanifold immersed in indefinite Kenmotsu manifold to be metric connection is given by the following result.Theorem 3.11.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifolds M¯. Then the induced connection ∇ on M is a metric connection if and only if
(3.41)TQ1AFQ2YX-BDs(X,FQ2Y)=0,
for any X∈Γ(TM) and Y∈Γ(RadTM).Proof.
ForX∈Γ(TM) and Y∈Γ(RadTM), from (3.1), we have
(3.42)∇¯XY=-ϕ∇¯XϕY-(∇¯Xϕ)ϕY.
Making use of (2.3), (2.7), and (2.9) in the above equation, we arrive at
(3.43)∇XY+hl(X,Y)+hs(X,Y)=-ϕ(-AFQ2YX+∇XlFQ2Y+Ds(X,FQ2Y))+g(ϕX,ϕY)V.
Using (3.9) and (3.10) in (3.43) and comparing the tangential components of the resulting equation, we obtain
(3.44)∇XY=TQ1AFQ2YX-B∇XlFQ2Y-BDs(X,FQ2Y),
from which our assertion follows.Theorem 3.12.
LetM be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then T is never parallel on M.Proof.
ForX∈Γ(TM), from (2.2) and (2.7), we have
(3.45)∇XV+hl(X,V)+hs(X,V)=-X+η(X)V.
Comparing the tangential Components of the above equation, we get
(3.46)∇XV=-X+η(X)V.
Since TV=0, so differentiating TV=0 covariantly with respect to X we obtain
(3.47)(∇XT)V+T(∇XV)=0.
Using (3.46) in the above equation, we get
(3.48)(∇XT)V=TX,
from which our assertion follows.Similar to the notion of curvature invariant submanifolds given by Atceken and Kilic [13], we define curvature invariant lightlike submanifolds as follows.Definition 3.13.
LetM be a 2q-lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is said to be curvature invariant lightlike submanifold of M¯ if the covariant derivatives of the second fundamental forms hl and hs of M satisfy
(3.49)(∇¯Xhl)(Y,Z)-(∇¯Yhl)(X,Z)=0,(∇¯Xhs)(Y,Z)-(∇¯Yhs)(X,Z)=0,
for any X,Y∈Γ(TM).Theorem 3.14.
There does not exist any curvature-invariant proper hemi-slant lightlike submanifold of an indefinite Kenmotsu space formM¯(c) with c≠-1.Proof.
Using (2.15) and (2.16), for any X,Y∈Γ(S(TM)) and Z∈Γ(RadTM), we obtain
(3.50)(∇¯Xhs)(Y,Z)-(∇¯Yhs)(X,Z)=0,(3.51)(∇¯Xhl)(Y,Z)-(∇¯Yhl)(X,Z)=-c+12{g(TQ1X,Y)FQ2Z}.
Thus, our assertion follows from (3.50) and (3.51).


## 4. Minimal Hemi-Slant Submanifolds

In this section, we study minimal hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold. First we recall the following definition of minimal lightlike submanifolds of a semi-Riemannian manifoldM¯ given by Bejan and Duggal [14].Definition 4.1.
A lightlike submanifold(M,g,S(TM)) isometrically immersed in a semi-Riemannian manifold (M¯,g¯) is said to be minimal if(i)
hs=0 on RadTM and(ii)
traceh=0,
where trace is written with respect tog restricted to S(TM).In [14], it has been shown that the above definition is independent of S(TM) and S(TM⊥), but it depends on the choice of transversal bundle tr(TM).For the existence of proper minimal hemi-slant lightlike submanifolds, we give the following example.Example 4.2.
LetM¯=(R29,g¯) be a semi-Euclidean space of signature (-,-,+,+,+,+,+,+,+) with respect to canonical basis (∂x1,∂y1,∂x2,∂y2,∂x3,∂y3,∂x4,∂y4,∂z).
Consider an almost contact structureϕ on R29 defined by ϕ(x1, y1, x2, y2, x3, y3, x4, y4, z)=(-y1,x1, -y2,x2, -x4cosα-y3sinα, -y4cosα+x3sinα, x3cosα+y4sinα, y3cosα-x4sinα,0).
For someα∈(0,π/2). Let M be a submanifold of (R29,ϕ) defined by the following equations:
(4.1)x1=u1,x2=u2,x3=u3,x4=u4,y1=u2,y2=u1,y3=sinu3sinhu4,y4=cosu3coshu4.
Then the tangent bundle TM of M is spanned by
(4.2)ξ1=e-z(∂x1+∂y2),ξ2=e-z(∂x2+∂y1),X1=e-z(∂x3+cosu3sinhu4∂y3-sinu3coshu4∂y4),X2=e-z(∂x4+sinu3coshu4∂y3+cosu3sinhu4∂y4),V=∂z.
Suppose that the distribution RadTM=span{ξ1,ξ2}. Then M is a 2-lightlike submanifold of M¯. If we take S(TM)=span{X1,X2⊥{V}}, then S(TM) is Riemannian vector subbundle and one can easily see that S(TM) is a slant distribution with slant angle α. A direct calculation shows that the screen transversal bundle is spanned by
(4.3)W1=e-z(-coshu4sinhu4∂x3+cosu3coshu4∂y3-sinu3cosu3∂x4+sinu3sinhu4∂y4),W2=e-z(cosu3sinu3∂x3+sinu3sinhu4∂y3-coshu4sinhu4∂x4+cosu3coshu4∂y4),
and ltr(TM) is spanned by
(4.4)N1=-e-z2(-∂y2+∂x1),N2=e-z2(-∂y1+∂x2).
For a vector field X tangent to M, from Gauss formula and after calculations, we get
(4.5)hl=0,hs(X,ξ1)=0,hs(X,ξ2)=0,hs(X1,X1)=-e-z(sinh2u4+cos2u3)cosh4u4-sin4u3W2,hs(X2,X2)=e-z(sinh2u4+cos2u3)cosh4u4-sin4u3W2,hs(X1,X2)=e-z(sinh2u4+cos2u3)cosh4u4-sin4u3W1.
Hence, M is a minimal proper hemi-slant lightlike submanifold of R29, and the induced connection on M is a metric connection. We note that M is neither totally geodesic nor totally umbilical.Now, we prove a lemma which will be helpful for the study of minimal hemi-slant lightlike submanifolds.Lemma 4.3.
LetM be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯ such that dim(Dθ)=dim(S(TM⊥)). If {e1,e2,…,em} is a local orthonormal basis of Γ(Dθ), then {cscθFe1,…,cscθFem} is an orthonormal basis of S(TM⊥).Proof.
Suppose that{e1,e2,…,em} is a local orthonormal basis of Dθ. From (3.25) together with the fact that Dθ is Riemannian, we conclude that
(4.6)g¯(cscθFei,cscθFej)=δij,
where i,j=1,2,3,…,m. Hence our assertion follows from (4.6).The existence of minimal hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold is given by the following two theorems.Theorem 4.4.
LetM be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is minimal if and only if
(4.7)traceAξj*|S(TM)=0,traceAWα|S(TM)=0,g(Dl(X,W),Y)=0,
for any X,Y∈Γ(RadTM) and W∈Γ(S(TM⊥)), where {ξj}j=1r is a basis of RadTM and {Wα}α=1m is a basis of S(TM⊥).Proof.
Using∇¯VV=0 and (2.2), we obtain hl(V,V)=hs(V,V)=0. It is known that hl=0 on RadTM [14, Proposition 4.1]. If we take {e1,e2,…,em} as an orthonormal basis of Dθ, then M is minimal if and only if
(4.8)Σk=1mh(ek,ek)=0
and hs=0 on RadTM. From (2.10) and (2.14), we have
(4.9)Σk=1mh(ek,ek)=Σk=1m1rΣj=1mg(Aξj*ek,ek)Nj+1mΣα=1mg(AWαek,ek)Wα.
On the other hand, from (2.10), we infer that hs=0 on RadTM if
(4.10)g¯(Dl(X,W),Y)=0,
for X,Y∈Γ(RadTM) and W∈Γ(S(TM⊥)). Thus, our assertion follows from (4.9) and (4.10).Theorem 4.5.
LetM be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold M¯ such that dim(Dθ)=dim(S(TM⊥)). Then M is minimal if and only if
(4.11)traceAξj*|S(TM)=0,traceAWei|S(TM)=0,g(Dl(X,Fei),Y)=0.
for X,Y∈Γ(RadTM), where {ξk}k=1r is a basis of RadTM and {ej}j=1m is a basis of Dθ.Proof.
From∇¯VV=0 and (2.2), we obtain hl(V,V)=hs(V,V)=0. Recall that hl=0 on RadTM [14, Proposition 4.1]. In view of Lemma 4.3, we observe that {cscθFe1,…,cscθFem} is an orthonormal basis of S(TM⊥). Then, for any X∈Γ(TM) and for some function Ai, i∈(1,2,…,m), we can write
(4.12)hs(X,X)=Σi=1mAicscθFei,
from which we have
(4.13)hs(X,X)=Σi=1mcscθg(AFeiX,X)Fei,
for any X∈Γ(S(TM)). Thus our assertion follows from the above equation and Theorem 4.4.



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*Source: 251213-2012-09-13.xml*