# Geodesic Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

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

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## Abstract

The aim of the present paper is to study geodesic contact screen Cauchy Riemannian (SCR-) lightlike submanifolds, geodesic screen transversal lightlike, and
geodesic transversal lightlike submanifolds of indefinite Kenmotsu manifolds.


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## Body

## 1. Introduction

The study of the geometry of submanifolds of a Riemannian or semi-Riemannian manifold is one of the interesting topics of differential geometry. Despite of some similarities between semi-Riemannian manifolds and Riemannian manifolds, the lightlike submanifolds are different since their normal vector bundle intersect with the tangent bundle making it more interesting and difficult to study. These submanifolds were introduced and studied by Duggal and Bejancu [1]. On the other hand, geodesic CR-lightlike submanifolds in Kähler manifolds were studied by Sahin and Gunes [2], and geodesic lightlike submanifolds of indefinite Sasakian manifolds were investigated by Dong and Liu [3]. In 2006, Sahin [4] initiated the study of transversal lightlike submanifolds of an indefinite Kähler manifold which are different from CR-lightlike [1] and screen CR-lightlike submanifolds [5]. Recently, Sahin [6] introduced the notion of screen transversal lightlike submanifolds of indefinite Kähler manifolds and obtained some useful results. In this paper, we study geometric conditions under which some lightlike submanifolds of an indefinite Kenmotsu manifold are totally geodesic.


## 2. Preliminaries

We follow [1] for the notation and fundamental equations for lightlike submanifolds used in this paper. A submanifold Mm immersed in a semi-Riemannian manifold (M¯m+n,g¯) is called a lightlike submanifold if it admits a degenerate metric g induced from g¯ whose radical distribution RadTM=TM∩TM⊥ is of rank r, where 1≤r≤m and
(2.1)TM⊥=∪{U∈TxM¯:g¯(U,V)=0,∀V∈TxM¯}.
Let S(TM) be a screen distribution which is a semi-Riemannian complementary distribution of RadTM in TM, that is,
(2.2)TM=RadTM⊥S(TM).
Consider a screen transversal vector bundle S(TM⊥), which is a semi-Riemannian complementary vector bundle of RadTM in TM⊥. Since for any local basis {ξi} of RadTM, there exists 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 and g¯(Ni,Nj)=0, it follows that there exists a lightlike transversal vector bundle ltr(TM) locally spanned by {Ni} [1, page 144]. Let tr(TM) be the complementary (but not orthogonal) vector bundle to TM in TM¯|M.Then,(2.3)tr(TM)=ltr(TM)⊥S(TM⊥),TM¯=S(TM)⊥[RadTM⊕ltr(TM)]⊥S(TM⊥).
Let ∇¯ be the Levi-Civita connection on M¯. Then, in view of the decomposition (2.3), the Gauss and Weingarten formulas are given by
(2.4)∇¯XY=∇XY+h(X,Y)∀X,Y∈Γ(TM),(2.5)∇¯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 connection on M and on the vector bundle tr(TM), respectively. Moreover, we have
(2.6)∇¯XY=∇XY+hl(X,Y)+hs(X,Y),(2.7)∇¯XN=-ANX+∇XlN+Ds(X,N),(2.8)∇¯XW=-AWX+∇XsW+Dl(X,W)
for all, 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 (2.6)–(2.8) and the fact that ∇¯ is a metric connection, we obtain
(2.9)g¯(hs(X,Y),W)+g¯(Y,Dl(X,W))=g(AWX,Y),(2.10)g¯(Ds(X,N),W)=g¯(N,AWX).
From the decomposition of the tangent bundle of a lightlike submanifold, we have
(2.11)∇XPY=∇X*PY+h*(X,PY),(2.12)∇Xξ=-Aξ*X+∇X*tξ
for any X,Y∈Γ(TM), and ξ∈Γ(RadTM). By using above equation we obtain
(2.13)g¯(hl(X,PY),ξ)=g(Aξ*X,PY),g¯(h*(X,PY),N)=g(ANX,PY),(2.14)g¯(hl(X,ξ),ξ)=0,Aξ*ξ=0.An odd dimensional semi-Riemannian manifold(M¯,g¯) is said to be an indefinite contact metric manifold [7] if there exists a (1,1) tensor field ϕ, a vector field V, called the characteristic vector field, and its 1-form η satisfying
(2.15)g¯(ϕX,ϕY)=g¯(X,Y)-ϵη(X)η(Y),g¯(V,V)=ϵ,(2.16)ϕ2X=-X+η(X)V,g¯(X,V)=ϵη(X),∀X,Y∈Γ(TM),
where ϵ=±1. It is not difficult to show that ϕV=0,ηoϕ=0,η(V)=ϵ.An indefinite almost contact metric manifoldM¯ is said to be an indefinite Kenmotsu manifold [8] if
(2.17)∇¯XV=-X+η(X)V,(2.18)(∇¯Xϕ)Y=-g¯(ϕX,Y)V+ϵ(Y)ϕX
for any X,Y∈Γ(TM¯).Without loss of generality, we takeϵ=1. For any vector field X tangent to M, we put
(2.19)ϕX=TX+ωX,
where TX and ωX are the tangential and transversal parts of ϕX, respectively.For anyU∈Γ(tr(TM)), we have
(2.20)ϕU=BU+CU,
where BU and CU are the tangential and transversal parts of ϕU, respectively.From now on, we denote(M,g,S(TM),S(TM⊥)) by M in this paper.


## 3. Geodesic ContactSCR-Lightlike Submanifolds

In this section, we study the geometric conditions under which the distributions involved in the definition of a contactSCR-lightlike submanifold M and the submanifold itself are totally geodesic. We recall the following definition of contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold given by Duggal and Sahin [5].Definition 3.1.
A lightlike submanifoldM, tangent to structure vector field V, immersed in an indefinite Kenmotsu manifold (M¯,g¯) is said to be contact SCR-lightlike submanifold of M¯ if the following conditions are satisfied. (a) There exists real nonnull distributionD and D⊥ such that
(3.1)S(TM)=D⊥D⊥⊥{V},ϕ(D⊥)⊂S(TM⊥),D
∩
D
⊥
=
0, where D⊥ is the orthogonal complementary to D⊥{V} in S(TM).(b)ϕD=D,ϕRadTM=RadTM,ϕltr(TM)=ltr(TM).The tangent bundle of a contactSCR-lightlike submanifold is decomposed as
(3.2)TM=D¯⊥D⊥⊥{V},D¯=D⊥RadTM.
We will use the symbol μ to denote the orthogonal complement of ϕD⊥ in S(TM⊥).Definition 3.2.
A contactSCR-lightlike submanifold M of an indefinite Kenmostsu manifold M¯ is said to be(i)
D⊥-totally geodesic contact SCR-lightlike submanifold if h(X,Y)=0 for any X,Y∈Γ(D⊥),(ii)
mixed totally geodesic contactSCR-lightlike submanifold if h(X,Y)=0 for any X∈Γ((D¯)⊥{V}) and Y∈Γ(D⊥).LetM be a contact SCR-lightlike submanifold of indefinite Kenmotsu manifold M¯ and let P and Q be the projection morphisms on D¯ and D⊥, respectively. Then for any vector field X tangent to M, we can write
(3.3)X=PX+QX+η(X)V.
Applying ϕ to (3.3) and using (2.19), we obtain
(3.4)ϕX=TPX+ωQX.
If we denote TPX by TX and ωQX by ωX, then (3.4) can be rewritten as
(3.5)ϕX=TX+ωX,
where TX∈Γ(D¯) and ωX∈Γ(ϕ(D⊥))⊂S(TM⊥).For anyW∈Γ(S(TM⊥)), we have
(3.6)ϕW=BW+CW,
where BW∈Γ(D⊥) and CW∈Γ(μ)⊂S(TM⊥).In view of the above arguments, we are in a position to prove the following characterization theorem for the existence of aD⊥-totally geodesic contact SCR-lightlike submanifold immersed in indefinite Kenmotsu manifolds.Theorem 3.3.
LetM be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifolds M¯. Then M is D⊥-totally geodesic if and only if(i)
hs(X,ϕξ)∈μ.(ii)
AωYX∉D⊥ and ∇XsωY∉μ for any X,Y∈Γ(D⊥).Proof.
Suppose that the contactSCR-lightlike submanifold M is totally geodesic. Then we see that g¯(hl(X,Y),ξ)=0 and g¯(hs(X,Y),W)=0  for all,X,Y∈Γ(D⊥).
Also, from (2.6) and (2.15), we obtain
(3.7)g¯(hl(X,Y),ξ)=g¯(∇¯XϕY-(∇¯Xϕ)Y,ϕξ),
from which we derive
(3.8)g¯(hl(X,Y),ξ)=g¯(Dl(X,ωY),ϕξ),
where we have used (2.8), (2.18), and (3.5). Using (2.9) in the above equation, we get
(3.9)g¯(hl(X,Y),ξ)=-g¯(hs(X,ϕξ),ωY).
On the other hand, making use of (2.6), (2.8), (2.15), (2.18), (3.5), and (3.6), we arrive at
(3.10)g¯(hs(X,Y),W)=-g(AωYX,BW)+g¯(∇XsωY,CW).
Hence, (i) and (ii) follows from (3.9) and (3.10) together with the fact that g¯(hl(X,Y),ξ)=0 and g¯(hs(X,Y),W)=0.
Converse part directly follow from (3.9) and (3.10).The necessary and sufficient conditions for contactSCR-lightlike submanifolds to be mixed totally geodesic is given by the following theorem.Theorem 3.4.
LetM be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is mixed totally geodesic if and only if(i)
hs(X,ϕξ)∈μ,(ii)
AωYX∉D⊥ and ∇XsωY∉μ for any X∈Γ(D¯⊥{V}) and Y∈Γ(D⊥).Proof.
Assume thatM is mixed totally geodesic. Then g¯(hl(X,Y),ξ)=0,g¯(hl(X,Y),W)=0, for any X∈Γ(D¯⊥{V}) and Y∈Γ(D⊥).
Moreover, using (2.6), (2.8), (2.15), (2.18), and (3.5) a direct calculation shows that
(3.11)g¯(hl(X,Y),ξ)=g¯(Dl(X,ωY),ϕξ).
From (2.9) and (3.11), we have
(3.12)g¯(hl(X,Y),ξ)=-g¯(hs(X,ϕξ),ωY).
On the other hand, using (2.6), (2.8), (2.15), (2.18), (3.5), and (3.6), we derive
(3.13)g¯(hs(X,Y),W)=-g(AωYX,BW)+g(∇XsωY,CW).
Thus, (i) and (ii) follow from (3.12) and (3.13) along with g¯(hl(X,Y),ξ)=0,g¯(hl(X,Y),W)=0.Converse part directly follows from (3.12) and (3.13).


## 4. Screen Transversal Lightlike Submanifolds

We begin this section by recalling the following definitions from [6].Definition 4.1.
Anr-lightlike submanifold M of an indefinite Kenmotsu manifold M¯ is said to be screen transversal (ST) lightlike submanifold of M¯ if there exists a screen transversal bundle S(TM⊥) such that
(4.1)ϕ(RadTM)⊂S(TM⊥).Definition 4.2.
AnST-lightlike submanifold of an indefinite Kenmotsu manifold M¯ is said to be (i)
radicalST-lightlike submanifold if S(TM) is invariant with respect to ϕ.(ii)
ST-anti-invariant lightlike submanifold if S(TM) is screen transversal with respect to ϕ, that is,
(4.2)ϕ((S(TM)))⊂S(TM⊥).For a radical screen transversal lightlike submanifoldsM immersed in indefinite Kenmotsu manifold M¯, we will denote the projection morphisms of S(TM) and RadTM by P and Q, respectively. Then for X∈Γ(TM), we can write
(4.3)X=PX+QX.
We apply ϕ to (4.3) and then using (2.19), we obtain
(4.4)ϕX=TPX+ωQX.
Denoting TPX by TX and ωQX by ωX, (4.4) can be we rewritten as
(4.5)ϕX=TX+ωX,
where TX∈Γ(S(TM)) and ωX∈Γ(ϕ(RadTM))⊂S(TM⊥).ForW∈Γ(S(TM⊥)), we write
(4.6)ϕW=BW+C1W+C2W,
where BW∈Γ(RadTM),C1W∈Γ(ltr(TM)), and C2W∈Γ(μ) (μ is the orthogonal complement of ϕ(RadTM)⊕ϕ(ltr(TM)) in S(TM⊥)).The geometric conditions under which the distributionRadTM is totally geodesic is given by the following theorem.Theorem 4.3.
LetM be a radical screen transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then RadTMis totally geodesic if and only if hs(ξ1,BW)+Ds(ξ1,C1W)+∇ξ1sC2W has no component in ϕ(ltr(TM))⊥μ.Proof.
If the distributionRadTM is totally geodesic, then
(4.7)g¯(hl(ξ1,ξ2),ξ3)=0,g¯(hs(ξ1,ξ2),W)=0
for any ξ1,ξ2,ξ3∈Γ(RadTM),W∈Γ(S(TM⊥)) and hl=0 on RadTM [9]. On the other hand, using (2.6), (2.15), (2.18), and (4.6), we arrive at
(4.8)g¯(hs(ξ1,ξ2),W)=-g¯(ϕξ2,hs(ξ1,BW)+Ds(ξ1,C1W)+∇ξ1sC2W).
Thus, our assertion follows from (4.8) and (4.7).
Converse part directly follows from (4.8) and (4.7).For the screen distributionS(TM) to be totally geodesic in M, we have the following.Theorem 4.4.
LetM be a radical screen transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then, S(TM) is totally geodesic if and only if AωξX,ABW*X+AC1WX+AC2WX∉S(TM)  for all  X∈Γ(S(TM)),ξ∈Γ(RadTM)  and  W∈Γ(S(TM⊥)).Proof.
Suppose that the distributionS(TM) is totally geodesic. Then
(4.9)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0
for any X,Y∈Γ(S(TM)),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)). Using (2.6), (2.8), (2.15), (2.18), and (4.5), a direct calculation shows that
(4.10)g¯(hl(X,Y),ξ)=g¯(TY,AωξX).
On the other hand, from (2.6), (2.7), (2.12), (2.15), (2.18), (4.5), and (4.6), we obtain
(4.11)g¯(hs(X,Y),W)=g¯(TY,ABW*X+AC1WX+AC2WX).
Thus, our assertion follows from (4.9),(4.10),  and (4.11).
Converse part directly follows from (4.10) and (4.11).In respect of a radical screen transversal lightlike submanifold to be mixed totally geodesic and totally geodesic, we have the following two theorems.Theorem 4.5.
LetM be a radical screen transversal lightlike submanifold of M¯. Then M is mixed totally geodesic if and only if AωξX∉(RadTM),∇Xsωξ∉μ and Dl(X,ωξ)=0forall,X∈Γ(S(TM)),ξ∈Γ(RadTM).Proof.
Assume that the submanifoldM is mixed geodesic. Then
(4.12)g¯(hl(X,ξ),ξ)=0,g¯(hs(X,ξ),W)=0
for any X∈Γ(S(TM)),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)). Also, from (2.14) we have
(4.13)g¯(hl(X,ξ),ξ)=0.
On the other hand, by the use of (2.6), (2.8), (2.15), (2.18), and (4.6), we obtain
(4.14)g¯(hs(X,ξ),W)=-g¯(AωξX,C1W)+g¯(∇Xsωξ,C2W)+g¯(Dl(X,ωξ),BW).
Thus, our assertion follows from (4.12) and (4.14).
Converse part directly follows from (4.14) and the fact that g¯(hl(X,ξ),ξ)=0.Theorem 4.6.
LetM be a radical screen transversal lightlike submanifold of M¯. Then M is totally geodesic if and only if(i)
AϕξX∉Γ(S(TM)) and ∇Xsϕξ∉ϕ(ltr(TM))⊥μ,(ii)
hl(X,TY)+Dl(X,ωY)∉(ltr(TM)),h*(X,TY)-AωYX∉RadTM  and  hs(X,TY)+∇XsωY∉μfor any X,Y∈Γ(TM) and ξ∈Γ(RadTM).Proof.
If the submanifoldM is totally geodesic, then
(4.15)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0
for any X,Y∈Γ(TM),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)). Also, making use of (2.6), (2.8), (2.15), (2.17), (2.18), and (4.5), a direct calculation shows that
(4.16)g¯(hl(X,Y),ξ)=g¯(TY,AϕξX)-g(∇Xsϕξ,ωY).
On the other hand, from (2.6), (2.8), (2.11), (2.15), (2.18), (4.5), and (4.6), we obtain
(4.17)g¯(hs(X,Y),W)=g¯(hl(X,TY)+Dl(X,ωY),BW)+g(h*(X,TY)-AωYX,C1W)+g¯(hs(X,TY)+∇XsωY,C2W).
Thus, (i) and (ii) follow from (4.16), (4.17), and (4.15).
Converse part directly follows from (4.16) and (4.17).
For aST-anti-invariant lightlike submanifold M immersed in M¯, if we denote the projection morphism of S(TM) and RadTM by P and Q, respectively, then for any vector field tangent to M we can write
(4.18)X=PX+QX.
By applying ϕ to (4.18) and then using (2.19), we obtain
(4.19)ϕX=ωPX+ωQX.
DenotingωPX by ω2 and ωQX by ω1. Then (4.19) can be rewritten as
(4.20)ϕX=ω1X+ω2X,
where ω1X∈Γ(ltr(TM)) and ω2X∈Γ(S(TM⊥)).
ForW∈Γ(S(TM⊥)), writing
(4.21)ϕW=B1W+B2W+C1W+C2W,
where B1W∈Γ(RadTM),B2W∈Γ(S(TM)),C1W∈Γ(ltr(TM)), and C2W∈Γ(μ) (μ is the orthogonal complement of {ϕ(RadTM)⊕ϕ(ltr(TM))}⊕orthoϕ(S(TM) in S(TM⊥)).In view of the above discussions, the conditions under which the distributionRadTM of a ST-anti-invariant lightlike submanifold immersed in indefinite Kenmotsu manifolds to be totally geodesic is given by the following result.Theorem 4.7.
LetM be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the distribution RadTM is totally geodesic if and only if Aωξ2ξ1=0,∇ξ1sωξ2∉μ and Dl(ξ1,ϕξ2)=0, for any ξ1,ξ2∈Γ(RadTM).Proof.
Suppose that the distributionRadTM is totally geodesic. Then we see that
(4.22)g¯(hl(ξ1,ξ2),ξ)=0,g¯(hs(ξ1,ξ2),W)=0
for any ξ1,ξ2∈Γ(RadTM) and W∈Γ(S(TM⊥). Recall that hl=0 on RadTM [9]. On the other hand, by the use of (2.6), (2.8), (2.15), (4.20), and (4.21), we obtain
(4.23)g¯(hs(ξ1,ξ2),W)=-g¯(Aωξ2ξ1,B2W+C1W)+g¯(∇ξ2sωξ2,C2W)+g¯(Dl(ξ1,ϕξ2),B1W).
Thus, our assertion follows from (4.22) and (4.23).
Converse part directly follows from (4.22) and (4.23).For the screen distributionS(TM) of a ST-anti-invariant lightlike submanifold to be totally geodesic, we have the following.Theorem 4.8.
LetM be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the distribution S(TM) is totally geodesic if and only if ∇XsωY∉ϕ(ltr(TM))⊥μ,AωYX=0 and Dl(X,ωY)=0, for any X,Y∈Γ(S(TM)).Proof.
IfS(TM) is totally geodesic, then
(4.24)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0,
for any X,Y∈Γ(S(TM)),ξ∈Γ(RadTM) and W∈Γ(S(TM⊥)).
On the other hand, using (2.6), (2.15), (2.18), and (4.20), we obtain
(4.25)g¯(hl(X,Y),ξ)=g¯(∇XsωY,ωξ).
Also,
(4.26)g¯(hs(X,Y),W)=g¯(-AωYX,B2W+C1W)+g(∇XsωY,C2W)+g¯(Dl(X,ωY),B1W),
where we have used (2.6), (2.8), (2.15), (2.18), (4.20), and (4.21). Thus, our assertion follows from (4.25), (4.26), and (4.24).
Converse part directly follows from (4.25) and (4.26).The necessary and sufficient conditions for aST-anti-invariant lightlike submanifold to be mixed totally geodesic, we have the following.Theorem 4.9.
LetM be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is mixed totally geodesic if and only if AωξX=0,Dl(X,ωξ)=0 and ∇Xsωξ∉μ. for any X∈Γ(S(TM)) and ξ∈Γ(RadTM).Proof.
Assume that the submanifoldM is mixed gedesic. Then
(4.27)g¯(hl(X,ξ),ξ)=0,g¯(hl(X,ξ),W)=0
for any X∈Γ(S(TM)),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)). By virtue of (2.14), we have
(4.28)g¯(hl(X,ξ),ξ)=0.
On the other hand, using (2.6), (2.8), (2.15), (2.18), (4.20), and (4.21), we get
(4.29)g¯(hs(X,ξ),W)=g-(-AωξX,B2W+C1W)+g-(∇Xsωξ,C2W)+g-(Dl(X,ωξ),B1W).
Thus, our assertion follows from (4.27) and (4.29).
Converse part directly follows from (4.29).Now, we prove the following.Theorem 4.10.
LetM be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is totally geodesic if and only if ∇XsωY∉μ,AωYX=0 and Dl(X,ωY)=0, for any X,Y∈Γ(TM).Proof.
The submanifoldM is totally geodesic if and only if
(4.30)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0
for any X,Y∈Γ(TM),ξ∈Γ(RadTM) and W∈Γ(S(TM⊥)). By the use of (2.6), (2.8), (2.15), (2.18), and (4.20), we obtain
(4.31)g¯(hl(X,Y),ξ)=g¯(∇XsωY,ωξ).
On the other hand, from (2.6), (2.8), (2.15), (4.20), and (4.21), we have
(4.32)g¯(hs(X,Y),W)=-g¯(AωYX,B2W+C1W)+g¯(∇XsωY,C2W)+g¯(Dl(X,ωY),B1W).
Thus, our assertion follows from (4.31), (4.32), and (4.30).
Converse part directly follows from (4.31) and (4.32).


## 5. Transversal Lightlike Submanifolds

The purpose of this section is to study transversal and radical transversal lightlike submanifolds in an indefinite Kenmotsu manifold. We recall here the definitions of these submanifolds given by Yıldırım and Sahin [10].Definition 5.1.
A lightlike submanifoldM tangent to structure vector field V immersed in an indefinite Kenmotsu manifold M¯ is said to be(i)
radical transversal lightlike submanifold ofM¯ if
(5.1)ϕ(RadTM)=ltr(TM),ϕ(S(TM))=S(TM),(ii)
transversal lightlike submanifold ofM¯ if
(5.2)ϕ(RadTM)=ltr(TM),ϕ(S(TM))⊆S(TM⊥).
For a radical transversal lightlike submanifoldM of an indefinite Kenmotsu manifold M¯, if P and Q are the projection morphism on S(TM) and RadTM, respectively, then any vector field X tangent to M can be written as
(5.3)X=PX+QX.
We apply ϕ to (5.3) and then using (2.19), we get
(5.4)ϕX=TPX+ωQX.
If we denote TPX by TX and ωQX by ωX, then (5.4) can be rewritten as
(5.5)ϕX=TX+ωX,
where TX∈Γ(S(TM)) and ωX∈Γ(ltr(TM)).
Moreover, ifW∈Γ(S(TM⊥)), then
(5.6)ϕW=CW,
from which we observe that ϕW∈Γ(S(TM⊥)).Using the above notations, one can prove the following.Theorem 5.2.
LetM be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the distribution RadTM is totally geodesic if and only if ACWξ1∉Γ(RadTM) for any ξ1∈Γ(RadTM) and W∈Γ(S(TM⊥)).Proof.
Sincehl=0 on RadTM [9], we observe that the distribution RadTM is totally geodesic if and only if
(5.7)g¯(hl(ξ1,ξ2),ξ)=0,g¯(hs(ξ1,ξ2),W)=0
for any ξ1∈Γ(RadTM) and W∈Γ(S(TM⊥)). On the other hand, using (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we arrive at
(5.8)g¯(hs(ξ1,ξ2),W)=g¯(ωξ2,ACWξ1).
Thus, our assertion follows from (5.7) and (5.8).
Converse part directly follows from (5.8).A screen distributionS(TM) of a radical transversal lightlike submanifold in indefinite Kenmotsu manifolds to be totally geodesic, we have the following.Theorem 5.3.
LetM be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the distribution S(TM) is totally geodesic if and only if h*(X,TY)=0 and ACWX∉Γ(S(TM)) for any X,Y∈Γ(S(TM)) and W∈Γ(S(TM⊥)).Proof.
We note that the distributionS(TM) is totally geodesic if and only if
(5.9)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0
for any X,Y∈Γ(S(TM)),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)). Making use of (2.6), (2.11), (2.15), (2.18), and (5.5), we get
(5.10)g¯(hl(X,Y),ξ)=g¯(h*(X,TY),ωξ).
On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we have
(5.11)g¯(hs(X,Y),W)=g¯(TY,ACWX).
Thus, our assertion follows from (5.10), (5.11), and (5.9).
Converse part directly follows from (5.10) and (5.11).The conditions under which a radical transversal lightlike submanifold immersed in indefinite Kenmotsu manifolds to be mixed totally geodesic is given by the following theorem.Theorem 5.4.
LetM be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is mixed totally geodesic if and only if ACWX∉Γ(RadTM) for any X∈Γ(S(TM)) and W∈Γ(S(TM⊥)).Proof.
The submanifoldM is mixed totally geodesic if and only if
(5.12)g¯(hl(X,Y),ξ)=0,g¯(hs(X,ξ),W)=0
for any X∈Γ(S(TM)),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)). From (2.14), we have
(5.13)g¯(hl(X,ξ),W)=0.
On the other hand, using (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we obtain
(5.14)g¯(hs(X,Y),W)=g¯(ωξ,ACWX).
Thus, our assertion follows from (5.12) and (5.14).
Converse part directly follows from (5.14).Theorem 5.5.
LetM be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the submanifold M is totally geodesic if and only if (i)
h*(X,TY)-AωYX∉Γ(RadM).(ii)
ACWX=0
for anyX,Y∈Γ(TM).Proof.
We observe that the submanifoldM is totally geodesic if and only if
(5.15)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0
for any X,Y∈Γ(TM),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)).
By the use of (2.6), (2.8), (2.11), (2.15), (2.18), (5.5), and (5.6), we arrive at
(5.16)g¯(hl(X,Y),ξ)=g¯(h*(X,TY),ωξ)-g(AωYX,ωξ).
On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we have
(5.17)g¯(hs(X,Y),W)=g(TY,ACWX)+g¯(ωY,ACWX).
Thus, our assertion follows from (5.15), (5.16), and (5.17).
Converse part directly follows from (5.16), and (5.17).If we denote the projections on the distributionsS(TM) and RadTM involved with the definition of a transversal lightlike submanifold M immersed in indefinite Kenmotsu manifold M¯ by P and Q, respectively, then any vector field X tangent to M can be written as
(5.18)X=PX+QX.
Applying ϕ to (5.18) and then using (2.19), we get
(5.19)ϕX=ωPX+ωQX.
If we denote ωQX by ω1 and ωPX by ω2, then (5.19) can be written as
(5.20)ϕX=ω2X+ω1X,
where ω1∈Γ(ltr(TM)) and ω2X∈Γ(S(TM⊥)).ForW∈Γ(S(TM⊥)), we have
(5.21)ϕW=BW+CW,
where BW∈Γ(S(TM)) and CW∈Γ(μ) (μ is the orthogonal complement of ϕ(S(TM)) in S(TM⊥)).Theorem 5.6.
LetM be a transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the distribution RadTM is totally geodesic if and only if Aωξ2ξ1∉S(TM) and Ds(ξ1,ϕξ2)∈(μ) for any ξ1,ξ2∈Γ(RadTM).Proof.
The distributionRadTM is totally geodesic if and only if
(5.22)g¯(hl(ξ1,ξ2),ξ)=0,g¯(hs(ξ1,ξ2),ξ)=0
for any ξ1,ξ2,ξ∈Γ(RadTM). In view of hl=0 on RadTM[9], we have
(5.23)g¯(hl(ξ1,ξ2),ξ)=0.
On the other hand, making use of (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we get
(5.24)g¯(hs(ξ1,ξ2),W)=-g(Aωξ2ξ1,BW)+g¯(Ds(ξ1,ϕξ2),CW).
Thus, our assertion follows from (5.22), (5.23), and (5.24).
Converse part directly follows from (5.23) and (5.24).Theorem 5.7.
LetM be a transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then S(TM) is totally geodesic if and only if
(5.25)Ds(X,ω1ξ),∇Xsω2Y∉μ,Aω2YX∉Γ(S(TM))
for any X,Y∈Γ(S(TM)).Proof.
We note that the distributionS(TM) is totally geodesic if and only if g¯(hl(X,Y),ξ)=0 and g¯(hs(X,Y),W)=0 for any X,Y∈Γ(S(TM)),ξ∈Γ(RadTM), and W∈Γ(S(TM⊥)).
Combining (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we obtain
(5.26)g¯(hl(X,Y),ξ)=-g¯(ω2Y,Ds(X,ω1ξ)).
On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.20), and (5.21), we have
(5.27)g¯(hs(X,Y),W)=-g¯(-Aω2YX,BW)+g¯(∇Xsω2Y,CW).
Thus, our assertion follows from (5.25), (5.26), and (5.27).
Converse part directly follows from (5.26) and (5.27).Theorem 5.8.
LetM be a transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is mixed totally geodesic if and only if Aω1ξX∉Γ(S(TM)) and Ds(X,ω1ξ)∉(μ) for any X∈Γ(S(TM)) and ξ∈Γ(RadTM).Proof.
We observe that the submanifoldM is mixed totally geodesic if and only if
(5.28)g¯(hl(X,ξ),ξ)=0,g¯(hs(X,ξ),W)=0
for all X∈Γ(S(TM)),ξ∈Γ(RadTM) and W∈Γ(S(TM⊥)).
From (2.14), we infer that
(5.29)g¯(hl(X,ξ),ξ)=0.
On the other hand, by the use of (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we arrive at
(5.30)g¯(hs(X,ξ),W)=-g(Aω1ξX,BW)+g¯(Ds(X,ω1ξ),CW).
Thus, our assertion follows from (5.28), (5.29), and (5.30).
Converse part directly follows from (5.29) and (5.30).Theorem 5.9.
LetM be a transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is totally geodesic if and only AωYX=0 and Ds(X,ω1Y)+∇Xsω2Y∉Γ(μ)for all X,Y∈Γ(TM).Proof.
The submanifoldM is totally geodesic if and only if
(5.31)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0
for any X,Y∈Γ(TM),ξ∈Γ(RadTM) and W∈Γ(S(TM⊥)).
By virtue of (2.6), (2.7), (2.8), (2.15), (2.18), (5.20) and (5.21), we have
(5.32)g¯(hl(X,Y),ξ)=g¯(-AωYX,ϕξ).
On the other hand, by the use of (2.6), (2.7), (2.8), (2.15), (2.18), (5.20), and (5.21), we get
(5.33)g¯(hs(X,Y),ξ)=g(-AωYX,BW)+g¯(Ds(X,ω1Y)+∇Xsω2Y,CW).
Thus, our assertion follows from (5.31), (5.32), and (5.33).
Converse part directly follows from (5.32), and (5.33).



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*Source: 804520-2012-11-14.xml*