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.
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 manifold M¯ 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 take ϵ=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 any U∈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 Contact SCR-Lightlike Submanifolds
In this section, we study the geometric conditions under which the distributions involved in the definition of a contact SCR-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 submanifold M, 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 distribution D 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 contact SCR-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 contact SCR-lightlike submanifold M of an indefinite Kenmostsu manifold M¯ is said to be
D⊥-totally geodesic contact SCR-lightlike submanifold if h(X,Y)=0 for any X,Y∈Γ(D⊥),
mixed totally geodesic contact SCR-lightlike submanifold if h(X,Y)=0 for any X∈Γ((D¯)⊥{V}) and Y∈Γ(D⊥).
Let M 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 any W∈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 a D⊥-totally geodesic contact SCR-lightlike submanifold immersed in indefinite Kenmotsu manifolds.
Theorem 3.3.
Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifolds M¯. Then M is D⊥-totally geodesic if and only if
hs(X,ϕξ)∈μ.
AωYX∉D⊥ and ∇XsωY∉μ for any X,Y∈Γ(D⊥).
Proof.
Suppose that the contact SCR-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 contact SCR-lightlike submanifolds to be mixed totally geodesic is given by the following theorem.
Theorem 3.4.
Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then M is mixed totally geodesic if and only if
hs(X,ϕξ)∈μ,
AωYX∉D⊥ and ∇XsωY∉μ for any X∈Γ(D¯⊥{V}) and Y∈Γ(D⊥).
Proof.
Assume that M 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.
An r-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.
An ST-lightlike submanifold of an indefinite Kenmotsu manifold M¯ is said to be
radical ST-lightlike submanifold if S(TM) is invariant with respect to ϕ.
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 submanifolds M 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⊥).
For W∈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 distribution RadTM is totally geodesic is given by the following theorem.
Theorem 4.3.
Let M 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 distribution RadTM 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 distribution S(TM) to be totally geodesic in M, we have the following.
Theorem 4.4.
Let M 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 distribution S(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.
Let M 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 submanifold M 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.
Let M be a radical screen transversal lightlike submanifold of M¯. Then M is totally geodesic if and only if
AϕξX∉Γ(S(TM)) and ∇Xsϕξ∉ϕ(ltr(TM))⊥μ,
hl(X,TY)+Dl(X,ωY)∉(ltr(TM)),h*(X,TY)-AωYX∉RadTM and hs(X,TY)+∇XsωY∉μ
for any X,Y∈Γ(TM) and ξ∈Γ(RadTM).
Proof.
If the submanifold M 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 a ST-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.
Denoting ωPX 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⊥)).
For W∈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 distribution RadTM 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.
Let M 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 distribution RadTM 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 distribution S(TM) of a ST-anti-invariant lightlike submanifold to be totally geodesic, we have the following.
Theorem 4.8.
Let M 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.
If S(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 a ST-anti-invariant lightlike submanifold to be mixed totally geodesic, we have the following.
Theorem 4.9.
Let M 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 submanifold M 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.
Let M 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 submanifold M 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 submanifold M tangent to structure vector field V immersed in an indefinite Kenmotsu manifold M¯ is said to be
radical transversal lightlike submanifold of M¯ if
(5.1)ϕ(RadTM)=ltr(TM),ϕ(S(TM))=S(TM),
transversal lightlike submanifold of M¯ if
(5.2)ϕ(RadTM)=ltr(TM),ϕ(S(TM))⊆S(TM⊥).
For a radical transversal lightlike submanifold M 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, if W∈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.
Let M 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.
Since hl=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 distribution S(TM) of a radical transversal lightlike submanifold in indefinite Kenmotsu manifolds to be totally geodesic, we have the following.
Theorem 5.3.
Let M 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 distribution S(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.
Let M 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 submanifold M 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.
Let M be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold M¯. Then the submanifold M is totally geodesic if and only if
h*(X,TY)-AωYX∉Γ(RadM).
ACWX=0
for any X,Y∈Γ(TM).
Proof.
We observe that the submanifold M 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 distributions S(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⊥)).
For W∈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.
Let M 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 distribution RadTM 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.
Let M 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 distribution S(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.
Let M 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 submanifold M 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.
Let M 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 submanifold M 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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