We define GCR-lightlike submanifolds of indefinite cosymplectic manifolds and give an example. Then, we study mixed geodesic GCR-lightlike submanifolds of indefinite cosymplectic manifolds and obtain some characterization theorems for a GCR-lightlike submanifold to be a GCR-lightlike product.

1. Introduction

To fill the gaps in the general theory of submanifolds, Duggal and Bejancu [1] introduced lightlike (degenerate) geometry of submanifolds. Since the geometry of CR-submanifolds has potential for applications in mathematical physics, particularly in general relativity, and the geometry of lightlike submanifolds has extensive uses in mathematical physics and relativity, Duggal and Bejancu [1] clubbed these two topics and introduced the theory of CR-lightlike submanifolds of indefinite Kaehler manifolds and then Duggal and Sahin [2], introduced the theory of CR-lightlike submanifolds of indefinite Sasakian manifolds, which were further studied by Kumar et al. [3]. But CR-lightlike submanifolds do not include the complex and real subcases contrary to the classical theory of CR-submanifolds [4]. Thus, later on, Duggal and Sahin [5] introduced a new class of submanifolds, generalized-Cauchy-Riemann- (GCR-) lightlike submanifolds of indefinite Kaehler manifolds and then of indefinite Sasakian manifolds in [6]. This class of submanifolds acts as an umbrella of invariant, screen real, contact CR-lightlike subcases and real hypersurfaces. Therefore, the study of GCR-lightlike submanifolds is the topic of main discussion in the present scenario. In [7], the present authors studied totally contact umbilical GCR-lightlike submanifolds of indefinite Sasakian manifolds.

In present paper, after defining GCR-lightlike submanifolds of indefinite cosymplectic manifolds, we study mixed geodesic GCR-lightlike submanifolds of indefinite cosymplectic manifolds. In [8, 9], Kumar et al. obtained some necessary and sufficient conditions for a GCR-lightlike submanifold of indefinite Kaehler and Sasakian manifolds to be a GCR-lightlike product, respectively. Thus, in this paper, we obtain some characterization theorems for a GCR-lightlike submanifold of indefinite cosymplectic manifold to be a GCR-lightlike product.

2. Lightlike Submanifolds

Let V be a real m-dimensional vector space with a symmetric bilinear mapping g:V×V→ℜ. The mapping g is called degenerate on V if there exists a vector ξ≠0 of V such that
(2.1)g(ξ,v)=0,∀v∈V,
otherwise g is called nondegenerate. It is important to note that a non-degenerate symmetric bilinear form on V may induce either a non-degenerate or a degenerate symmetric bilinear form on a subspace of V. Let W be a subspace of V and g∣w degenerate; then W is called a degenerate (lightlike) subspace of V.

Let (M-,g-) be a real (m+n)-dimensional semi-Riemannian manifold of constant index q such that m,n≥1, 1≤q≤m+n-1, and let (M,g) be an m-dimensional submanifold of M- and g the induced metric of g- on M. Thus, if g- is degenerate on the tangent bundle TM of M, then M is called a lightlike (degenerate) submanifold of M- (for detail see [1]). For a degenerate metric g on M, TM⊥ is also a degenerate n-dimensional subspace of TxM-. Thus, both TxM and TxM⊥ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace RadTxM=TxM∩TxM⊥, which is known as radical (null) subspace. If the mapping RadTM:x∈M→RadTxM defines a smooth distribution on M of rank r>0, then the submanifold M of M- is called an r-lightlike submanifold and RadTM is called the radical distribution on M. Then, there exists a non-degenerate screen distribution S(TM) which is a complementary vector subbundle to RadTM in TM. Therefore,
(2.2)TM=RadTM⊥S(TM),
where ⊥ denotes orthogonal direct sum. Let S(TM⊥), called screen transversal vector bundle, be a non-degenerate complementary vector subbundle to RadTM in TM⊥. Let tr(TM) and ltr(TM) be complementary (but not orthogonal) vector bundles to TM in TM-|M and to RadTM in S(TM⊥)⊥, called transversal vector bundle and lightlike transversal vector bundle of M, respectively. Then, we have
(2.3)tr(TM)=ltr(TM)⊥S(TM⊥),(2.4)TM-|M=TM⊕tr(TM)=(RadTM⊕ltr(TM))⊥S(TM)⊥S(TM⊥).

Let u be a local coordinate neighborhood of M and consider the local quasiorthonormal fields of frames of M- along M on u as {ξ1,…,ξr,Wr+1,…,Wn,N1,…,Nr,Xr+1,…,Xm}, where {ξ1,…,ξr}and{N1,…,Nr} are local lightlike bases of Γ(RadTM|u) and Γ(ltr(TM)|u) and {Wr+1,…,Wn}and{Xr+1,…,Xm} are local orthonormal bases of Γ(S(TM⊥)|u) and Γ(S(TM)|u), respectively. For these quasiorthonormal fields of frames, we have the following theorem.

Theorem 2.1 (see [<xref ref-type="bibr" rid="B4">1</xref>]).

Let (M,g,S(TM),S(TM⊥)) be an r-lightlike submanifold of a semi-Riemannian manifold (M-,g-). Then there, exist a complementary vector bundle ltr(TM) of Rad TM in S(TM⊥)⊥ and a basis of Γ(
ltr
(TM)|u) consisting of smooth section {Ni} of S(TM⊥)⊥|u, where u is a coordinate neighborhood of M, such that
(2.5)g-(Ni,ξj)=δij,g-(Ni,Nj)=0,foranyi,j∈{1,2,…,r},
where {ξ1,…,ξr} is a lightlike basis of Γ(
Rad
(TM)).

Let ∇- be the Levi-Civita connection on M-. Then, according to decomposition (2.4), the Gauss and Weingarten formulas are given by
(2.6)∇-XY=∇XY+h(X,Y),∇-XU=-AUX+∇X⊥U,
for any X,Y∈Γ(TM) and U∈Γ(tr(TM)), where {∇XY,AUX} and {h(X,Y),∇X⊥U} belong to Γ(TM) and Γ(tr(TM)), respectively. Here ∇ is a torsion-free linear connection on M, h is a symmetric bilinear form on Γ(TM) that is called second fundamental form, and AU is a linear operator on M, known as shape operator.

According to (2.3), considering the projection morphisms L and S of tr(TM) on ltr(TM) and S(TM⊥), respectively, then (2.6) gives
(2.7)∇-XY=∇XY+hl(X,Y)+hs(X,Y),∇-XU=-AUX+DXlU+DXsU,
where we put hl(X,Y)=L(h(X,Y)),hs(X,Y)=S(h(X,Y)),DXlU=L(∇X⊥U), DXsU=S(∇X⊥U).

As hl and hs are Γ(ltr(TM))-valued and Γ(S(TM⊥))-valued, respectively, they are called the lightlike second fundamental form and the screen second fundamental form on M. In particular,
(2.8)∇-XN=-ANX+∇XlN+Ds(X,N),∇-XW=-AWX+∇XsW+Dl(X,W),
where X∈Γ(TM),N∈Γ(ltr(TM)), and W∈Γ(S(TM⊥)). By using (2.3)-(2.4) and (2.7)-(2.8), we obtain
(2.9)g-(hs(X,Y),W)+g-(Y,Dl(X,W))=g(AWX,Y),(2.10)g-(hl(X,Y),ξ)+g-(Y,hl(X,ξ))+g(Y,∇Xξ)=0,
for any ξ∈Γ(RadTM), W∈Γ(S(TM⊥)), and N,N′∈Γ(ltr(TM)).

Let P be the projection morphism of TM on S(TM). Then, using (2.2), we can induce some new geometric objects on the screen distribution S(TM) on M as
(2.11)∇XPY=∇X*PY+h*(X,Y),∇Xξ=-Aξ*X+∇X*tξ,
for any X,Y∈Γ(TM) and ξ∈Γ(RadTM), where {∇X*PY,Aξ*X} and {h*(X,Y),∇X*tξ} belong to Γ(S(TM)) and Γ(RadTM), respectively. ∇* and ∇*t are linear connections on complementary distributions S(TM) and RadTM, respectively. Then, using (2.7), (2.8), and (2.11), we have
(2.12)g-(hl(X,PY),ξ)=g(Aξ*X,PY),g-(h*(X,PY),N)=g(ANX,PY).

Next, an odd-dimensional semi-Riemannian manifold M- is said to be an indefinite almost contact metric manifold if there exist structure tensors (ϕ,V,η,g-), where ϕ is a (1,1) tensor field, V is a vector field called structure vector field, η is a 1-form, and g- is the semi-Riemannian metric on M- satisfying (see [10])
(2.13)g-(ϕX,ϕY)=g-(X,Y)-η(X)η(Y),g-(X,V)=η(X),ϕ2X=-X+η(X)V,η∘ϕ=0,ϕV=0,η(V)=1,
for any X,Y∈Γ(TM).

An indefinite almost contact metric manifold M- is called an indefinite cosymplectic manifold if (see [11])
(2.14)∇-Xϕ=0,(2.15)∇-XV=0.

Calin [12] proved that if the characteristic vector field V is tangent to (M,g,S(TM)), then it belongs to S(TM). We assume that the characteristic vector V is tangent to M throughout this paper. Thus, we define the generalized Cauchy-Riemann lightlike submanifolds of an indefinite cosymplectic manifold as follows.

Definition 3.1.

Let (M,g,S(TM),S(TM⊥)) be a real lightlike submanifold of an indefinite cosymplectic manifold (M-,g-) such that the structure vector field V is tangent to M; then M is called a generalized-Cauchy-Riemann- (GCR-) lightlike submanifold if the following conditions are satisfied:

there exist two subbundles D1 and D2 of Rad(TM) such that
(3.1)Rad(TM)=D1⊕D2,ϕ(D1)=D1,ϕ(D2)⊂S(TM),

there exist two subbundles D0 and D- of S(TM) such that
(3.2)S(TM)={ϕD2⊕D-}⊥D0⊥V,ϕ(D-)=L⊥S,
where D0 is invariant nondegenerate distribution on M, {V} is one-dimensional distribution spanned by V, and L and S are vector subbundles of ltr(TM) and S(TM)⊥, respectively.

Therefore, the tangent bundle TM of M is decomposed as
(3.3)TM={D⊕D-⊕{V}},D=Rad(TM)⊕D0⊕ϕ(D2).
A contact GCR-lightlike submanifold is said to be proper if D0≠{0},D1≠{0},D2≠{0}, and L≠{0}. Hence, from the definition of GCR-lightlike submanifolds, we have that

condition (A) implies that dim(RadTM)≥3,

condition (B) implies that dim(D)≥2s≥6 and dim(D2)=dim(S), and thus dim(M)≥9 and dim(M-)≥13.

any proper 9-dimensional contact GCR-lightlike submanifold is 3-lightlike,

(a) and contact distribution (η=0) imply that index (M-)≥4.

The following proposition shows that the class of GCR-lightlike submanifolds is an umbrella of invariant, contact CR and contact SCR-lightlike submanifolds.
Proposition 3.2.

A GCR-lightlike submanifold M of an indefinite cosymplectic manifold M- is contact CR-submanifold (resp., contact SCR-lightlike submanifold) if and only if D1={0} (resp., D2={0}).

Proof.

Let M be a contact CR-lightlike submanifold; then ϕRadTM is a distribution on M such that RadTM⋂ϕRadTM={0}. Therefore, D2=RadTM and D1={0}. Since ltr(TM)⋂ϕ(ltr(TM))={0}, this implies that ϕ(ltr(TM))⊂S(TM). Conversely, suppose that M is a GCR-lightlike submanifold of an indefinite Cosymplectic manifold such that D1={0}. Then, from (3.1), we have D2=Rad(TM), and therefore RadTM⋂ϕRadTM={0}. Hence, ϕRadTM is a vector subbundle of S(TM). This implies that M is a contact CR-lightlike submanifold of an indefinite cosymplectic manifold. Similarly the other assertion follows.

The following construction helps in understanding the example of GCR-lightlike submanifold. Let (Rq2m+1,ϕ0,V,η,g-) be with its usual Cosymplectic structure and given by
(3.4)η=dz,V=∂z,g-=η⊗η-∑i=1q/2(dxi⊗dxi+dyi⊗dyi)+∑i=q+1m(dxi⊗dxi+dyi⊗dyi),ϕ0(X1,X2,…,Xm-1,Xm,Y1,Y2,…,Ym-1,Ym,Z)=(-X2,X1,…,-Xm,Xm-1,-Y2,Y1,…,-Ym,Ym-1,0),
where (xi;yi;z) are the Cartesian coordinates.

Example 3.3.

Let M-=(R413,g-) be a semi-Euclidean space and M a 9-dimensional submanifold of M- that is given by
(3.5)x4=x1cosθ-y1sinθ,y4=x1sinθ+y1cosθ,x2=y3,x5=1+(y5)2,
where g- is of signature (-,-,+,+,+,+,-,-,+,+,+,+,+) with respect to the canonical basis {∂x1,∂x2,∂x3,∂x4,∂x5,∂x6,∂y1,∂y2,∂y3,∂y4,∂y5,∂y6,∂z}. Then, the local frame of TM is given by
(3.6)ξ1=∂x1+cosθ∂x4+sinθ∂y4,ξ2=-sinθ∂x4+∂y1+cosθ∂y4,ξ3=∂x2+∂y3,X1=∂x3-∂y2,X2=∂x6,X3=∂y6,X4=y5∂x5+x5∂y5,X5=∂x3+∂y2,X6=V=∂z.
Hence, M is a 3-lightlike as RadTM=span{ξ1,ξ2,ξ3}. Also, ϕ0ξ1=-ξ2 and ϕ0ξ3=X1; these imply that D1=span{ξ1,ξ2} and D2=span{ξ3}, respectively. Since ϕ0X2=-X3, D0=span{X2,X3}. By straightforward calculations, we obtain
(3.7)S(TM⊥)=span{W=x5∂x5-y5∂y5},
where ϕ0(W)=X4; this implies that S=S(TM⊥). Moreover, the lightlike transversal bundle ltr(TM) is spanned by
(3.8)N1=12(-∂x1+cosθ∂x4+sinθ∂y4),N2=12(-sinθ∂x4-∂y1+cosθ∂y4),N3=12(-∂x2+∂y3),
where ϕ0(N1)=-N2 and ϕ0(N3)=X5. Hence, L=span{N3}. Therefore, D-=span{ϕ0(N3),ϕ0(W)}. Thus, M is a GCR-lightlike submanifold of R413.

Let Q, P1, P2 be the projection morphism on D, ϕS=M2, ϕL=M1, respectively; therefore
(3.9)X=QX+V+P1X+P2X,
for X∈Γ(TM). Applying ϕ to (3.9), we obtain
(3.10)ϕX=fX+ωP1X+ωP2X,
where fX∈Γ(D), ωP1X∈Γ(L), and ωP2X∈Γ(S), or, we can write (3.10) as
(3.11)ϕX=fX+ωX,
where fX and ωX are the tangential and transversal components of ϕX, respectively.

Similarly,
(3.12)ϕU=BU+CU,U∈Γ(tr(TM)),
where BU and CU are the sections of TM and tr(TM), respectively. Differentiating (3.10) and using (2.8)–(2.10) and (3.12), we have
(3.13)Ds(X,ωP2Y)=-∇XsωP1Y+ωP1∇XY-hs(X,fY)+Chs(X,Y),Dl(X,ωP1Y)=-∇XlωP2Y+ωP2∇XY-hl(X,fY)+Chl(X,Y),
for all X,Y∈Γ(TM). By using, cosymplectic property of ∇- with (2.7), we have the following lemmas.

Lemma 3.4.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-; then one has
(3.14)(∇Xf)Y=AωYX+Bh(X,Y),(∇Xtω)Y=Ch(X,Y)-h(X,fY),
where X,Y∈Γ(TM) and
(3.15)(∇Xf)Y=∇XfY-f∇XY,(∇Xtω)Y=∇XtωY-ω∇XY.

Lemma 3.5.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-; then one has
(3.16)(∇XB)U=ACUX-fAUX,(∇XtC)U=-ωAUX-h(X,BU),
where X∈Γ(TM) and U∈Γ(tr(TM)) and
(3.17)(∇XB)U=∇XBU-B∇XtU,(∇XtC)U=∇XtCU-C∇XtU.

A GCR-lightlike submanifold of an indefinite cosymplectic manifold is called mixed geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h(X,Y)=0, for any X∈Γ(D⊕V) and Y∈Γ(D-).

Definition 4.2.

A GCR-lightlike submanifold of an indefinite cosymplectic manifold is called D- geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h(X,Y)=0, for any X,Y∈Γ(D-).

Theorem 4.3.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, M is mixed geodesic if and only if Aξ*X and AWX∉Γ(M2⊥ϕD2), for any X∈Γ(D⊕V),W∈Γ(S(TM⊥)) and ξ∈Γ(
Rad
(TM)).

Proof.

Using, definition of GCR-lightlike submanifolds, M is mixed geodesic if and only if g-(h(X,Y),W)=g-(h(X,Y),ξ)=0, for X∈Γ(D⊕V),Y∈Γ(D-),W∈Γ(S(TM⊥)), and ξ∈Γ(Rad(TM)). Using (2.8) and (2.11), we get
(4.1)g-(h(X,Y),W)=g-(∇-XY,W)=-g(Y,∇-XW)=g(Y,AWX),g-(h(X,Y),ξ)=g-(∇-XY,ξ)=-g(Y,∇Xξ)=g(Y,Aξ*X).
Therefore, from (4.1), the proof is complete.

Theorem 4.4.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, M is D- geodesic if and only if Aξ*X and AWX∉Γ(M2⊥ϕD2), for any X∈Γ(D-),ξ∈Γ
Rad
(TM), and W∈Γ(S(TM⊥)).

Proof.

The proof is similar to the proof of Theorem 4.3.

Lemma 4.5.

Let M be a mixed geodesic GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then Aξ*X∈Γ(ϕD2), for any X∈Γ(D-), ξ∈Γ(D2).

Proof.

For X∈Γ(D-) and ξ∈Γ(D2), using (2.7) we have
(4.2)h(ϕξ,X)=∇-Xϕξ-∇Xϕξ=ϕ∇Xξ+ϕh(X,ξ)-∇Xϕξ.
Since M is mixed geodesic, we obtain ϕ∇Xξ=∇Xϕξ. Here, using (2.11), we get ϕ(-Aξ*X+∇X*tξ)=∇X*ϕξ+h*(X,ϕξ), and then, by virtue of (3.11), we obtain -fAξ*X-ωAξ*X+ϕ(∇X*tξ)=∇X*ϕξ+h*(X,ϕξ). Comparing the transversal components, we get ωAξ*X=0; this implies that
(4.3)Aξ*X∈Γ(D0⊕{V}⊥ϕ(D2)).
If Aξ*X∈D0, then the nondegeneracy of D0 implies that there must exist a Z0∈D0 such that g-(Aξ*X,Z0)≠0. But using the hypothesis that M is a mixed geodesic with (2.7) and (2.11), we get
(4.4)g-(Aξ*X,Z0)=-g-(∇Xξ,Z0)=g-(ξ,∇-XZ0)=g-(ξ,∇XZ0+h(X,Z0))=0.
Therefore,
(4.5)Aξ*X∉Γ(D0).
Also using (2.13), and (2.15), we get
(4.6)g-(Aξ*X,V)=-g-(∇Xξ,V)=g-(ξ,∇-XV)=0.
Therefore,
(4.7)Aξ*X∉{V}.
Hence, from (4.3), (4.5), and (4.7), the result follows.

Corollary 4.6.

Let M be a mixed geodesic GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, g-(hl(X,Y),ξ)=0, for any X∈Γ(D-),Y∈Γ(M2) and ξ∈Γ(D2).

Proof.

The result follows from (2.12) and Lemma 4.5.

Theorem 4.7.

Let M be a mixed geodesic GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, AUX∈Γ(D⊕{V}) and ∇XtU∈Γ(L⊥S), for any X∈Γ(D⊕{V}) and U∈Γ(L⊥S).

Proof.

Since M is mixed geodesic GCR-lightlike submanifold h(X,Y)=0 for any X∈Γ(D⊕{V}),Y∈Γ(D-), and thus (2.6) implies that
(4.8)0=∇-XY-∇XY.
Since D- is an anti-invariant distribution there exists a vector field U∈Γ(L⊥S) such that ϕU=Y. Thus, from (2.8), (2.14), (3.11), and (3.12), we get
(4.9)0=∇-XϕU-∇XY=ϕ(-AUX+∇XtU)-∇XY=-fAUX-ωAUX+B∇XtU+C∇XtU-∇XY.
Comparing the transversal components, we get ωAUX=C∇XtU. Since ωAUX∈Γ(L⊥S) and C∇XtU∈Γ(L⊥S)⊥, this implies that ωAUX=0 and C∇XtU=0. Hence, AUX∈Γ(D⊕{V}) and ∇XtU∈Γ(L⊥S).

GCR-lightlike submanifold M of an indefinite cosymplectic manifold M- is called GCR-lightlike product if both the distributions D⊕{V} and D- define totally geodesic foliation in M.

Theorem 5.2.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, the distribution D⊕{V} define a totally geodesic foliation in M if and only if Bh(X,ϕY)=0, for any X,Y∈D⊕{V}.

Proof.

Since D-=ϕ(L⊥S), D⊕{V} defines a totally geodesic foliation in M if and only if g(∇XY,ϕξ)=g(∇XY,ϕW)=0, for any X,Y∈Γ(D⊕{V}), ξ∈Γ(D2), and W∈Γ(S). Using (2.7) and (2.14), we have
(5.1)g(∇XY,ϕξ)=-g-(∇-XϕY,ξ)=-g-(hl(X,fY),ξ),(5.2)g(∇XY,ϕW)=-g-(∇-XϕY,W)=-g-(hs(X,fY),W).
Hence, from (5.1) and (5.2), the assertion follows.

Theorem 5.3.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, the distribution D- defines a totally geodesic foliation in M if and only if ANX has no component in ϕS⊥ϕD2 and AωYX has no component in D2⊥D0, for any X,Y∈Γ(D-) and N∈Γ(
ltr
(TM)).

Proof.

From the definition of a GCR-lightlike submanifold, we know that D- defines a totally geodesic foliation in M if and only if
(5.3)g(∇XY,N)=g(∇XY,ϕN1)=g(∇XY,V)=g(∇XY,ϕZ)=0,
for X,Y∈Γ(D-),N∈Γ(ltr(TM)),Z∈Γ(D0) and N1∈Γ(L). Using (2.7) and (2.8), we have
(5.4)g(∇XY,N)=g-(∇-XY,N)=-g-(Y,∇-XN)=g(Y,ANX).
Using (2.7), (2.15), and (2.14), we obtain
(5.5)g(∇XY,ϕN1)=-g(ϕ∇-XY,N1)=-g(∇-XωY,N1)=g(AωYX,N1),(5.6)g(∇XY,ϕZ)=-g(ϕ∇-XY,Z)=-g(∇-XωY,Z)=g(AωYX,Z),(5.7)g(∇XY,V)=g(∇-XY,V)=-g(Y,∇-XV)=0.
Thus, from (5.4)–(5.7), the result follows.

Theorem 5.4.

Let M be a GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. If (∇Xf)Y=0, then M is a GCR lightlike product.

Proof.

Let X,Y∈Γ(D-); therefore fY=0. Then using (3.15) with the hypothesis, we get f∇XY=0. Therefore the distribution D- defines a totally geodesic foliation. Next, let X,Y∈D⊕{V}; therefore ωY=0. Then using (3.14), we get Bh(X,Y)=0. Therefore, D⊕{V} defines a totally geodesic foliation in M. Hence, M is a GCR lightlike product.

Definition 5.5.

A lightlike submanifold M of a semi-Riemannian manifold is said to be an irrotational submanifold if ∇-Xξ∈Γ(TM), for any X∈Γ(TM) and ξ∈ΓRad(TM). Thus, M is an irrotational lightlike submanifold if and only if hl(X,ξ)=0 and hs(X,ξ)=0.

Theorem 5.6.

Let M be an irrotational GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, M is a GCR lightlike product if the following conditions are satisfied:

∇-XU∈Γ(S(TM⊥)),forallX∈Γ(TM),andU∈Γ(tr(TM)),

Aξ*Y∈Γ(ϕ(S)),forallY∈Γ(D).

Proof.

Let (A) hold; then, using (2.8), we get ANX=0,AWX=0, Dl(X,W)=0, and ∇XlN=0 for X∈Γ(TM). These equations imply that the distribution D- defines a totally geodesic foliation in M, and, with (2.9), we get g-(hs(X,Y),W)=0. Hence, the non degeneracy of S(TM⊥) implies that hs(X,Y)=0. Therefore, hs(X,Y) has no component in S. Finally, from (2.10) and the hypothesis that M is irrotational, we have g-(hl(X,Y),ξ)=g-(Y,Aξ*X), for X∈Γ(TM) and Y∈Γ(D). Assume that (B) holds; then hl(X,Y)=0. Therefore, hl(X,Y) has no component in L. Thus, the distribution D⊕{V} defines a totally geodesic foliation in M. Hence, M is a GCR lightlike product.

Definition 5.7 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

If the second fundamental form h of a submanifold, tangent to characteristic vector field V, of a Sasakian manifold M- is of the form
(5.8)h(X,Y)={g(X,Y)-η(X)η(Y)}α+η(X)h(Y,V)+η(Y)h(X,V),
for any X,Y∈Γ(TM), where α is a vector field transversal to M, then M is called a totally contact umbilical submanifold of a Sasakian manifold.

Theorem 5.8.

Let M be a totally contact umbilical GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Then, M is a GCR-lightlike product if Bh(X,Y)=0, for any X,Y∈Γ(TM).

Proof.

Let X,Y∈Γ(D⊕{V}); then the hypothesis that Bh(X,Y)=0 implies that the distribution D⊕{V} defines a totally geodesic foliation in M.

If we assume that X,Y∈Γ(D-), then, using (3.14), we have -f∇XY=AωYX+Bh(X,Y), and taking inner product with Z∈Γ(D0) and using (2.6) and (2.14), we obtain
(5.9)-g(f∇XY,Z)=g(AωYX+Bh(X,Y),Z)=g(∇-XY,ϕZ)=-g(Y,∇XZ′),
where ϕZ=Z′∈Γ(D0). For any X∈Γ(D-) from (3.14), we have ωP∇XZ=h(X,fZ)-Ch(X,Z). Therefore, using the hypothesis with (5.8), we get ωP∇XZ=0; this implies that ∇XZ∈Γ(D), and thus (5.9) becomes g(f∇XY,Z)=0. Then, the nondegeneracy of the distribution D0 implies that the distribution D- defines a totally geodesic foliation in M. Hence, the assertion follows.

Theorem 5.9.

Let M be a totally geodesic GCR-lightlike submanifold of an indefinite cosymplectic manifold M-. Suppose that there exists a transversal vector bundle of M which is parallel along D- with respect to Levi-Civita connection on M, that is, ∇-XU∈Γ(tr(TM)), for any U∈Γ(tr(TM)), X∈Γ(D-). Then, M is a GCR-lightlike product.

Proof.

Since M is a totally geodesic GCR-lightlike Bh(X,Y)=0, for X,Y∈Γ(D⊕{V}); this implies D⊕{V} defines a totally geodesic foliation in M.

Next ∇-XU∈Γ(tr(TM)) implies AUX=0, and hence, by Theorem 5.3, the distribution D- defines a totally geodesic foliation in M. Hence, the result follows.

Acknowledgment

The authors would like to thank the anonymous referee for his/her comments that helped them to improve this paper.

DuggalK. L.BejancuA.DuggalK. L.SahinB.Lightlike submanifolds of indefinite Sasakian manifoldsKumarR.RaniR.NagaichR. K.On contact CR-lightlike submanifolds of indefinite Sasakian manifoldsBejancuA.CR submanifolds of a Kaehler manifold. IDuggalK. L.SahinB.Generalized Cauchy-Riemann lightlike submanifolds of Kaehler manifoldsDuggalK. L.SahinB.Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifoldsJainV.KumarR.NagaichR. K.Totally contact umbilical GCR-lightlike submanifolds indefinite Sasakian manifoldsKumarR.KumarS.NagaichR. K.GCR-lightlike product of indefinite Kaehler manifoldsKumarR.JainV.NagaichR. K.GCR-lightlike product of indefinite Sasakian manifoldsKumarR.RaniR.NagaichR. K.On sectional curvatures of (ϵ)-Sasakian manifoldsBlairD. E.CalinC.On the existence of degenerate hypersurfaces in Sasakian manifoldsYanoK.KonM.