# Some Remarks on Quasi-Generalized CR-Null Geometry in Indefinite Nearly Cosymplectic Manifolds

Authors: Fortuné Massamba; Samuel Ssekajja
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

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## Abstract

Attention is drawn to some distributions on ascreen Quasi-Generalized Cauchy-Riemannian (QGCR) null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifolds.


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## 1. Introduction

One of the current interesting research areas in semi-Riemannian geometry is the theory of null (or lightlike) submanifolds. An intrinsic approach to the theory of null submanifolds was advanced by Kupeli [1], yet an extrinsic counterpart had to wait for Duggal and Bejancu [2], and later by Duggal and Sahin [3]. Since then, many researchers have laboured to extend their theories with evidence from the following few selected papers: [3–11] and other references therein. The rapid increase in research on this topic, since 1996, is inspired by the numerous applications of the theory to mathematical physics, particularly in general relativity. More precisely, in general relativity, null submanifolds represent different models of black hole horizons (see [2, 3] for details).In [5], the authors initiated the study of generalized CR- (GCR-) null submanifolds of an indefinite Sasakian manifold, which aretangent to the structure vector field, ξ, of the almost contact structure (ϕ¯,ξ,η). Moreover, when ξ is tangent to the submanifold, Calin [12] proved that it belongs to its screen distribution. This assumption is widely accepted and it has been applied in many papers on null contact geometry, for instance, [3, 5, 8–11, 13]. It is worth mentioning that ξ is aglobal vector field defined on the entire tangent bundle of the ambient almost contact manifold. Thus, restricting it to the screen distribution is only one of those cases in which it can be placed. In the study of Riemannian CR-submanifolds of Sasakian manifolds, Yano and Kon [14, p. 43] proved that making ξ a normal vector field in such scenario leads to an anti-invariant submanifold, and hence ξ was kept tangent to the CR-submanifold. Their proof leans against the fact that the shape operator on such CR-submanifold is naturallysymmetric with respect to the induced Riemannian metric g. On the other hand, the shape operators of any r-null submanifold are generallynot symmetric with respect to the induced degenerate metric g (see [2, 3] for details).In an attempt to generalizeξ, we introduced a special class of CR-null submanifolds of a nearly Sasakian manifold, known asquasi-generalized CR- (QGCR-) null submanifold [15], for which the classical GCR-null submanifolds [3] form part. Among other benefits, generalizing ξ leads to QGCR-null submanifolds of lower dimensions and with quite different geometric properties compared to respective GCR-null submanifolds.The purpose of this paper is to investigate the geometry of distributions on ascreen QGCR-null submanifolds of indefinite nearly cosymplectic manifolds. A null submanifoldM of an indefinite nearly cosymplectic manifold is called ascreen if the structure vector field ξ belongs to Rad(TM)⊕ltr(TM) [16]. The paper is organized as follows. In Section 2, we present the basic notions of null submanifolds and nearly cosymplectic manifolds. More details can be found in [17–22]. In Section 3, we review the basic notions of QGCR-null submanifolds and we give an example of ascreen QGCR-null submanifold. In Section 4, we discuss totally umbilical, totally geodesic, and irrotational ascreen QGCR-null submanifolds of an indefinite nearly cosymplectic space form M¯(c¯). Finally, in Section 5 we investigate the geodesity of the distributions D and D^.


## 2. Preliminaries

LetMm be a codimension n submanifold of a semi-Riemannian manifold (M¯,g¯) of constant index ν, 1≤ν≤m+n, with m,n≥1. Then, M is said to be anull submanifold of M¯ if the tangent and normal bundles of M have a nontrivial intersection. This intersection defines a smooth distribution on M, called theradical distribution [2]. More precisely, consider p∈M; one defines the orthogonal complement TpM⊥ of the tangent space TpM by (1)TpM⊥=X∈TpM:g¯X,Y=0,∀Y∈TpM. If we denote the radical distribution on M by RadTpM, then RadTpM=RadTpM⊥=TpM∩TpM⊥. The submanifold M of M¯ is said to be r-null submanifold (one supposes that the index of M¯ is ν≥r), if the mapping RadTM:p∈M→RadTpM defines asmooth distribution on M of rank r>0.In this paper,r-null submanifold will simply be called anull submanifold and g=g¯|TM is anull metric, unless we need to specify r.LetS(TM) be a screen distribution which is a semi-Riemannian complementary distribution of Rad⁡TM in TM; that is,(2)TM=Rad⁡TM⊥STM.Consider a screen transversal bundle S(TM⊥), which is semi-Riemannian and complementary to RadTM in TM⊥. For any local basis {E1,…,Er} of RadTM, there exists a local null frame {N1,…,Nr}⊂S(TM⊥) in S(TM)⊥ such that g(Ei,Nj)=δij and g¯(Ni,Nj)=0. It follows that there exists a null transversal vector bundle ltr(TM) locally spanned by {N1,…,Nr} (see details in [2, 3]). If tr(TM) denotes the complementary (but not orthogonal) vector bundle to TM in TM¯, then,(3)trTM=ltrTM⊥STM⊥,(4)TM¯=STM⊥STM⊥⊥RadTM⊕ltrTM.It is important to note that the screen distribution S(TM) is not unique and is canonically isomorphic to the factor vector bundle TM/RadTM [1].Given a null submanifoldM, then the following classifications of M are well-known [2]: (i) M is r-null if 1≤r<min⁡{m,n}; (ii) M is coisotropic if 1≤r=n<m, S(TM⊥)={0}; (iii) M is isotropic if 1≤r=m<n, S(TM)={0}; (iv) M is totally null if r=n=m, S(TM)=S(TM⊥)={0}.Where necessary, the following range of indices will be used(5)i,j,k∈1,…,r,α,β,γ∈r+1,…,n. Consider a local quasi-orthonormal fields of frames of M¯ along M as (6)E1,…,Er,N1,…,Nr,Xr+1,…,Xm,W1+r,…,Wn, where {Xr+1,…,Xm} and {W1+r,…,Wn} are, respectively, orthonormal bases of Γ(S(TM)) and Γ(S(TM⊥)).Throughout the paper we considerΓ(Ξ) to be a set of smooth sections of the vector bundle Ξ.LetP be the projection morphism of TM onto S(TM). Then, the following are the Gauss-Weingarten equations of r-null submanifold M and S(TM) (see [2, 3] for detailed explanations):(7)∇¯XY=∇XY+∑i=1rhilX,YNi+∑α=r+1nhαsX,YWα,(8)∇¯XNi=-ANiX+∑j=1rτijXNj+∑α=r+1nρiαXWα,(9)∇¯XWα=-AWαX+∑i=1rφαiXNi+∑β=r+1nσαβXWβ,(10)∇XPY=∇X∗PY+∑i=1rhi∗X,PYEi,(11)∇XEi=-AEi∗X-∑j=1rτjiXEj,∀X,Y∈ΓTM,where ∇ and ∇∗ are the induced connections on TM and S(TM), respectively and hil and hαs are symmetric bilinear forms known aslocal null andscreen fundamental forms of TM, respectively. Furthermore, hi∗ are thesecond fundamental forms of S(TM). ANi, AEi∗, and AWα are linear operators on TM while τij, ρiα, φαi, and σαβ are 1-forms on TM. Note that thesecond fundamental tensor of M is given by(12)hX,Y=∑i=1rhilX,YNi+∑α=r+1nhαsX,YWα,for any X,Y∈Γ(TM). The connection ∇∗ is a metric connection on S(TM) while ∇ is generally not a metric connection and is given by (13)∇XgY,Z=∑i=1rhilX,YλiZ+hilX,ZλiY, for any X,Y∈Γ(TM) and λi are 1-forms given by λi(X)=g¯(X,Ni), for all X∈Γ(TM). By using (7), (8), and (9), the curvature tensors R¯ and R of M¯ and M, respectively, are related as, for any X,Y,Z,W∈Γ(TM),(14)R¯X,W,Z,Y=g¯RX,WZ,Y+g¯AhlX,ZW,Y-g¯AhlW,ZX,Y+g¯AhsX,ZW,Y-g¯AhsW,ZX,Y+g¯∇XhlW,Z,Y-g¯∇WhlX,Z,Y+g¯DlX,hsW,Z,Y-g¯DlW,hsX,Z,Y+g¯∇XhsW,Z,Y-g¯∇WhsX,Z,Y+g¯DsX,hlW,Z,Y-g¯DsW,hlX,Z,Y.A null submanifold (M,g) of an indefinite manifold (M¯,g¯) is said to be totally umbilical in M¯ [3] if there is a smooth transversal vector field H∈Γ(tr(TM)), called the transversal curvature vector of M such that(15)h=g⊗H.Moreover, it is easy to see that M is totally umbilical in M¯, if and only if on each coordinate neighborhood U there exist smooth vector fields Hl∈Γ(ltr(TM)) and Hs∈Γ(S(TM⊥)) and smooth functions Hil∈F(ltr(TM)) and Hαs∈F(S(TM⊥)) such that,(16)hlX,Y=HlgX,Y,hsX,Y=HsgX,Y,hilX,Y=HilgX,Y,hαsX,Y=HαsgX,Y, for all X,Y∈Γ(TM).Let us now considerM¯ to be a (2n+1)-dimensional manifold endowed with an almost contact structure (ϕ¯,ξ,η); that is, ϕ¯ is a tensor field of type (1,1), ξ is a vector field, and η is a 1-form satisfying(17)ϕ¯2=-I+η⊗ξ,ηξ=1,η∘ϕ¯=0,ϕ¯ξ=0.Then (ϕ¯,ξ,η,g¯) is called anindefinite almost contact metric structure on M¯ if (ϕ¯,ξ,η) is an almost contact structure on M¯ and g¯ is a semi-Riemannian metric on M¯ such that [19], for any vector field X¯, Y¯ on M¯,(18)g¯ϕ¯X¯,ϕ¯Y¯=g¯X¯,Y¯-ηX¯ηY¯,ηX¯=g¯ξ,X¯.An indefinite almost contact metric manifold (M¯,ϕ¯,ξ,η,g¯) is said to benearly cosymplectic if(19)∇¯X¯ϕ¯Y¯+∇¯Y¯ϕ¯X¯=0,∀X¯,Y¯∈ΓTM¯,where ∇¯ is the Levi-Civita connection for g¯. Taking Y¯=ξ in (19), we get(20)∇¯X¯ξ=-H¯X¯,∀X¯∈ΓTM¯.It is easy to verify the following properties of H¯:(21)H¯ϕ¯+ϕ¯H¯=0,H¯ξ=0,η∘H¯=0,∇¯Xϕ¯ξ=ϕ¯H¯X,g¯H¯X¯,Y¯=-g¯X¯,H¯Y¯i.e.,H¯is skew-symmetric, for all X¯,Y¯∈Γ(TM¯). Let Ω denote the fundamental 2-form of M¯ defined by(22)ΩX¯,Y¯=g¯X¯,ϕ¯Y¯,X¯,Y¯∈ΓTM¯. Then the 1-form η and tensor H¯ are related as follows.Lemma 1.
Let(M¯,ϕ¯,ξ,η,g¯) be indefinite nearly cosymplectic. Then,(23)dηX¯,Y¯=g¯X¯,H¯Y¯,∀X¯,Y¯∈ΓTM¯.Moreover, M¯ is cosymplectic if and only if H¯ vanishes identically on M¯.Note that, for allX¯, Y¯, Z¯∈Γ(TM¯),(24)g¯∇¯Z¯ϕ¯X¯,Y¯=-g¯X¯,∇¯Z¯ϕ¯Y¯, which means that the tensor ∇¯ϕ¯ is skew-symmetric. The following lemma is fundamental to the sequel.Lemma 2.
LetM¯ be a nearly cosymplectic manifold. Then(25)∇¯X¯ϕ¯ϕ¯Y¯=-ϕ¯∇¯X¯ϕ¯Y¯-g¯Y¯,H¯X¯ξ-ηY¯H¯X¯,(26)∇¯ϕ¯X¯ϕ¯ϕ¯Y¯=-∇¯X¯ϕ¯Y¯-ηX¯ϕ¯H¯Y¯+ηY¯ϕ¯H¯X¯,for all X¯,Y¯∈Γ(TM¯).Proof.
The proof follows from a straightforward calculation.


## 3. Quasi-Generalized CR-Null Submanifolds

We recall some basic notions on QGCR-null submanifolds (see [15] for details).The structure vector fieldξ of an indefinite almost contact manifold (M¯,g¯) can be written according to decomposition (4) as follows:(27)ξ=ξS+∑i=1raiEi+∑i=1rbiNi+∑α=r+1ncαWα,where ξS is a smooth vector field of S(TM) while ai=η(Ni), bi=η(Ei), and cα=ϵαη(Wα), all smooth functions on M¯. Here ϵα=g¯(Wα,Wα).We adopt the definition of quasi-generalized CR- (QGCR-) null submanifolds given in [15] for indefinite nearly cosymplectic manifolds.Definition 3.
Let(M,g,S(TM),S(TM⊥)) be a null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). We say that M is quasi-generalized CR- (QGCR-) null submanifold of M¯ if the following conditions are satisfied.
(i) There exist two distributionsD1 and D2 of RadTM such that(28)Rad⁡TM=D1⊕D2,ϕ¯D1=D1,ϕ¯D2⊂STM.
(ii) There exist vector bundlesD0 and D¯ over S(TM) such that(29)STM=ϕ¯D2⊕D¯⊥D0,withϕ¯D0⊆D0,D¯=ϕ¯S⊕ϕ¯L,where D0 is a nondegenerate distribution on M and L and S are, respectively, vector subbundles of ltr(TM) and S(TM⊥).IfD1≠{0}, D0≠{0}, D2≠{0}, and S≠{0}, then M is called aproper QGCR-null submanifold.A proof of the following proposition uses similar arguments as in [15].Proposition 4.
A QGCR-null submanifoldM of an indefinite nearly cosymplectic manifold M¯ tangent to the structure vector field ξ is a GCR-null submanifold.Using (2), the tangent bundle of any QGCR-null submanifold, TM, can be decomposed as (30)TM=D⊕D^,withD=D0⊥D1,D^=D2⊥ϕ¯D2⊕D¯.Unlike a GCR-null submanifold, in a QGCR-null submanifold, D is invariant with respect to ϕ¯ while D^ is not generally anti-invariant.Throughout this paper, we suppose that(M,g,S(TM),S(TM⊥)) is a proper QGCR-null submanifold. From the above definition, we can easily deduce the following:(1)
Condition (i) implies thatdim⁡(RadTM)=s≥3.(2)
Condition (ii) implies thatdim⁡(D)≥4l≥4 and dim(D2)=dim(L).Definition 5 (see [16]).
A null submanifoldM of a semi-Riemannian manifold M¯ is said to beascreen if the structure vector field, ξ, belongs to RadTM⊕ltr(TM).From Definition3.3, Lemma 3.6, and Theorem 3.7 of [15], we have the following.Theorem 6.
Let(M,g,S(TM),S(TM⊥)) be an ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold M¯; then ξ∈Γ(D2⊕L). If M is a 3-null QGCR submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯), then M is ascreen null submanifold if and only if ϕ¯L=ϕ¯D2.Proof.
The proof follows from straightforward calculation as in [15].It is crucial to note the following aspects with ascreen QGCR-null submanifold: item (2) of Definition3 implies that dim(D)≥4l≥4 and dim(D2)=dim(L). Thus dim(M)≥7 and dim(M¯)≥11, and any 7-dimensional ascreen QGCR-null submanifold is 3-null.In what follows, we construct an ascreen QGCR-null submanifold of a special nearly cosymplectic manifold withH¯=0 (i.e., M¯ is a cosymplectic manifold). Thus, let (Rq2m+1,ϕ¯0,ξ,η,g¯) denote the manifold Rq2m+1 with its usual cosymplectic structure given by(31)η=dz,ξ=∂z,g¯=η⊗η-∑i=1q/2dxi⊗dxi+dyi⊗dyi+∑i=q+1mdxi⊗dxi+dyi⊗dyi,ϕ0∑i=1mXi∂xi+Yi∂yi+Z∂z=∑i=1mYi∂xi-Xi∂yi, where (xi,yi,z) are Cartesian coordinates and ∂tk=∂/∂tk, for t∈R2m+1.Now, we use the above structure to construct the following example.Example 7.
LetM¯=(R411,g¯) be a semi-Euclidean space, with g¯ being of signature (-,-,+,+,+,-,-,+,+,+,+) with respect to the canonical basis (32)∂x1,∂x2,∂x3,∂x4,∂x5,∂y1,∂y2,∂y3,∂y4,∂y5,∂z. Let (M,g) be a submanifold of M¯ given by (33)x1=y4,y1=-x4,z=x2sin⁡θ+y2cos⁡θ,y5=x51/2,where θ∈(0,π/2). By direct calculations, we can see that the vector fields (34)E1=∂x4+∂y1,E2=∂x1-∂y4,E3=sin⁡θ∂x2+cos⁡θ∂y2+∂z,X1=2y5∂x5+∂y5,X2=-cos⁡θ∂x2+sin⁡θ∂y2,X3=∂y3,X4=∂x3, form a local frame of TM. Then RadTM is spanned by {E1,E2,E3}, and, therefore, M is 3-null. Further, ϕ¯0E1=E2; therefore we set D1=Span{E1,E2}. Also ϕ¯0E3=-X2 and thus D2=Span{E3}. It is easy to see that ϕ¯0X3=X4, so we set D0=Span{X3,X4}. On the other hand, following direct calculations, we have(35)N1=12∂x4-∂y1,N2=12-∂x1-∂y4,N3=12-sin⁡θ∂x2-cos⁡θ∂y2+∂z,W=∂x5-2y5∂y5, from which ltr(TM)=Span{N1,N2,N3} and S(TM⊥)=Span{W}. Clearly, ϕ¯0N2=-N1. Further, ϕ¯0N3=1/2X2 and thus L=Span{N3}. Notice that ϕ¯0N3=-1/2ϕ¯0E3 and therefore ϕ¯0L=ϕ¯0D2. Also, ϕ¯0W=-X1 and therefore S=Span{W}. Finally, we calculate ξ as follows. Using Theorem 6 we have ξ=aE3+bN3. Applying ϕ¯0 to this equation we obtain aϕ¯0E3+bϕ¯0N3=0. Now, substituting for ϕ¯0E3 and ϕ¯0N3 in this equation we get 2a=b, from which we get ξ=1/2(E3+2N3). Since ϕ¯0ξ=0 and g¯(ξ,ξ)=1, we conclude that (M,g) is an ascreen QGCR-null submanifold of M¯.Proposition 8.
There exist no coisotropic, isotropic, or totally null proper QGCR-null submanifolds of an indefinite nearly cosymplectic manifold.


## 4. Umbilical and Geodesic Ascreen QGCR-Null Submanifolds

In this section, we prove two main theorems concerning totally umbilical, totally geodesic and irrotational ascreen QGCR-null submanifolds ofM¯. An indefinite nearly cosymplectic manifold M¯ is called anindefinite nearly cosymplectic space form, denoted by M¯(c¯), if it has the constant ϕ¯-sectional curvature c¯. The curvature tensor R¯ of the indefinite nearly cosymplectic space form M¯(c¯) is given by [21]:(36)4R¯X¯,W¯,Z¯,Y¯=g¯∇¯W¯ϕ¯Z¯,∇¯X¯ϕ¯Y¯-g¯∇¯W¯ϕ¯Y¯,∇¯X¯ϕ¯Z¯-2g¯∇¯W¯ϕ¯X¯,∇¯Y¯ϕ¯Z¯+g¯H¯W¯,Z¯g¯H¯X¯,Y¯-g¯H¯W¯,Y¯g¯H¯X¯,Z¯-2g¯H¯W¯,X¯g¯H¯Y¯,Z¯-ηW¯ηY¯g¯H¯X¯,H¯Z¯+ηW¯ηZ¯g¯H¯X¯,H¯Y¯+ηX¯ηY¯g¯H¯W¯,H¯Z¯-ηX¯ηZ¯g¯H¯W¯,H¯Y¯+c¯g¯X¯,Y¯g¯Z¯,W¯-g¯Z¯,X¯g¯Y¯,W¯+ηZ¯ηX¯g¯Y¯,W¯-ηY¯ηX¯g¯Z¯,W¯+ηY¯ηW¯g¯Z¯,X¯-ηZ¯ηW¯g¯Y¯,X¯+g¯ϕ¯Y¯,X¯g¯ϕ¯Z¯,W¯-g¯ϕ¯Z¯,X¯g¯ϕ¯Y¯,W¯-2g¯ϕ¯Z¯,Y¯g¯ϕ¯X¯,W¯,for all X¯,Y¯,Z¯,W¯∈Γ(TM¯).Notice thatD0 and ϕ¯S are orthogonal and nondegenerate subbundles of TM and that when M is ascreen QGCR-null submanifold, we observe that(37)ηX=ηZ=0,∀X∈ΓD0,Z∈Γϕ¯S.Theorem 9.
Let(M,g,S(TM), S(TM⊥)) be a totally umbilical or totally geodesic ascreen QGCR-null submanifold of an indefinite nearly cosymplectic space form M¯(c¯), of pointwise constant ϕ¯-sectional curvature c¯, such that D0 and ϕ¯S are space-like and parallel distributions with respect to ∇. Then, c¯≥0. Equality occurs when M¯(c¯) is an indefinite cosymplectic space form.Proof.
LetX and Z be vector fields in D0 and ϕ¯S, respectively. Replacing W¯ with ϕ¯X and Y¯ with ϕ¯Z in (36), we get(38)4R¯X,ϕ¯X,Z,ϕ¯Z=g¯∇¯ϕ¯Xϕ¯Z,∇¯Xϕ¯ϕ¯Z-g¯∇¯ϕ¯Xϕ¯ϕ¯Z,∇¯Xϕ¯Z-2g¯∇¯ϕ¯Xϕ¯X,∇¯ϕ¯Zϕ¯Z+g¯H¯ϕ¯X,Zg¯H¯X,ϕ¯Z-g¯H¯ϕ¯X,ϕ¯Zg¯H¯X,Z-2g¯H¯ϕ¯X,Xg¯H¯ϕ¯Z,Z-2c¯gϕ¯Z,ϕ¯Zgϕ¯X,ϕ¯X.Considering the first three terms on the right hand side of (38), we have(39)g¯∇¯ϕ¯Xϕ¯Z,∇¯Xϕ¯ϕ¯Z=-g¯∇¯Zϕ¯ϕ¯X,∇¯Xϕ¯ϕ¯Z.Applying (25) of Lemma 2 on (39) we derive(40)g¯∇¯ϕ¯Xϕ¯Z,∇¯Xϕ¯ϕ¯Z=-g¯∇¯Zϕ¯ϕ¯X,∇¯Xϕ¯ϕ¯Z=g¯∇¯Xϕ¯Z,∇¯Xϕ¯Z-g¯ϕ¯Z,H¯X2+g¯Z,H¯X2.In a similar way, using (26) of Lemma 2, we get(41)-g¯∇¯ϕ¯Xϕ¯ϕ¯Z,∇¯Xϕ¯Z=g¯∇¯Xϕ¯Z,∇¯Xϕ¯Z,-2g¯∇¯ϕ¯Xϕ¯X,∇¯ϕ¯Zϕ¯Z=0.Now substituting (40) and (41) into (38), we get (42)4R¯X,ϕ¯X,Z,ϕ¯Z=2g¯∇¯Xϕ¯Z,∇¯Xϕ¯Z-g¯ϕ¯Z,H¯X2+g¯Z,H¯X2+g¯H¯ϕ¯X,Zg¯H¯X,ϕ¯Z-g¯H¯ϕ¯X,ϕ¯Zg¯H¯X,Z-2g¯H¯ϕ¯X,Xg¯H¯ϕ¯Z,Z-2c¯gϕ¯Z,ϕ¯Zgϕ¯X,ϕ¯X, from which we obtain(43)2R¯X,ϕ¯X,Z,ϕ¯Z=g¯∇¯Xϕ¯Z,∇¯Xϕ¯Z+g¯Z,H¯X2-c¯gZ,ZgX,X.Then using the facts that D0 and ϕ¯S are space-like and parallel with respect to ∇, we have (∇¯Zϕ¯)X=(∇Zϕ¯)X∈Γ(D0), and (43) reduces to(44)2R¯X,ϕ¯X,Z,ϕ¯Z=∇Zϕ¯X2+g¯Z,H¯X2-c¯X2Z2,where · denotes the norm on D0⊥ϕ¯S with respect to g.
On the other hand, if we setW¯=ϕ¯X and Y¯=ϕ¯Z in (14), we have(45)R¯X,ϕ¯X,Z,ϕ¯Z=g¯∇Xhsϕ¯X,Z,ϕ¯Z-g¯∇ϕ¯XhsX,Z,ϕ¯Z,where(46)∇Xhsϕ¯X,Z=∇Xshsϕ¯X,Z-hs∇Xϕ¯X,Z-hsϕ¯X,∇XZ.By the fact that M is totally umbilical in M¯, we have hs(ϕ¯X,Z)=0. Thus using (16), (46) becomes(47)∇Xhsϕ¯X,Z=-hs∇Xϕ¯X,Z-hsϕ¯X,∇XZ=-g∇Xϕ¯X,ZHs-gϕ¯X,∇XZHs.Differentiating g¯(ϕ¯X,Z)=0 covariantly with respect to X and then applying (7), we obtain(48)g∇Xϕ¯X,Z+gϕ¯X,∇XZ=0.Substituting (48) into (47) gives(49)∇Xhsϕ¯X,Z=0.Similarly,(50)∇ϕ¯XhsX,Z=0.Then, substituting (49) and (50) into (45), we get(51)R¯X,ϕ¯X,Z,ϕ¯Z=0.Substituting (51) into (44), gives(52)c¯X2Z2=∇Zϕ¯X2+g¯Z,H¯X2≥0,which implies that c¯≥0. When the ambient manifold is cosymplectic, then ∇ϕ¯=0 and dη=0 [19] and in this case c¯=0.Example 10.
LetM be an ascreen QGCR-null submanifold in Example 7. Applying (7) and Koszul’s formula (see [2]) to Example 7 we obtain(53)hilX,Y=0∀X,Y∈ΓTM,wherei=1,2,3,ϵ4h4sX1,X1=2,h4sX,Y=0,∀X≠X1,Y≠X1.Using (12), (53), and ϵ4=g¯(W,W)=1+4(y5)2, we also derive(54)hX1,X1=21+4y52W.We remark that M is not totally geodesic. From (54) and (15) we note that M is totally umbilical with (55)H=21+4y522W. By straightforward calculations we also have (56)∇X1X1=4y5X1,∇XiXj=0∀i,j≠1. Thus, D0 and ϕ¯S are parallel distributions with respect to ∇. Hence, M satisfies Theorem 9 and c¯=0.Corollary 11.
Let(M,g,S(TM), S(TM⊥)) be a totally umbilical or totally geodesic ascreen QGCR-null submanifold of an indefinite cosymplectic space form M¯(c¯) of pointwise constant ϕ¯-sectional curvature c¯. Then, c¯=0.A null submanifoldM of a semi-Riemannian manifold (M¯,g¯) is called irrotational [3] if ∇¯XE∈Γ(TM), for any E∈Γ(RadTM) and X∈Γ(TM). Equivalently, M is irrotational if(57)hlX,E=hsX,E=0,for all X∈Γ(TM) and E∈Γ(RadTM).Theorem 12.
Let(M,g,S(TM), S(TM⊥)) be an irrotational ascreen QGCR-null submanifold of an indefinite nearly cosymplectic space form M¯(c) of pointwise constant ϕ¯-sectional curvature c¯. Then, c¯≤0 or c¯≥0. Equality holds when M¯(c¯) is an indefinite cosymplectic space form.Proof.
By settingY=E, Z=ϕ¯E, and W=E in (14), we get(58)R¯X,E,ϕ¯E,E=g¯∇XhlE,ϕ¯E,E-g¯∇EhlX,ϕ¯E,E+g¯∇XhsE,ϕ¯E,E-g¯∇EhsX,ϕ¯E,Efor any X∈Γ(TM) and E∈Γ(RadTM). Then, using the fact that M is irrotational, (58) reduces to(59)R¯X,E,ϕ¯E,E=0,∀X∈ΓTM.On the other hand, setting Y¯=W¯=E and Z¯=ϕ¯E in (36) and simplifying, we get(60)R¯X,E,ϕ¯E,E=-3g¯∇¯Eϕ¯ϕ¯E,∇¯Eϕ¯X-ηE2g¯H¯X,H¯ϕ¯E+4c¯ηE2g¯X,ϕ¯E.
Now, using (59) and (60), we get(61)4c¯ηE2g¯X,ϕ¯E=3g¯∇¯Eϕ¯ϕ¯E,∇¯Eϕ¯X+ηE2g¯H¯X,H¯ϕ¯E.Replacing X with ϕ¯E in (61) and using (25) of Lemma 2 to the resulting equation gives(62)c¯ηE2g¯ϕ¯E,ϕ¯E=ηE2g¯H¯ϕ¯E,H¯ϕ¯E.Since M is ascreen QGCR-null submanifold, there exists E∈Γ(D2) such that η(E)=b≠0, and thus (62) simplifies to(63)c¯=-1b2g¯H¯E,H¯E=1b2dηE,H¯E.We observe that c¯=0 if either dη=0 (i.e., M¯(c) is cosymplectic space form [19]) or H¯E is a null vector field. The second case implies that H¯E belongs to RadTM or ltr(TM). If H¯E∈Γ(RadTM), then there exists a nonzero smooth function κ such that H¯E=κE, for some arbitrary E∈Γ(RadTM). Taking the g¯-product of H¯E=κE with ξ leads to 0=κη(E), from which η(E)=0. Since M is ascreen QGCR-null submanifold, then, there is E∈Γ(D2) such that η(E)≠0, hence a contradiction. Similar reasoning can be applied if H¯E∈Γ(ltr(TM)). Therefore, c¯=0 only if H¯E=0 (i.e., dη=0) which occurs when M¯(c) is cosymplectic space form [19]. It turns out that c¯≤0 or c¯≥0 depending on whether H¯E is space-like or time-like vector field, respectively.Corollary 13.
Let(M,g,S(TM), S(TM⊥)) be an irrotational ascreen QGCR-null submanifold of an indefinite cosymplectic space form M¯(c¯) of pointwise constant ϕ¯-sectional curvature c¯. Then, c¯=0.It is easy to see from (54) that hl(X,E)=hs(X,E)=0 and hence M given in Example 10 is an irrotational ascreen QGCR-null submanifold of an indefinite cosymplectic space form M¯(c¯). As is proved in that example c¯=0.


## 5. Mixed Totally Geodesic QGCR-Null Submanifolds

Definition 14.
A QGCR-null submanifold of an indefinite nearly cosymplectic manifold(M¯,g¯) is calledmixed totally geodesic if its second fundamental form, h, satisfies h(X,Y)=0, for any X∈Γ(D) and Y∈Γ(D^).We will need the following lemma in the next theorem.Lemma 15.
Let(M,g,S(TM),S(TM⊥)) be any 3-null proper ascreen QG CR-null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). Then, (64)2ηEηN=1, for any E∈Γ(D2) and N∈Γ(L).Proof.
The proof follows from straightforward calculations usingg¯(ξ,ξ)=1 and ξ=η(N)E+η(E)N.Theorem 16.
Let(M,g,S(TM),S(TM⊥)) be a 3-null proper ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). Then, M is mixed totally geodesic if and only if hαs(X,Y)=0 and AEi∗X=0, for all X∈Γ(D), Y∈Γ(D^), Wα∈Γ(S(TM⊥)), and Ei∈Γ(RadTM).Proof.
By the definition of ascreen QGCR-null submanifold,M is mixed geodesic if(65)g¯hX,Y,Wα=g¯hX,Y,Ei=0,for all X∈Γ(D), Y∈Γ(D^), Wα∈Γ(S(TM⊥)), and Ei∈Γ(RadTM). Now, by virtue of (12) and the first equation of (65), we have (66)0=g¯hX,Y,Wα=ϵαhαsX,Y, from which hαs(X,Y)=0, since ϵα≠0. On the other hand, using (65), (7), and (11) we derive(67)g¯hX,Y,Ei=g¯∇¯XY,Ei=-g¯Y,∇¯XEi=gY,AEi∗X=0.Since D=D0⊥D1 and D^={D2⊥ϕ¯D2}⊕D¯, we observe that AEi∗X∉Γ(ϕ¯D2) or ϕ¯L. In fact, suppose that AEi∗X∉Γ(ϕ¯D2); then there exists a nonvanishing smooth function l such that AEi∗X=lϕ¯E, for E∈Γ(D2). Thus,(68)0=gY,AEi∗X=lgY,ϕ¯E,∀Y∈ΓD^.Taking Y=ϕ¯N in (68), where N∈Γ(L) and using Lemma 15, we have(69)0=gY,AEi∗X=lgϕ¯N,ϕ¯E=l1-ηEηN=12l, which is a contradiction, since l≠0. Hence AEi∗X∉Γ(ϕ¯D2⊕ϕ¯L). Moreover, AEi∗X∉Γ(ϕ¯S) since if AEi∗X∈Γ(ϕ¯S), then there is a nonvanishing smooth function ω such that AEi∗X=ωϕ¯Wα. Taking the g¯-product of this equation with respect to Y=ϕ¯Wα and using the fact that η(Wα)=0, we get(70)0=gY,AEi∗X=ωg¯ϕ¯Wα,ϕ¯Wα=ωg¯Wα,Wα=ωϵα, which is a contradiction, since ϵα≠0 and ω≠0. Hence, AEi∗X∉Γ({ϕ¯D2⊕ϕ¯L}⊥ϕ¯S), which implies that AEi∗X∈Γ(D0). Since AEi∗X∈Γ(D0), then the nondegeneracy of D0 implies that there exists some Z∈Γ(D0) such that g(AEi∗X,Z)≠0. But using (11) and (7), together with the fact that M is mixed geodesic, we derive(71)gAEi∗X,Z=-g∇XEi,Z=g¯Ei,∇¯XZ=g¯Ei,∇XZ=0,which is a contradiction. Thus AEi∗X∉Γ({ϕ¯D2⊕ϕ¯L}⊥ϕ¯S⊥D0); that is, AEi∗X=0. The converse is obvious.Corollary 17.
Let(M,g,S(TM),S(TM⊥)) be a proper ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). Then, if M is mixed totally geodesic then hil(X,Ei)=0 and φαi(X)=0, for all X∈Γ(D) and Ei∈Γ(D2).Definition 18.
A QGCR-null submanifold of an indefinite nearly cosymplectic manifold(M¯,g¯) is called D-totally geodesic if its second fundamental form h satisfies h(X,Y)=0,∀X,Y∈Γ(D).SinceM is ascreen QGCR-null submanifold, we have g¯(X,ξ)=0, for all X∈Γ(D). Applying ∇¯Y to g¯(X,ξ)=0 we get(72)η∇¯YX=-g¯X,∇¯Yξ=g¯X,H¯Y.Interchanging X and Y in (72) and then adding the resulting equation to (72) gives(73)η∇¯XY+η∇¯YX=g¯Y,H¯X+g¯X,H¯Y=0.Theorem 19.
Let(M,g,S(TM),S(TM⊥)) be a proper ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). Then, M is D-totally geodesic if and only if ϕ¯hl(X,ϕ¯E) and ϕ¯hs(X,ϕ¯W), respectively, have no components along ltr(TM) and S(TM⊥), while both ∇Xϕ¯E and ∇Xϕ¯W∉Γ(D0) for all X∈Γ(D), E∈Γ(RadTM), and W∈Γ(S).Proof.
By the definition of ascreen QGCR-null submanifold,M is D geodesic if and only if g¯(h(X,Y),E)=g¯(h(X,Y),W)=0, for all X,Y∈Γ(D), Wα∈Γ(S(TM⊥)), and E∈Γ(RadTM).
Using (7) and (18), we derive (74)g¯hX,Y,E=g¯∇¯XY,E=g¯ϕ¯∇¯XY,ϕ¯E-g¯Y,∇¯Xξg¯E,ξ, from which when we apply (20) we get(75)g¯hX,Y,E=g¯ϕ¯∇¯XY,ϕ¯E+g¯Y,H¯Xg¯E,ξ.Interchanging X and Y in (75) and considering the fact that h is symmetric we get(76)g¯hX,Y,E=g¯ϕ¯∇¯YX,ϕ¯E+g¯X,H¯Yg¯E,ξ.Summing (75) and (76) and then applying (73), we have(77)2g¯hX,Y,E=g¯ϕ¯∇¯XY,ϕ¯E+g¯ϕ¯∇¯YX,ϕ¯E.Now, applying the nearly cosymplectic condition in (19) to (77) leads to(78)2g¯hX,Y,E=g¯∇¯Xϕ¯Y,ϕ¯E+g¯∇¯Yϕ¯X,ϕ¯E.From (78) and (7) we derive(79)2g¯hX,Y,E=g¯∇¯Xϕ¯Y,ϕ¯E+g¯∇¯Yϕ¯X,ϕ¯E-g¯ϕ¯Y,hX,ϕ¯E-g¯ϕ¯X,hY,ϕ¯E.If we let X,Y∈Γ(D1) in (79), we obtain(80)2g¯hX,Y,E=g¯Y,ϕ¯hX,ϕ¯E+g¯X,ϕ¯hY,ϕ¯E.On the other hand, for X,Y∈Γ(D0), we get(81)2g¯hX,Y,E=-g¯ϕ¯Y,∇Xϕ¯E-g¯ϕ¯X,∇Yϕ¯E.It is easy to see from (80) and (81) that if ϕ¯h(X,ϕ¯E)∉Γ(ltr(TM)) and ∇Xϕ¯E∉Γ(D0), then g¯(h(X,Y),E)=0. The other assertions follows in the same way. The converse is obvious.Corollary 20.
Let(M,g,S(TM),S(TM⊥)) be a proper ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). If M is D-totally geodesic then ∇X∗ϕ¯E and ∇X∗ϕ¯W∉Γ(D0), for all X∈Γ(D), E∈Γ(D2), and W∈Γ(S).Corollary 21.
Let(M,g,S(TM),S(TM⊥)) be a proper ascreen QGCR-null submanifold of an indefinite nearly cosymplectic manifold (M¯,g¯). If M is D-totally geodesic, then D defines a totally geodesic foliation in M.



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