Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold

Samuel Ssekajja

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Open Access. Article publication date: 5 April 2021

Issue publication date: 11 January 2022

782

Abstract

Purpose

The author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M of its screen distribution S(TM) are, also, spaces of constant curvature 1.

Design/methodology/approach

The author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.

Findings

The author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).

Originality/value

To the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

Keywords

Citation

Ssekajja, S. (2022), "Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 100-111. https://doi.org/10.1108/AJMS-10-2020-0097

Publisher

:

Emerald Publishing Limited

Copyright © 2021, Samuel Ssekajja

License

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

Unlike non-degenerate submanifolds, lightlike submanifolds are quite complicated to study. One of the main reasons is that the tangent and normal bundles of a lightlike submanifold have, in general, a non-trivial intersection. It follows that one may not be able to use the well-known structural equations for non-degenerate submanifolds on lightlike submanifolds. In trying to overcome such difficulties, K. L. Duggal and A. Bejancu published their work [] on lightlike submanifolds of semi-Riemannian manifolds. Later, it was updated by K. L. Duggal and B. Sahin to reference []. In the above two books, the authors make use of a non-degenerate screen distribution on the submanifold, which gives rise to a four-factor breakdown of the ambient space. Unfortunately, the screen distribution is generally not unique and up to now there is no preferred technique of finding one. However, with some geometric conditions, one can secure a unique screen distribution, and some classes of lightlike submanifolds have been discussed, in the above books, with canonical screens, like the Monge lightlike hypersurfaces and many more. The foundations set in the books above motivated many other scholars to investigate the geometry of lightlike submanifolds. They include, amongst others, [].

Theory of invariant non-degenerate submanifolds of almost-contact manifolds has extensively been studied and many interesting results are currently known about them. Some of the notable results on the topic can be found in references [] and many more references cited therein. On the other hand, the invariant lightlike submanifolds have not yet been given the necessary attention. In fact, all the work presently known on this topic are limited to the scope set by K. L. Duggal and B. Sahin in the paper [, pp. 4–6] as well as in the book [, Chapter 7, p. 318]. Since invariant lightlike submanifolds are a part of many other general classes of lightlike submanifolds, such as the contact SCR (see [, p. 11]), generalised CR [, p. 334], amongst others, it would be important to understand their geometries well before any attempt is made to generalise them. The present paper is dedicated to the study of invariant lightlike submanifolds of indefinite Sasakian manifolds, whose transversal bundle is flat. The rest of the paper is arranged as follows: in , we quote some basics notions on almost-contact manifolds as well as lightlike geometry required in the rest of the paper. In , we focus on invariant submanifolds and some basic results. In , we discuss invariant submanifolds whose transversal bundles are flat in indefinite Sasakian space form.

2. Preliminaries

A (2n¯+1)-dimensional semi-Riemannian manifold M¯=(M¯,g¯,φ¯,ζ,η) is said to be an indefinite Sasakian manifold [] if it admits an almost-contact structure (φ¯,ζ,η), that is φ¯ is a tensor of type (1,1) of rank 2n¯, ζ is a unit spacelike vector field and η is a 1-form satisfying

(2.1)ϕ2=I+ηζ,η(ζ)=1,η(X)=g(X,ζ),φζ=0,η°φ=0,
(2.2)g¯(ϕ¯X,φ¯Y)=g¯(X,Y)η(X)η(Y),(¯Xφ¯)Y=g¯(X,Y)ζη(Y)X,
(2.3)¯Xζ=φ¯X,R¯(X,Y)ζ=η(Y)Xη(X)Y,
for all X and Y tangent to M¯. Here, ¯ is the Levi-Civita connection for a semi-Riemannian metric g¯. Furthermore, R¯ is the curvature tensor of M¯. Next, a plane section π in TxM¯ of a Sasakian manifold M¯ is called a φ¯-section if it is spanned by a unit vector X orthogonal to ζ and φ¯X, where X is a non-null vector field on M¯. The sectional curvature κ¯(X,φ¯X) of a φ¯-section is called a φ¯-sectional curvature. When c does not depend on the φ¯-section at each point, then c constant in M¯ and M¯ is called a Sasakian space form, denoted by M¯(c). Moreover, the curvature tensor R¯ of M¯ satisfies (see [, Theorem 7.1.3, p. 307])
(2.4)4R(X,Y,Z,W)=(c+3){g(Y,Z)g(X,W)g(X,Z)g(Y,W)}+(c1){η(X)η(Z)g(Y,W)η(Y)η(Z)g(X,W)+g(X,Z)η(Y)η(W)g(Y,Z)η(X)η(W)+g(φY,Z)g(φX,W)g(φX,Z)g(φY,W)2g(φX,Y)g(φZ,W)},X,Y,Z,WT(TM)

Let (M¯,g¯) be a real (m+n)-dimensional semi-Riemannian manifold, where m>1 and n1, with g¯ a semi-Riemannian metric of index q, such that 1qm+n1. It follows that M¯ is never a Riemannian manifold. Let M be an m-dimensional submanifold of M¯. For each pM, we consider TpM={UpTpM¯:g¯p(Up,Xp)=0,XpTpM}. If M is a lightlike submanifold, then there exists a smooth distribution RadTpM, called the radical distribution, such that RadTpM=TpMTpM{0}, for all pM. Denote by r the rank of RadTM. If r>0, then M is called an r-lightlike submanifold [, p. 191]. There are four possible classes of lightlike submanifolds, according to

  1. r-lightlike submanifold, 0<r<min{m,n},

  2. co-isotropic submanifold, 1<r=n<m,

  3. isotropic submanifold, 1<r=m<n,

  4. totally lightlike submanifold, 1<r=n=m.

Next, we consider a complementary distribution to RadTM in TM, called the screen distribution and denoted by S(TM). Such a screen is always secured due to the fact that M is paracompact. Moreover, S(TM) is orthogonal to RadTM and non-degenerate with respect to g¯. Thus, we have the decomposition TM=S(TM)RadTM. Obviously, S(TM) is not unique; however, it is canonically isomorphic to the factor bundle TM/RadTM []. Let us consider the vector bundle TM=pMTpM. In a lightlike case, TM is not complementary to TM in TM¯|M due to the fact that RadTM=TMTM is a distribution on M of rank r>1. Next, let us consider a non-degenerate complementary vector bundle S(TM) to RadTM in TM. Then, TM=S(TM)RadTM. We call S(TM) the screen transversal bundle of M. Furthermore, using the fact that S(TM) is non-degenerate, we have the decomposition TM¯|M=S(TM)S(TM), where S(TM) is the complementary orthogonal vector bundle to S(TM) in TM¯|M. Note that S(TM) is a vector subbundle of S(TM), and since both are non-degenerate, we have the orthogonal decomposition S(TM)=S(TM)S(TM). The theory of lightlike submanifolds largely depends on the vector bundles S(TM) and S(TM), a lightlike submanifold is often denoted as (M,g,S(TM),S(TM)). The following characterisation result of lightlike submanifolds is well known:

Theorem 2.1.

(Duggal-Sahin []). Let (M,g,S(TM),S(TM)) be an r-lightlike submanifold of semi-Riemannian manifold (M¯,g¯). Suppose is a coordinate neighbourhood of M. There exists a complementary vector bundle ltr(TM), called the lightlike transversal bundle of RadTM in S(TM) and a basis of Γ(ltr(TM)|) consists of smooth sections {N1,,Nr} of S(TM)| such that g¯(ξi,Nj)=δij, g¯(Ni,Nj)=0, i,j=1,,r, where {ξ1,,ξr} is a basis of Γ(RadTM).

The above theorem shows that there exists a complementary (but not orthogonal) vector bundle tr(TM) to TM in TM¯|M, called the transversal bundle, such that tr(TM)=ltr(TM)S(TM) and TM¯|M=TMtr(M).

From now on, we denote by M an m-dimensional lightlike submanifold instead of (M,g,S(TM),S(TM)) and (m+n)-dimensional semi-Riemannian manifold by M¯. Let us denote by F(M) the algebra of smooth functions on M and Γ(E) the F(M) module of smooth sections of a vector bundle E (the same notation for any other vector bundle) over M. Then, we have

(2.5)¯XY=XY+h(X,Y),X,YΓ(TM),
(2.6)¯XU=AUX+XtU,XΓ(TM),UΓ(tr(TM)),
where {XY,AUX} and {h(X,Y),XtU} belong to Γ(TM) and Γ(tr(TM)), respectively. Further, and t are linear connections on M and tr(TM), respectively. The second fundamental form h is a symmetric F(M)-bilinear form on Γ(TM) with values in Γ(tr(TM)) and the shape operator AV is a linear endomorphism of Γ(TM). Moreover, and lead to (see [, pp. 196–198]).
(2.7)¯XY=XY+hl(X,Y)+hs(X,Y),
(2.8)¯XN=ANX+XlN+Ds(X,N),
(2.9)¯XW=AWX+XsW+Dl(X,W),
for all X,YΓ(TM), NΓ(ltr(TM)) and WΓ(S(TM)). Here, AN and AW are called the shape operators of M. We call hl and hs the lightlike second fundamental form and the screen second fundamental form, respectively. Furthermore, l and s are, respectively, linear connections on ltr(TM) and S(TM), called the lightlike connection and the screen transversal connection. Note that Dl and Ds are Otsuki connections on ltr(TM) and S(TM), respectively. Denote the projection of TM on S(TM) by P. Then, we have
(2.10)XPY=XPY+h(X,PY),Xξ=AξX+Xtξ,
for all X,YΓ(TM) and ξΓ(RadTM). Here, * and Aξ* are, respectively, the linear connection and shape operator of S(TM). Furthermore, h* and *t stand for the second fundamental form and a linear connection on RadTM, respectively. Furthermore, by using , , we obtain
(2.11)g¯(hs(X,Y),W)+g¯(Y,Dl(X,W))=g(AWX,Y),
(2.12)g¯(hl(X,Y),ξ)+g¯(Y,hl(X,ξ))+g(Y,Xξ)=0,
(2.13)g(h(X,PY),N)=g(ANX,PY),g(hl(X,ξ),ξ)=0,Aξξ=0,
(2.14)g(Ds(X,N),W)=g(N,AW,X),g(XPY,N)=g(PY,ANX),
where X,YΓ(TM),ξΓ(RadTM)andWΓ(S(TM)). In general, the induced connection on M is not a metric connection. Since ¯ is a metric connection, by using , we get (Xg)(Y,Z)=g¯(hl(X,Y),Z)+g¯(hl(X,Z),Y), for all X,Y,ZΓ(TM). However, it is important to note that * is a metric connection on S(TM). Denoted by R, Rl and Rs, the curvature tensors of M, ltr(TM) and S(TM), respectively. Then we have (see [, p. 171] for more details)
(2.15)R¯(X,Y)Z=R(X,Y)Z+Ahl(X,Z)YAhl(Y,Z)X+Ahs(X,Z)YAhs(Y,Z)X+(~Xhl)(Y,Z)(~Yhl)(X,Z)+Dl(X,hs(Y,Z))Dl(Y,hs(X,Z))+(~Xhs)(Y,Z)(~Yhs)(X,Z)+Ds(X,hl(Y,Z))Ds(Y,hl(X,Z)),
(2.16)R¯(X,Y)N=Rl(X,Y)N+hl(Y,ANX)hl(X,ANY)+Dl(X,Ds(Y,N))Dl(Y,Ds(X,N))+(YA)(N,X)(XA)(N,Y)+ADs(X,N)YADs(Y,N)X+(XDs)(Y,N)(YDs)(X,N)+hs(Y,ANX)hs(X,ANY),
(2.17)R¯(X,Y)W=Rs(X,Y)W+hs(Y,AWX)hs(X,AWY)+Ds(X,Dl(Y,W))Ds(Y,Dl(X,W))+(YA)(W,X)(XA)(W,Y)+ADl(X,W)YADl(Y,W)X+(XDl)(Y,W)(YDl)(X,W)+hl(Y,AWX)hl(X,AWY),
where ~hl,~hs,Dl,Ds,(XA)(Y,N)and(XA)(Y,W) are given by
(2.18)(~Xhl)(Y,Z)=Xlhl(Y,Z)hl(XY,Z)hl(Y,XZ),
(2.19)(~Xhs)(Y,Z)=Xshs(Y,Z)hs(XY,Z)hs(Y,XZ),
(2.20)(XDl)(Y,W)=XlDl(Y,W)Dl(XY,W)Dl(Y,XsW),
(2.21)(XDs)(Y,N)=XsDs(Y,N)Ds(XY,N)Ds(Y,XlN),
(XA)(N,Y)=XA(N,Y)A(XlN,Y)A(N,XY),
(XA)(W,Y)=XA(W,Y)A(XlW,Y)A(W,XY),
for all X,Y,ZΓ(TM),NΓ(tr(TM))andWΓ(S(TM)). Furthermore, we say that the screen transversal bundle S(TM) is flat if s is a flat linear connection. In this case, the corresponding curvature tensor Rs vanishes. Similarly, the lightlike transversal bundle ltr(TM) is flat if l is a flat linear connection, which also implies that Rl vanishes. Next, we end this section by defining the parallelism of the connections Dl and Ds.
Definition 2.2.

We say that the Otsuki connection Dl (resp. Ds) is parallel if Dl=0 (resp. Ds=0).

It follows from relations , and that Dl and Ds are parallel if and only if

(2.22) XlDl(Y,W)=Dl(XY,W)+Dl(Y,XsW),
(2.23) andXsDs(Y,N)=Ds(XY,N)+Ds(Y,XlN),
for all X,YΓ(TM), respectively.

3. Definitions and basic results

Let M be a lightlike submanifold of an indefinite almost-contact metric manifold M¯=(M¯,ζ,η,φ¯,g¯). If ζ is tangent to M, then ζ does not belong to the lightlike distribution RadTM. Thus, by ζ tangent, we shall mean ζΓ(S(TM)) []. With the above note in mind, we have the following definition:

Definition 3.1.

(Duggal-Sahin []). Let M be a lightlike submanifold of an indefinite Sasakian manifold M¯, tangent to ζ, that is, ζΓ(S(TM)). We call M an invariant lightlike submanifold if both S(TM) and RadTM are invariant with respect to φ¯. That is, φ¯S(TM)=S(TM) and φ¯RadTM=RadTM.

It is easy to see, from above, that ltr(TM) and S(TM) are also invariant with respect to φ¯. That is, φ¯ltr(TM)=ltr(TM) and φ¯S(TM)=S(TM). Also the following, about an invariant lightlike submanifold, holds:

Proposition 3.2.

There exist no any isotropic or totally lightlike invariant submanifold M of an indefinite Sasakian manifold M¯.

According to , by an invariant lightlike submanifold M of an indefinite Sasakian manifold M¯, we shall always mean M to be an r-lightlike or a co-isotropic in M¯.

Lemma 3.3.

Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold. Then, the following holds:

(3.1) Xζ=φ¯X,hl(X,ζ)=hs(X,ζ)=0,
(3.2)h(X,φY)=φh(X,Y),h(φX,φY)=h(X,Y),
(3.3) (Xφ¯)Y=g(X,Y)ζη(Y)X,R(X,Y)ζ=η(Y)Xη(X)Y,
for all X,YΓ(TM).

Proof: The relations in , and the first in follow easily from , and . Turning to the second relation in . Setting Z=ζ in and then considering in the resulting relation, we get

(3.4) R¯(X,Y)ζ=R(X,Y)ζ+(Xhl)(Y,ζ)(Yhl)(X,ζ)+(Xhs)(Y,ζ)(Yhs)(X,ζ),
for any X,YΓ(TM). Now, using and , we derive (Xhl)(Y,ζ)=hl(Y,Xζ)=hl(Y,φ¯X). It then follows
(3.5) (Xhl)(Y,ζ)(Yhl)(X,ζ)=0,
in which we have used and the symmetry of hl. In a similar way, but this time using and , we have (Xhs)(Y,ζ)=hs(Y,φ¯X), from which
(3.6)(Xhs)(Y,ζ)(Yhs)(X,ζ)=0.

Then, putting and in , we obtain R¯(X,Y)ζ=R(X,Y)ζ. It then follows from that R(X,Y)ζ=η(Y)Xη(X)Y. ∎

Considering , we have the following.

Theorem 3.4.

The sectional curvature of any non-degenerate plane spanned by ζ and a non-null vector field on M orthogonal to ζ is 1.

Lemma 3.5.

On any invariant lightlike submanifold M of an indefinite Sasakian manifold M¯, we have the following:

  1. Aφ¯NX=φ¯ANXg¯(X,N)ζ,Xlφ¯N=φ¯XlN,Ds(X,φ¯N)=φ¯Ds(X,N);

  2. Aφ¯WX=φ¯AWX,Xsφ¯W=φ¯XsW,Dl(X,φ¯W)=φ¯Dl(X,W), for any XΓ(TM).

Proof: Taking Y=N in the second relation of , we have (¯Xφ¯)N=g¯(X,N)ζ, for any XΓ(TM). Then, applying to this relation leads to
(3.7) Aφ¯NX+Xlφ¯N+Ds(X,φ¯N)+φ¯ANXφ¯XlNφ¯Ds(X,N)=g¯(X,N)ζ.
The relations in (1), then, follow from by comparing tangential and transversal parts. On the other hand, using and , we derive
(3.8) Aφ¯WX+Xsφ¯W+Dl(X,φ¯W)+φ¯AWXφ¯XsWφ¯Dl(X,W)=0,
for all XΓ(TM). Finally, the relations in (2) follow from by comparing tangential and transversal parts. ∎

From the first relation in (1) of , the following holds:

Proposition 3.6.

There exists no any invariant lightlike submanifold of an indefinite Sasakian manifold such that AN vanishes on RadTM.

It is well known [, Eq. 4.20, p. 62] that when S(TM) is totally umbilic, then

(3.9) PANX=λPXandh(ξ,PX)=0,
for any XΓ(TM), where λ is smooth function on each coordinate neighbourhood M. Thus, in case M is invariant submanifold, with a totally umbilic screen, the first relation in (1) of and the first relation of gives
(3.10)Pφ¯ANξ=g¯(ξ,N)ζ.

Taking the inner product of with ζ leads to g¯(ξ,N)=g(Pφ¯ANξ,ζ)=g(φ¯ANξ,ζ)=0. This is clearly a contradiction.

On the other hand, when S(TM) is parallel, with respect to , it is known [, p. 89] that h(X,PY)=0, for all X,YΓ(TM). From this relation and the first one in , we see that

(3.11) g(ANξ,PX)=0,
for each XΓ(TM) and NΓ(ltr(TM)). Then, the first relation in (1) of , leads to
(3.12)g(Aφ¯Nξ,PY)=g(ANξ,φ¯PY)g¯(ξ,N)g(ζ,PY).

Thus, from and , we have g¯(ξ,N)g(ζ,PY)=0, for any YΓ(TM). As g¯(ξ,N)0, it follows that g(ζ,PY)=0. Now, replacing PY with ζ (this is possible since ζ belongs to S(TM) by ) in the last relation, we get g(ζ,ζ)=0, which is a contradiction to g(ζ,ζ)=η(ζ)=1 (see the second relation in . With the above discussion, we have the following result:

Theorem 3.7.

There exists no any invariant lightlike submanifold of an indefinite Sasakian manifold with a totally umbilic or parallel screen distribution.

Lemma 3.8.

Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold M¯. Then, the following holds:

  1. Dl(ζ,W)=0.

  2. Ds(ζ,N)=0 if and only if AWζ=0.

Proof: From and , we have g¯(Y,Dl(ζ,W))=g(AWζ,Y), for all YΓ(TM). Taking Y=ξ in this relation, we get g¯(ξ,Dl(ζ,W))=0. It follows from the last relation that Dl(ζ,W)=0. On the other hand, from , we have g¯(hs(X,PY),W)=g(AWX,PY), for any X,YΓ(TM). Taking X=ζ in this relation and then applying in , we get g(AWζ,PY)=0. It follows from this relation that AWζ is Γ(RadTM)valued. Hence, using this information in the first relation of , we conclude that Ds(ζ,N)=0 if and only if AWζ=0. ∎

Proposition 3.9.

Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold M¯. Then,

  1. Dl is parallel if and only if Dl=0. Moreover, AW is a symmetric operator.

  2. Ds is parallel and AWζ=0 if and only if Ds=0.

Proof: Using and , we have Dl(Xζ,W)=0, for any XΓ(TM). Now, applying the first relation of to this, we get Dl(φ¯X,W)=0. Replacing X with φ¯X, we get Dl(X,W)+η(X)D(ζ,W)=Dl(X,W)=0. Hence, Dl=0. Now from , we have g¯(hs(X,Y),W)=g(AWX,Y), for any X,YΓ(TM). Since hs is symmetric, it follows that AW is symmetric on TM, which proves (1). The proof of (2) follows similar steps, while considering , which completes the proof. ∎

4. Main results

In this section, we characterise an invariant lightlike submanifold M of an indefinite Sasakian manifold M¯, whose transversal bundle is flat. In line with the above, we start with a few characterisation results.

Proposition 4.1.

Let M be an invariant lightlike submanifold of an indefinite Sasakian space form M¯(c), with a flat screen transversal bundle S(TM). Then,

(c1)g(W,W)g(ϕX,ϕY)=2{g(hs(Y,AWX),W)+g(hs(X,AϕWϕY),W)g(Ds(X,Dl(ϕY,ϕW)),W)+g(Ds(ϕY,Dl(X,ϕW)),W)},
for all X,YΓ(TM) and WΓ(S(TM)).

Proof: Replacing Y with φ¯Y, Z with W and W with φ¯W in (2), we get

(4.1) R¯(X,φ¯Y,W,φ¯W)=c12g¯(W,W)g(φ¯X,φ¯Y),
for all X,YΓ(TM). On the other hand, since Rs=0, leads to
(4.2)R(X,φY,W,φW)=g(hs(φY,AWX)hs(X,AWφY),φW)+g(Ds(X,Dl(φY,W))Ds(φY,Dl(X,W)),φW).

Then, applying the relations of to , we get

(4.3)R(X,φY,W,φW)=g(hs(Y,AWX),W)+g(hs(X,AφWφY),W)g(Ds(X,Dl(φY,φW)),W)+g(Ds(φY,Dl(X,φW)),W),
for any X,YΓ(TM) and WΓ(S(TM)). Finally, our claim follows from and , which completes the proof. ∎

Proposition 4.2.

Let M be an invariant r-lightlike submanifold of an indefinite Sasakian space form M¯(c), with a flat screen transversal bundle S(TM). If Dl is parallel, then M¯(c) is a space of constant curvature c=1 if and only if AW has no components in S(TM).

Proof: Suppose that Dl is parallel. It follows from that AW is a symmetric operator. Hence, applying and , we derive,

(4.4) g¯(hs(X,Aφ¯Wφ¯Y),W)=g(AWX,Aφ¯Wφ¯Y)=g(AWX,AWY),
for all X,YΓ(TM). Applying to the relation of , we get,
(4.5) (c1)g¯(W,W)g(φ¯X,φ¯Y)=4g(AWX,AWY).

Now, from , we see that when c=1 , then g(AWX,AWY)=0. This shows that PAWX=0. On the other hand, when PAWX=0, for each XΓ(TM), then gives (c1)g¯(W,W)g(φ¯X,φ¯Y)=0. Clearly, c=1 since S(TM) and S(TM) are non-degenerate subbundles, which completes the proof. ∎

Proposition 4.3.

Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold M¯, such that Dl is parallel. If the lightlike transversal bundle ltr(TM) is flat, then c=1 if and only if the operator AN is symmetric with respect to the lightlike second fundamental form hl.

Proof: As Dl is parallel, suggests that Dl=0. Since ltr(TM) is flat, leads to

(4.6) R¯(X,Y,N,ξ)=g¯(hl(Y,ANX)hl(X,ANY),ξ),
for any X,YΓ(TM). Next, applying to (2), we get
(4.7) (c1)2g¯(X,φ¯Y)g¯(φ¯N,ξ)=g¯(hl(Y,ANX)hl(X,ANY),ξ).

If c=1, leads to g¯(hl(Y,ANX)hl(X,ANY),ξ)=0, from which we get hl(Y,ANX)=hl(X,ANY). Hence, AN is symmetric with respect to hl. On the other hand, when AN is sympathetic with respect to hl, gives (c1)g¯(X,φ¯Y)g¯(φ¯N,ξ)=0. Since S(TM) is non-degenerate, we deduce that c=1, which completes the proof. ∎

A lightlike submanifold M of a semi-Riemannian manifold M¯ is said to be totally umbilic [, Definition 1, p. 58], in M¯, if there is a smooth transversal vector field HΓ(tr(TM)), called the transversal curvature vector field of M, such that h(X,Y)=g(X,Y)H, for all X,YΓ(TM). Moreover, M is totally umbilic if and only if on each coordinate neighbourhood there exist smooth vector fields HlΓ(ltr(TM)) and HsΓ(S(TM)) such that hl(X,Y)=g(X,Y)Hl and hs(X,Y)=g(X,Y)Hs. Furthermore, Theorem 4.1 of [, p. 59] indicates that when M is totally umbilic, then the Otsuki connection Dl, on ltr(TM), vanishes, that is, Dl=0. We say that M is totally geodesic if H vanishes, equivalently when both Hl and Hs vanish.

Theorem 4.4.

Let M be a totally umbilic invariant lightlike submanifold of an indefinite Sasakian space form M¯(c). If the lightlike transversal bundle ltr(TM) or the screen transversal bundle S(TM) is flat, then c=1. Moreover,

  1. M is a space of constant curvature 1.

  2. RadTM is a flat distribution on M.

  3. Any leaf M of S(TM) is minimal in M¯ and has constant curvature 1.

Proof: When M is totally umbilic, the last two relations in indicate that g(X,ζ)Hl=0 and g(X,ζ)Hs=0, for any XΓ(TM). It follows from these two relations that Hl=Hs=0 and hence, M is totally geodesic. This was also proved in [, Theorem 2.5, p. 6]. Therefore, in view of , we easily conclude that R¯(X,Y)Z=R(X,Y)Z, for any X, Y and Z tangent to M. Moreover, Dl=0 , so it is trivially a metric connection in this case. By the flatness assumption of ltr(TM) or S(TM), we see, from and , that c=1. It follows from (2) that
(4.8) R(X,Y)Z=R¯(X,Y)Z=g(Y,Z)Xg(X,Z)Y,
for all X,Y,ZΓ(TM). We can easily see, from , that M is a space of constant curvature 1, which proves (1). Next, let R*t denote the curvature tensor of RadTM with respect to *t. Then, as per (3.8) of [, p. 57] and the fact that M is totally geodesic and of constant curvature 1, we obtain
(4.9) g¯(Rt(X,Y)ξ,N)=g¯(R(X,Y)ξ,N)=g(Y,ξ)g¯(X,N)g(X,ξ)g¯(Y,N)=0,
for all X,YΓ(TM). From , we deduce that R*t=0. Hence, RadTM is flat, which proves (2). Next, assume that S(TM) is integrable and let M be its leaf. In this case, AN is symmetric on S(TM); hence, the first relation in (1) of helps us to get
(4.10) h(φ¯X,φ¯Y)=h(X,Y),
for any X,YΓ(S(TM)). Since M is totally geodesic, the second fundamental form of M in M¯ is h(X,Y)=h(X,Y)+hl(X,Y)+hs(X,Y)=h(X,Y), for any X and Y tangent to M. Therefore, using , we get trace|TMh=0, which shows that M is minimal. Denote by R* the curvature tensor of S(TM) with respect to the semi-Riemannian connection *. Then, through a direct calculation while considering , and the fact M is totally geodesic, we derive
g(Y,Z)Xg(X,Z)Y=R(X,Y)Z=R(X,Y)Z+h(X,YZ)h(Y,XZ)h([X,Y],Z)+Xth(Y,Z)Yth(X,Z),
for any X, Y and Z tangent to M. It follows from the relation above that
R(X,Y)Z=g(Y,Z)Xg(X,Z)Y,
where R=R|M* is the curvature tensor of M and g=gM. Hence, M is a space of constant curvature 1, which completes the proof. ∎

Corollary 4.5.

There does not exist any totally umbilic invariant lightlike submanifold of an indefinite Sasakian space form M¯(c1), with a flat lightlike transversal bundle or flat screen transversal bundle.

We wind up this section by giving an example of an invariant lightlike submanifold M of an indefinite Sasakian manifold M¯.

Example 4.6.

(An invariant lightlike submanifold). Let M¯=(27,φ¯,ζ,η,g¯) be the manifold endowed with the usual Sasakian structure (see, for example, [, p. 321] for such a structure), in which g¯ has signature (,+,+,,+,+,+), with respect to the canonical basis {x1,x2,x3,y1,y2,y3,z}. Suppose that M is a submanifold of M¯ given by

x1=υ1coshθ,y1=υ2coshθ,x2=υ1sinhθυ2,y2=υ1+υ2sinhθ,x3=sinυ3sinhυ4,y3=cosυ3coshυ4,z=υ5.

It is easy to see that the vector fields ξ1,ξ2,ζ,Z1,Z2, and given by

ξ1=coshθx1+sinhθx2+y2+(y1coshθ+y2sinhθ)z,ξ2=x2+coshθy1+sinhθy2y2z,ζ=2z,Z1=cosυ3sinhυ4x3sinυ3coshυ4y3+y3cosυ3sinhυ4z,Z2=sinυ3coshυ4x3+cosυ3sinhυ4y3+y3sinυ3coshυ4z,
spans TM. Moreover, one can see that RadTM=Span{ξ1,ξ2} and S(TM)=Span{Z1,Z2,ζ}. Furthermore, we note that φ¯ξ2=ξ1 and φ¯Z2=Z1. It follows that RadTM and S(TM) are invariant under φ¯. On the other hand, ltr(TM) is spanned by N1 and N2, where
N1=2{coshθx1sinhθx2+y2(y1coshθ+y2sinhθ)z},N2=2{x2coshθy1sinhθy2y2z}.

Note that φ¯N2=N1; hence, ltr(TM) is invariant under φ¯. Therefore, M is a five-dimensional invariant lightlike submanifold of M¯.

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Acknowledgements

The author wishes to thank the University of Witwatersrand for its generous financial support through a start-up research funding. The author also wishes to thank the anonymous referees for their comments and suggestions that greatly improved this paper.

Corresponding author

Samuel Ssekajja can be contacted at: samuel.ssekajja@wits.ac.za

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