Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold

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 [1] on lightlike submanifolds of semi-Riemannian manifolds. Later, it was updated by K. L. Duggal and B. Sahin to reference [2]. 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, [3–16].

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 [17–19] 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 [20, pp. 4–6] as well as in the book [2, 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 [20, p. 11]), generalised CR [2, 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 Section 2, we quote some basics notions on almost-contact manifolds as well as lightlike geometry required in the rest of the paper. In Section 3, we focus on invariant submanifolds and some basic results. In Section 4, we discuss invariant submanifolds whose transversal bundles are flat in indefinite Sasakian space form.

A $(2n¯+1)$-dimensional semi-Riemannian manifold $M¯=(M¯,g¯,φ¯,ζ,η)$ is said to be an indefinite Sasakian manifold [21] 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

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 [2, Theorem 7.1.3, p. 307])

Let $(M¯,g¯)$ be a real $(m+n)$-dimensional semi-Riemannian manifold, where $m>1$ and $n≥1$⁠, with $g¯$ a semi-Riemannian metric of index q, such that $1≤q≤m+n−1$⁠. It follows that $M¯$ is never a Riemannian manifold. Let M be an m-dimensional submanifold of $M¯$⁠. For each $p∈M$⁠, we consider $TpM⊥={Up∈TpM¯:g¯p(Up,Xp)=0,∀Xp∈TpM}$⁠. If M is a lightlike submanifold, then there exists a smooth distribution $RadTpM$⁠, called the radical distribution, such that $RadTpM=TpM∩TpM⊥≠{0}$⁠, for all $p∈M$⁠. Denote by r the rank of $RadTM$⁠. If $r>0$⁠, then M is called an r-lightlike submanifold [2, 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 $TM/RadTM$ [22]. Let us consider the vector bundle $TM⊥=∪p∈MTpM⊥$⁠. In a lightlike case, $TM⊥$ is not complementary to $TM$ in $TM¯|M$ due to the fact that $RadTM=TM∩TM⊥$ 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:

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 $TM¯|M=TM⊕tr(M)$⁠.

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

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, (2.5) and (2.6) lead to (see [2, pp. 196–198]).

for all $X,Y∈Γ(TM)$⁠, $N∈Γ(ltr(TM))$ and $W∈Γ(S(TM⊥))$⁠. 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

for all $X,Y∈Γ(TM)$ and $ξ∈Γ(RadTM)$⁠. 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 (2.5), (2.7)–(2.10), we obtain

where $X,Y∈Γ(TM),ξ∈Γ(RadTM)andW∈Γ(S(TM⊥))$⁠. In general, the induced connection $∇$ on M is not a metric connection. Since $∇¯$ is a metric connection, by using (2.7), we get $(∇Xg)(Y,Z)=g¯(hl(X,Y),Z)+g¯(hl(X,Z),Y)$⁠, for all $X,Y,Z∈Γ(TM)$⁠. 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 [1, p. 171] for more details)

where $∇~hl,∇~hs,∇Dl,∇Ds,(∇XA)(Y,N)and(∇XA)(Y,W)$ are given by

for all $X,Y,Z∈Γ(TM),N∈Γ(tr(TM))andW∈Γ(S(TM⊥))$⁠. 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 $Ds$⁠.

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 $ζ∈Γ(S(TM))$ [8]. With the above note in mind, we have the following definition:

It is easy to see, from Definition 3.1 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:

From the first relation in (1) of Lemma 3.5, the following holds:

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.