Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold

Purpose – The author considers an invariant lightlike submanifold M , whose transversal bundle tr ð TM Þ is flat, in an indefinite Sasakian manifold M ð c Þ of constant f -sectional curvature c . Under some geometric conditions, the author demonstrates that c ¼ 1, that is, M is a space of constant curvature 1. Moreover, M and any leaf M 0 of its screen distribution S ð TM Þ 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.


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 wellknown 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 nondegenerate 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][4][5][6][7][8][9][10][11][12][13][14][15][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][18][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.

Preliminaries
A ð2n þ 1Þ-dimensional semi-Riemannian manifold M ¼ ðM ; g; f; ζ; ηÞ is said to be an indefinite Sasakian manifold [21] if it admits an almost-contact structure ðf; ζ; ηÞ, that is f is a tensor of type ð1; 1Þ 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 T x M of a Sasakian manifold M is called a f-section if it is spanned by a unit vector X orthogonal to ζ and fX, where X is a non-null vector field on M. The sectional curvature κðX; fXÞ of a f-section is called a f-sectional curvature. When c does not depend on the f-section at each point, then c constant in M and M is called a Sasakian space form, denoted by M ðcÞ. Moreover, the curvature tensor R of M satisfies (see [2,Theorem 7 Let ðM ; gÞ be a real ðm þ nÞ-dimensional semi-Riemannian manifold, where m > 1 and n ≥ 1, with g a semi-Riemannian metric of index q, such that 1 ≤ q ≤ m þ 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 T p M ⊥ ¼ fU p ∈ T p M : g p ðU p ; X p Þ ¼ 0; ∀ X p ∈ T p M g. If M is a lightlike submanifold, then there exists a smooth distribution Rad T p M, called the radical distribution, such that Rad T p M ¼ T p M ∩ T p M ⊥ ≠ f0g, for all p ∈ M. Denote by r the rank of Rad TM. 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 < minfm; ng, (2) co-isotropic submanifold, 1 < r ¼ n < m, The above theorem shows that there exists a complementary (but not orthogonal) vector bundle trðTM Þ to TM in TM jM , called the transversal bundle, such that trðTM Þ ¼ ltrðTMÞ ⊥ SðTM ⊥ Þ and TM jM ¼ TM ⊕ trðM Þ.
From now on, we denote by M an m-dimensional lightlike submanifold instead of ðM ; g; SðTM Þ; SðTM ⊥ ÞÞ and ðm þ nÞ-dimensional semi-Riemannian manifold by M. Let us denote by FðM Þ the algebra of smooth functions on M and ΓðEÞ the FðM Þ module of smooth sections of a vector bundle E (the same notation for any other vector bundle) over M. Then, we have (2.5) Invariant lightlike submanifolds where f∇ X Y ; A U X g and fhðX ; Y Þ; ∇ t X U g belong to ΓðTM Þ and ΓðtrðTM ÞÞ, respectively. Further, ∇ and ∇ t are linear connections on M and trðTM Þ, respectively. The second fundamental form h is a symmetric FðM Þ-bilinear form on ΓðTM Þ with values in ΓðtrðTM ÞÞ and the shape operator A V is a linear endomorphism of ΓðTMÞ. Moreover, (2.5) and (2.6) lead to (see [2, pp. 196-198]).
for all X ; Y ∈ ΓðTMÞ, N ∈ ΓðltrðTMÞÞ and W ∈ ΓðSðTM ⊥ ÞÞ. Here, A N and A W are called the shape operators of M. We call h l and h s the lightlike second fundamental form and the screen second fundamental form, respectively. Furthermore, ∇ l and ∇ s are, respectively, linear connections on ltrðTMÞ and SðTM ⊥ Þ, called the lightlike connection and the screen transversal connection. Note that D l and D s are Otsuki connections on ltrðTMÞ and SðTM ⊥ Þ, respectively. Denote the projection of TM on SðTM Þ by P. Then, we have for all X ; Y ∈ ΓðTMÞ and ξ ∈ ΓðRad TM Þ. Here, ∇ Ã and A Ã ξ are, respectively, the linear connection and shape operator of SðTM Þ. Furthermore, h Ã and ∇ Ãt stand for the second fundamental form and a linear connection on Rad TM, respectively. Furthermore, by using (2.5), (2.7)-(2.10), we obtain where X ; Y ∈ ΓðTMÞ; ξ ∈ ΓðRad TMÞ and W ∈ ΓðSðTM ⊥ ÞÞ. In general, the induced connection ∇ on M is not a metric connection. Since ∇ is a metric connection, by using However, it is important to note that ∇ Ã is a metric connection on SðTMÞ. Denoted by R, R l and R s , the curvature tensors of M, ltrðTMÞ and SðTM ⊥ Þ, respectively. Then we have (see [1, p. 171] for more details) where e ∇h l ; e ∇h s ; ∇D l ; ∇D s ; ð∇ X AÞðY ; N Þ and ð∇ X AÞðY ; W Þ are given by for all X ; Y ; Z ∈ ΓðTMÞ; N ∈ ΓðtrðTM ÞÞ and W ∈ ΓðSðTM ⊥ ÞÞ. Furthermore, we say that the screen transversal bundle SðTM ⊥ Þ is flat if ∇ s is a flat linear connection. In this case, the corresponding curvature tensor R s vanishes. Similarly, the lightlike transversal bundle ltrðTMÞ is flat if ∇ l is a flat linear connection, which also implies that R l vanishes. Next, we end this section by defining the parallelism of the connections D l and D s .
Proof: From (2.11) and (3.1), we have gðY ; D l ðζ; W ÞÞ ¼ gðA W ζ; Y Þ, for all Y ∈ ΓðTM Þ. Taking Y ¼ ξ in this relation, we get gðξ; D l ðζ; W ÞÞ ¼ 0. It follows from the last relation that D l ðζ; W Þ ¼ 0. On the other hand, from (2.11), we have gðh s ðX ; PY Þ; W Þ ¼ gðA W X ; PY Þ, for any X ; Y ∈ ΓðTM Þ. Taking X ¼ ζ in this relation and then applying in (2.15), we get gðA W ζ; PY Þ ¼ 0. It follows from this relation that A W ζ is ΓðRad TM Þvalued. Hence, using this information in the first relation of (2.14), we conclude that D s ðζ; N Þ ¼ 0 if and only if (2) D s is parallel and A W ζ ¼ 0 if and only if D s ¼ 0.
Proof: Using (2.22) and Lemma 3.8, we have D l ð∇ X ζ; W Þ ¼ 0, for any X ∈ ΓðTMÞ. Now, applying the first relation of (3.1) to this, we get D l ðfX ; W Þ ¼ 0. Replacing X with fX, we get Since h s is symmetric, it follows that A W is symmetric on TM, which proves (1). The proof of (2) follows similar steps, while considering (2.23), which completes the proof. ∎

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. for all X ; Y ∈ ΓðTMÞ and W ∈ ΓðSðTM ⊥ ÞÞ.