Abstract
Purpose
The author considers an invariant lightlike submanifold M, whose transversal bundle
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 [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, [316].
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 [1719] 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.
2. Preliminaries
A
Let
r-lightlike submanifold,
,co-isotropic submanifold,
,isotropic submanifold,
,totally lightlike submanifold,
.
Next, we consider a complementary distribution to
(Duggal-Sahin [2]). Let
The above theorem shows that there exists a complementary (but not orthogonal) vector bundle
From now on, we denote by M an m-dimensional lightlike submanifold instead of
We say that the Otsuki connection
It follows from relations (2.20), (2.21) and Definition 2.2 that
3. Definitions and basic results
Let M be a lightlike submanifold of an indefinite almost-contact metric manifold
(Duggal-Sahin [2]). Let M be a lightlike submanifold of an indefinite Sasakian manifold
It is easy to see, from Definition 3.1 above, that
There exist no any isotropic or totally lightlike invariant submanifold M of an indefinite Sasakian manifold
According to Proposition 3.2, by an invariant lightlike submanifold M of an indefinite Sasakian manifold
Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold. Then, the following holds:
Proof: The relations in (3.1), (3.2) and the first in (3.3) follow easily from (2.7), (2.2) and (2.3). Turning to the second relation in (3.3). Setting
Then, putting (3.5) and (3.6) in (3.4), we obtain
Considering Lemma 3.3, we have the following.
The sectional curvature of any non-degenerate plane spanned by ζ and a non-null vector field on M orthogonal to ζ is 1.
On any invariant lightlike submanifold M of an indefinite Sasakian manifold
for any .
From the first relation in (1) of Lemma 3.5, the following holds:
There exists no any invariant lightlike submanifold of an indefinite Sasakian manifold such that
It is well known [9, Eq. 4.20, p. 62] that when
Taking the inner product of (3.10) with ζ leads to
On the other hand, when
Thus, from (3.11) and (3.12), we have
There exists no any invariant lightlike submanifold of an indefinite Sasakian manifold with a totally umbilic or parallel screen distribution.
Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold
if and only if .
Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold
is parallel if and only if . Moreover, is a symmetric operator. is parallel and if and only if .
4. Main results
In this section, we characterise an invariant lightlike submanifold M of an indefinite Sasakian manifold
Let M be an invariant lightlike submanifold of an indefinite Sasakian space form
Proof: Replacing Y with
Then, applying the relations of Lemma 3.5 to (4.2), we get
Let M be an invariant r-lightlike submanifold of an indefinite Sasakian space form
Proof: Suppose that
Now, from (4.5), we see that when
Let M be an invariant lightlike submanifold of an indefinite Sasakian manifold
Proof: As
If
A lightlike submanifold M of a semi-Riemannian manifold
Let M be a totally umbilic invariant lightlike submanifold of an indefinite Sasakian space form
M is a space of constant curvature 1.
is a flat distribution on M.Any leaf
of is minimal in and has constant curvature 1.
There does not exist any totally umbilic invariant lightlike submanifold of an indefinite Sasakian space form
We wind up this section by giving an example of an invariant lightlike submanifold M of an indefinite Sasakian manifold
(An invariant lightlike submanifold). Let
It is easy to see that the vector fields
Note that
References
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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.