The geometry of slant lightlike submanifolds

Abstract

The present thesis entitled “The Geometry of Slant Lightlike Submanifolds” comprises certain investigations carried out by me at the School of Mathematics (SOM), Thapar University, Patiala, under the supervision of Dr. S. S. Bhatia, Professor, School of Mathematics, Thapar University, Patiala and Dr. Rakesh Kumar, Assistant Professor, Department of Basic and Applied Sciences, Punjabi University, Patiala. The core of differential geometry and modern geometrical dynamics represents the concept of a manifold. Manifolds are the higher-dimensional analogues of surfaces. The local and global properties of smooth manifolds equipped with a metric tensor encodes its geometry. To study the geometric aspects of a manifold, it is more convenient to first embed it into a known manifold and then study the geometry which is induced on it. This approach gives impetus to the study of submanifolds which later developed into a fascinating study of the theory of submanifolds. The geometry of submanifolds of an almost Hermitian manifold depends upon the behaviour of the tangent bundle of the submanifold with respect to the almost complex structure J¯ of the manifold. The action of the almost complex structure give rise to the two well known classes of submanifolds namely, the holomorphic submanifolds and the totally real submanifolds. Let M¯ be an almost Hermitian manifold with almost complex structure J¯. Consider M as a submanifold of M¯ and let the tangent space and normal space of M at p ∈ M be denoted by TpM and TpM⊥, respectively. If TpM is invariant under the action of J¯ for each p ∈ M, that is, if JT¯ pM = TpM, for each p ∈ M, then M is called an invariant (or holomorphic) submanifold of M¯ . On the other hand, if JT¯ pM is contained in the normal space TpM⊥, for each p ∈ M, vi that is, if JT¯ pM ⊂ TpM⊥, for each p ∈ M, then M is called an anti-invariant (or totally real) submanifold of M¯ . In other words, a submanifold of an almost Hermitian manifold is a holomorphic or a totally real submanifold, if and only if, the angle between JX¯ and the tangent space TpM is 0 or π/2 respectively, for every non zero vector X tangent to any point p ∈ M. This paved the way for generalization of holomorphic and totally real submanifold to a new class of submanifold, known as slant submanifold, introduced by Chen [23] in 1990. Slant submanifolds are those for which the angle between JX¯ and the tangent space TpM is constant. Thus the theory of slantsubmanifolds is an umbrella over the theory of holomorphic submanifolds and totally real submanifolds. Initially, the slant submanifolds were studied with positive definite metric. This geometry of slant submanifolds could not be used extensively in mathematical physics, since the metric may not always be definite. The developments in theoretical physics require a deeper understanding of the geometric structure of higher dimensional manifolds with indefinite metric. Thus the geometry of manifolds with indefinite metric becomes the topic of interest due to the fact that the signature of the indefinite metric creates an interesting changes in the geometry of manifolds. From the mid of the 20th century, the Riemannian and semi-Riemannian geometries have been active and fruitful areas of research in differential geometry due to their applications to a variety of subjects in mathematics and physics. In the process of generalization of submanifolds theory from Riemannian manifolds to semi-Riemannian manifolds, lightlike submanifolds arise naturally in the semi-Riemannian category. The theory of submanifolds of Riemannian or semivii Riemannian manifolds is well known but its counter part lightlike submanifolds is relatively new and is in its developing stage. In the theory of submanifolds of semi-Riemannian manifolds, it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is non-trivial. Thus, the study becomes more interesting and remarkably different from the study of non-degenerate submanifolds. Moreover, the growing importance of lightlike geometry in mathematical physics, in particular, their extensive uses in theory of relativity motivated the geometers to do research on this subject matter. Sahin [74] clubbed the theory of slant submanifolds with lightlike geometry and introduced slant lightlike submanifolds of an almost Hermitian manifold. The odd dimensional version of almost Hermitian manifolds are also of equal importance in differential geometry. A very important mechanism for the progress in the field of science and technology is to generalize the existing ideas. The present thesis has been written on the same basis. In present thesis, the geometric aspects of slant lightlike submanifolds of indefinite almost complex manifolds (K¨ahler manifolds) and indefinite almost contact manifolds (Sasakian manifolds, Cosymplectic manifolds and Kenmotsu manifolds) have been explored and thereby many existing results have been extended and generalized. The thesis embodies six chapters. The sequence of chapters is arranged so that the understanding of a chapter stimulates interest in reading the next chapters.