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 and#8712; M be denoted by TpM and TpMand#8869;, respectively. If TpM is invariant under the action of J¯ for each p and#8712; M, that is, if JT¯ pM = TpM, for each p and#8712; 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 TpMand#8869;, for each p and#8712; M, vi that is, if JT¯ pM and#8834; TpMand#8869;, for each p and#8712; M, then M is called an anti-invariant (or totally real) submanifold of M¯ .