Geometry of CR-Submanifolds

Abstract

To study the geometry of a manifold, it is more convenient to first embed into a known manifold and then study its geometry. This approach gave impetus to study of submanifold which later developed into an independent and fascinating topic of study. The submanifold of an almost Hermitian manifold presents an interesting geometric study as its almost complex structure transform a vector into a vector perpendicular to it. Thus in turn, gives rise to two types of submanifold, namely invariant and anti-invariant submanifolds [1]. These are also known as holomorphic and totally real submanifold. These submanifolds were extensively studied by many differential geometrs. A Bejancu [1] in 1978 introduced the notion of CR-submanifolds of a Kaehler manifold which generalize the holomorphic and totally real submanifolds, after that B.Y. Chen [3] studied CR- submanifold of Kaehler manifold. Chapter I is introductory and serves. The purpose of developing the basic concepts keeping in view the pre-requisities of the subsequent chapters. Chapter II is the first technical chapter. Which deals with the CR- submanifold of a Kaehler manifold. It contains the results obtained by B.Y. Chen in his paper “CR-submanifold of a Kaehler manifold I” [3]. In this paper B.Y. Chen studied integrability conditions of the canonical distributions as well as the conditions for, their leaves to be totally geodesic. Chapter III deals CR-submanifold of nearly Kaehler manifold which is the generalization of CR-submanifold, in more generalized setting namely nearly Kaehler, the results in this chapter are due to K.A. Khan et.al. [5].