Geometry of Generic 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 submanifolds , namely invariant and anti-invariant submanifolds. 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 introductory chapter of CR-submanifold of a Kaehler manifold. This chapter contains some basic results of CR-submanifold of a Kaehler manifold . All the results of this chapter are by B.Y.Chen from his paper “CR-submanifold of a kaehler manifold’’. Chapter III is the second technical chapter which deals with the generic submanifold of a kaehler manifold. Its contents are essentially the results obtained by B.Y. Chen in his paper “ Differential geometry of real submanifold of a Kaehler manifold ’’[4]. The generic submanifold are natural generalization of holomorphic , totally real and CR-submanifold. In this paper B.Y. Chen studied integrability condition of holomorphic and totally real distribution and totally geodesicness of leaves of these distributions.