A Study on Contact Metric Manifolds

Abstract

1. Introduction Let M n be a topological space. If each point of M n has a neighbourhood homeomorphic to an open set in R n , then M n is called an n-dimensional topological manifold. Let M n be n-dimensional topological manifold. If U, an open set of M n containing x and#8712; M n is homeomorphic to an open set E of R n by homeomorphism and#966; :U and#8594; E such that and#966; x)( = xi = (x1, x 2,......, xn ), then the pair U and#966;),( is called a co-ordinate chart. newlineU is called a co-ordinate neighbourhood, and#966; is called co-ordinate map and newlinexi = (x1, x 2,......, xn ) are called local co-ordinates on M n at x. The adjective newlinelocal is to indicate that the co-ordinates are defined only on the part U of newlineM n . newline