Lifting Of Isometric Mapping On Infinite Dimensional Manifold

Using probabistic methods study analysis and geometry on infinite dimensional manifold is one of hot areas of stochastic analysis in recent twenty years.The purpose of this paper is to explore lifting of isometric mapping problem that from Riemannian manifold to the path space and the configration space over a Riemannian manifold.In chapter two, we present difinition of lifting of isometric mapping that from compact a Riemannian manifold to its path space, we prove invariance for Markovian connection,O-U operator and curvature on the path space over a compact Riemannian manifold in lifting of isometric mappong.In chapter three,we obtain invariance for Laplace operator on the configuration space over a Riemannian manifold in lifting of isometric mapping,we also generalization Heck operator of hyperbolic plane to the configuration space over a hyperbolic plane.