Smooth Approximation Of Lipschitz Function On Riemannnian Manifolds

In this paper,we study the complete Riemannian manifold with Ricci curvature bounded,and the smooth approximation of Lipschitz function.We have proof that while M is a complete Riemannnian manifold,f is a Lipschitz function with compact support defined on the manifold,it's Lipschitz constant denoted by Lip(f),for any ?>0 there exists a smooth Lipschitz function g on M which Lipschitz constant is Lip(g),for every p ? M we have |f(p)-g(p)|??:and Lip(g)?<Lip(f).We have two ways to achieve the aim,one is using the convolution,the other is using the heat kernel on Riemannian manifolds.At last,we have estimated the Lipschitz constant of the g.In order to do that,we should know some properties of heat kernel on Riemannian manifold.