The Spectral Geometry Of A Riemannian Manifold

The asymptotic expansion coefficients of the heat kernel depict geometric properties of the manifold,it is one of the main tools for processing index theorem.The heat kernel has also been in the case of manifolds with boundary or in the case of elliptic operators acting on sections of a vector bundle over the manifold.The heat kernel is not only used to compute heat flows but is also used in many areas of geometric and topological analysis.The asymptotic expansion has been used to study the spectrum of the Laplacian,the determinant of the Laplacian conformal classes of metrics,analytic torsion,modular forms,index theory,stochastic analysis,gauge theory and so on.This paper first introduces the basic concepts and basic results of heat equation,the asymptotic expansion of the heat kernel was described in detail.Main work is to calculate directly the heat kernel asymptotic expansion coefficient of the first three by using the measure the Taylor expansion,and then study the previous use of the Weyl invariant theory to calculate the asymptotic expansion coefficient of the heat kernel,finally give relevant characteristic values of the application in the S~1.