Isospectral Problems On Riemann Manifolds

Eigenvalue problems has always been the center of differential geometry and inverse problems is a difficult part of it,where the difficulty is manifested by a great amount of counter-examples.Due to the occurence of the counter-examples,this article has mainly two aims in studying,one is to study the spectral invariants between isospectral manifolds and how do them determine topological and geometrical properties on manifolds,the other is to study the methods in constructing isospectral manifolds.The studying method used in this article is mainly heat kernel method,but accompanied with different areas of knowledge,including group action on manifolds and the theory of hyperbolic surface.This article is particularly focusing on explaining the sources of the theorems.Three different trace formulas are proved in this article,they are length trace formula,Sunada trace formula and Gordon trace formula,repectively.