Spectral Geometry of Riemannian Submanifolds

Abstract

In this thesis we study the geometry of the fixed point set $\Sigma$ of a smooth mapping $\Phi: M\to M$ on a smooth compact Riemannian manifold $M$ without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace \Phi\exp(t\Delta)$ of the Laplace operator $\Delta$ on $M$. We assume that the fixed point set $\Sigma$ is a union of connected components, each of which is a smooth compact submanifold of $M$ without boundary. The deformed heat trace asymptotics is determined by contributions of each connected component, so each of them can be studied separately. We develop a generalized Laplace method for computing the coefficients of this asymptotic expansion and compute the first three coefficients explicitly in the following cases: 1) zero- and one-dimensional components of the fixed point set of $\Phi$ in a flat two-dimensional manifold; 2) zero-dimensional component of the fixed point set of $\Phi$ in a curved manifold.