Spectral Geometry of Riemannian Submanifolds
Abstract
Description
In this thesis we study the geometry of the fixed point set $Σ$ of a smooth mapping $Φ: M\to M$ on a smooth compact Riemannian manifold $M$ without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace Φ\exp(tΔ)$ of the Laplace operator $Δ$ on $M$. We assume that the fixed point set $Σ$ 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 $Φ$ in a flat two-dimensional manifold; 2) zero-dimensional component of the fixed point set of $Φ$ in a curved manifold.
103 pages, 0 figures, M.S. Thesis, uses nmtthes2000.sty
103 pages, 0 figures, M.S. Thesis, uses nmtthes2000.sty