O Isospectral Deformations on Two-Step Nilmanifolds

Two Riemannian manifolds are said to be isospectral if the associated Laplace operators have the same spectrum. C. Gordon and E. Wilson constructed isospectral deformations of left-invariant metrics on nilmanifolds. Chapter One of the thesis proves that for two-step nilmanifolds, the only isospectral deformations are the ones contructed by Gordon and Wilson. In Chapter Two, I reprove the Gordon-Wilson isospectral deformation theorem on two-step nilmanifolds from a point of view of analysis on line bundles. Given a line bundle over a nilmanifold together with a Riemannian metric on the base nilmanifold and a connection on the bundle, one obtains a Laplace operator acting on the sections of the line bundle. In Chapter Three, I investigate how the spectrum of this Laplace operator changes as the metric and connection are deformed. Several examples are studied.