Comparisons of Laplace spectra, length spectra and geodesic flows of some Riemannian manifolds

Abstract

The goal of this article is to discuss relationships between the Laplace spectrum, the length spectrum and the geodesic flow of compact Riemannian manifolds. We will say two compact Riemannian manifolds M1 and M2 are Laplace isospectral if the associated Laplace-Beltrami operators, acting on smooth functions, have the same eigenvalue spectrum. We will say they have C k -conjugate geodesic flows if there exists a C k -diffeomorphism F : S(M1) → S(M2) between the unit tangent bundles which interwines the geodesic flows of M1 and M2. Since the Laplacian may be viewed as the quantum analogue of the classical dynamics, i.e., the geodesic flow, one might expect that Laplace isospectral manifolds would have conjugate geodesic flows. However, this is not the case. For example, flat tori with C 0 -conjugate geodesic flows