Mean Curvature and the Wave Invariants of the Basic Spectrum for a Riemannian Foliation

Abstract

Given a (possibly singular) Riemannian foliation $\mathcal{F}$ with closed leaves on a compact manifold $M$ with an adapted metric, we investigate the wave trace invariants for the basic Laplacian about a non-zero period. We compare them to the wave invariants of the underlying Riemannian orbifold that exists when the leaves in the regular region are identified to points, equipped with the metric that is transverse to the leaves of the foliation. Recalling that the basic Laplacian differs from the underlying orbifold Laplacian by a term that is the mean curvature vector field associated to the foliation, we show that the first wave invariant about any non-zero period $T$ corresponding to geodesics perpendicular to the leaves that all lie entirely in the regular region of $M$ is independent of the mean curvature vector field and depends only on the underlying orbifold structure of the leaf space quotient. Similar results hold on the singular strata whenever the transverse geodesic flow remains confined to the strata defined by the leaf dimension of the foliation. Conversely, closed geodesics that pass through the exceptional leaves depend on the full laplacian, including the leaf-wise metric on the ambient space. We also discuss families of representations that yield the same leaf space and basic spectrum. We use this to give conditions under which non-trivial isotropy for orbifold quotients can be detected.