Integrability of geodesic flows and isospectrality of Riemannian manifolds

TL;DR

This paper constructs two isospectral 8D nilmanifolds, one with integrable geodesic flow and one without, revealing insights into the relationship between spectral properties and integrability.

Contribution

It provides explicit examples of isospectral manifolds with differing geodesic flow integrability, and analyzes their geometric and spectral structures.

Findings

One manifold has completely integrable geodesic flow.

The other manifold's geodesic flow is not integrable.

Both manifolds satisfy the Clean Intersection Hypothesis.

Abstract

We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds $M$ and $M'$ which are isospectral for the Laplace operator on functions and such that $M$ has completely integrable geodesic flow in the sense of Liouville, while $M'$ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by to maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in $M$, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for $M$, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both $M$ and $M'$ satisfy the so-called Clean Intersection Hypothesis.