Geodesic Conjugacy in two-step nilmanifolds

Abstract

Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is $C^2$ rigid. For special classes of 2-step nilmanifolds, we show that the geodesic flow is $C^0$ or $C^2$ rigid. In particular, there exist continuous families of 2-step nilmanifolds whose Laplacians are isospectral but whose geodesic flows are not $C^0$ conjugate.