GEODESIC CONJUGACY IN TWO-STEP NILMANIFOLDS

Carolyn Gordon Yiping Mao

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.

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