Geodesic Conjugacy in Two - Step Nilmanifolds

Two Riemannian manifolds are said to have C-conjugate geodesic flows if there exist an C 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 rigid. For special classes of 2-step nilmanifolds, we show that the geodesic flow is C or C rigid. In particular, there exist continuous families of 2-step nilmanifolds whose Laplacians are isospectral but whose geodesic flows are not C conjugate. Introduction Two Riemannian manifolds (M, g) and (N, h) are said to have C-conjugate geodesic flows if there is a C diffeomorphism F : S(M, g) → S(N, h) which intertwines the geodesic flows on S(M, g) and S(N, h). Here S(M, g) and S(N, h) are the unit tangent bundles of (M, g) and (N, h) respectively. We call F a C-geodesic conjugacy from M to N . A compact Riemannian manifold is said to be C-geodesically rigid within a given class of manifolds if any Riemannian manifold N in that class whose geodesic flow is C-conjugate to that of M is isometric to M . A. Weinstein ([W]) exhibited a zoll surface of non-constant curvature whose geodesic flow is conjugate to that of the round sphere. On the other hand, two flat tori with C-conjugate geodesic flows must be isometric. Therefore, a natural question arises: Question. Which compact Riemannian manifolds are C-geodesically rigid in a given class of manifolds? This question is central to the study of negatively curved manifolds. Many important open problems in the field will follow if all negatively curved manifolds can be shown to be geodesically rigid (see [BFL], [EHS], [Ka], [Kk]). For negatively curved surfaces , C. Croke ([C]) and J. Otal ([O]) have independently answered this question affirmatively. Recently, C. Croke and B. Kleiner ([CK]) proved that compact Riemannian manifolds with a parallel vector field are geodesically rigid. For negatively curved manifolds of higher dimension, the question is still open. Recently, G. Besson, G. Courtois and G. Gallot proved that if a manifold has a geodesic flow which is C-conjugate to the geodesic flow of a rank one locally symmetric space M , then it is isometric to M . 1991 Mathematics Subject Classification. 58G25, 53C22.