Rigidity of Equality of Lyapunov Exponents for Geodesic Flows

We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We also show under a curvature pinching condition that equality of all Lyapunov exponents with respect to volume on the unstable bundle also implies that the manifold has constant negative curvature. We then study the degree to which one can emulate these rigidity theorems for the hyperbolic spaces of nonconstant negative curvature when the Lyapunov exponents with respect to volume match those of the appropriate symmetric space and obtain rigidity results under additional technical assumptions. The proofs use new results from hyperbolic dynamics including the nonlinear invariance principle of Avila and Viana and the approximation of Lyapunov exponents of invariant measures by Lyapunov exponents associated to periodic orbits which was developed by Kalinin in his proof of the Livsic theorem for matrix cocycles. We also employ rigidity results of Capogna and Pansu on quasiconformal mappings of certain nilpotent Lie groups.