Ergodic Measures of Geodesic Flows on Compact Lie Groups

Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature （possibly strictly positive on some sections）, whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.