This thesis comprises three results relating the geometry of a closed Riemannian manifold of negative sectional curvature to the dynamics of the associated geodesic flow. The first result proves that generically in the space of 1/4-pinched negatively curved metrics on a closed manifold the Lyapunov exponents of the geodesic flow all have multiplicity 1, a property known as simple Lyapunov spectrum, with respect to all measures with local product structure. The second is a rigidity theorem which provides a dynamical counterpart to a theorem of Eberlein showing that the universal cover of a closed manifold of negative sectional curvatures admits a discrete group of isometries, unless the manifold is locally symmetric. In this setting, we define a transformation group of the unit tangent bundle of the universal cover which preserves the Anosov structure of the geodesic flow, and show, under a 1/4-pinching assumption of sectional curvatures, that it must be discrete unless the underlying manifold is locally symmetric. Lastly, the third result, which is joint work with K. Butt, A. Erchenko and T. Humbert, shows that for a closed surface of variable negative Gaussian curvature the Liouville entropy of the geodesic flow is strictly increasing along the normalized Ricci flow.