Hyperbolic Rank-Rigidity and Frame Flow.

Abstract

This thesis presents a result bridging the general areas of geometry and dynamics. This rank-rigidity result shows that any compact manifold (under suitable curvature conditions) which has higher hyperbolic rank is hyperbolic. For this result, the crucial tool is the dynamics of the frame flow. This result provides a new, simpler, proof of Hamenstadt's hyperbolic rank rigidity theorem, under certain curvature restrictions. It also provides new results in other curvature settings. It is analogous to other rank-rigidity results in non-positive and positive curvature.