Journal of Modern Dynamics (JMD)

2-Frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature

Pages: 719 - 740, Issue 4, October 2008      doi:10.3934/jmd.2008.2.719

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David Constantine - Department of Mathematics,University of Michigan, Ann Arbor,MI 48109, United States (email)

Abstract: This paper presents hyperbolic rank-rigidity results for nonpositively curved spaces of rank 1. Let $M$ be a compact, rank-1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd-dimensional or if $M$ is even-dimensional and has sectional curvature pinched as follows: $-\Lambda^2 < K < -\lambda^2$ where $\lambda/\Lambda >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank-rigidity theorem of Hamenst├Ądt, subject to the pinching condition in even dimension; in all other cases it is a new result. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. The proof uses the dynamics of the frame flow, developed by Brin for negatively curved manifolds; portions of his work are adapted here for use in the nonpositively curved, rank-1 situation.

Keywords:  Rank-rigidity, hyperbolic rank, frame flow
Mathematics Subject Classification:  Primary: 53C24, 53C35; Secondary: 37D40

Received: May 2008;      Revised: September 2008;      Available Online: October 2008.