  This paper presents hyperbolic rank rigidity results for rank 1,
nonpositively curved spaces. 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\"{a}dt, 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 of the main
theorem is simplified considerably by assuming strict negative curvature; in
fact, in all dimensions but 7 and 8 it is then an immediate consequence of
ergodicity of the $(dim(M)-1)$-frame flow. In these exceptional dimensions,
recourse to the dynamics of the 2-frame flow must be made and the scheme of
proof developed there can be generalized to deal with rank 1, nonpositively
curved spaces.
