  This paper presents a rank rigidity result for negatively curved spaces. Let
$M$ be a compact manifold with negative 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 $a$ is extremal, i.e. $-a^2$ is the curvature
minimum or maximum for the manifold, this result is analogous to rank rigidity
results in various other curvature settings where higher rank implies that the
space is locally symmetric. In particular, this result is the first positive
result for lower rank (i.e. when $-a^2$ is minimal), and in the upper rank case
gives a shorter proof of the hyperbolic rank rigidity theorem of
Hamenst\"{a}dt, subject to the pinching condition in even dimension. We also
present a rigidity result using only an assumption on maximal Lyapunov
exponents in direct analogy with work done by Connell. Our proof of the main
theorem uses the ergodic theory of the frame flow developed by Brin and others
- in particular the transitivity group associated to this flow.
