  We study smooth volume-preserving perturbations of the time-1 map of the
geodesic flow $\psi_{t}$ of a closed Riemannian manifold of dimension at least
three with constant negative curvature. We show that such a perturbation has
equal extremal Lyapunov exponents with respect to volume within both the stable
and unstable bundles if and only if it embeds as the time-1 map of a smooth
volume-preserving flow that is smoothly orbit equivalent to $\psi_{t}$. Our
techniques apply more generally to give an essentially complete classification
of smooth, volume-preserving, dynamically coherent partially hyperbolic
diffeomorphisms which satisfy a uniform quasiconformality condition on their
stable and unstable bundles and have either compact center foliation with
trivial holonomy or are obtained as perturbations of the time-1 map of an
Anosov flow.
