  Let $S$ be a compact surface of genus $\geq 2$ equipped with a metric that is
flat everywhere except at finitely many cone points with angles greater than
$2\pi$. We examine the geodesic flow on $S$ and prove local product structure
for a wide class of equilibrium states. Using this, we establish the Bernoulli
property for these systems. We also establish local product structure for a
similar class of equilibrium states for geodesic flows on rank 1, nonpositively
curved manifolds.
