  Let $M$ be a smooth compact surface of nonpositive curvature, with genus
$\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent
bundle of $M$ with respect to the Liouville measure under the condition that
the set of points with negative curvature on $M$ has finitely many connected
components. Under the same condition, we prove that a non closed "flat"
geodesic doesn't exist, and moreover, there are at most finitely many flat
strips, and at most finitely many isolated closed "flat" geodesics.
