  We consider here a recently proposed geometrical criterion for local
instability based on the geodesic deviation equation. Although such a criterion
can be useful in some cases, we show here that, in general, it is neither
necessary nor sufficient for the occurrence of chaos. To this purpose, we
introduce a class of chaotic two-dimensional systems with Gaussian curvature
everywhere positive and, hence, locally stable. We show explicitly that chaotic
behavior arises from some trajectories that reach certain non convex parts of
the boundary of the effective Riemannian manifold. Our result questions, once
more, the viability of local, curvature-based criteria to predict chaotic
behavior.
