Convexity of domains of Riemannian manifolds

In this paper the problem of the geodesic connectedness and convexity of incomplete Riemannian manifolds is analyzed. To this aim, a detailed study of the notion of convexity for the associated Cauchy boundary is carried out. In particular, under widely discussed hypotheses, we prove the convexity of open domains (whose boundaries may be nondifferentiable) of a complete Riemannian manifold. Variational methods are mainly used. Examples and applications are provided, including a result for dynamical systems on the existence of trajectories with fixed energy.