  The main goal of this paper is to present results of existence and
non-existence of convex functions on Riemannian manifolds and, in the case of
the existence, we associate such functions to the geometry of the manifold.
Precisely, we prove that the conservativity of the geodesic flow on a Rieman-
nain manifold with infinite volume is an obstruction to the existence of convex
functions. Next, we present a geometric condition that ensures the existence of
(strictly) convex functions on a particular class of complete non-compact man-
ifolds, and, we use this fact to construct a manifold whose sectional curvature
assumes any real value greater than a negative constant and admits a strictly
convex function. In the last result we relate the geometry of a Riemannian
manifold of positive sectional curvature with the set of minimum points of a
convex function defined on the manifold.
