  We revisit certain path-lifting and path-continuation properties of abstract
maps as described in the work of F. Browder and R. Rheindboldt in 1950-1960s,
and apply their elegant theory to exponential maps. We obtain thereby a number
of novel results of existence and multiplicity of geodesics joining any two
points of a connected affine manifold, as well as ${\it causal}$ geodesics
connecting any two causally related points on a Lorentzian manifold. These
results include a generalization of the well-known Hadamard-Cartan theorem of
Riemannian geometry to the affine manifold context, as well as a new version of
the so-called Lorentzian Hadamard-Cartan theorem using weaker assumptions than
global hyperbolicity and timelike 1-connectedness required in the extant
version. We also include a general discription of ${\it pseudoconvexity}$ and
${\it disprisonment}$ of broad classes of geodesics in terms of suitable
restrictions of the exponential map. The latter description sheds further light
on the the relation between pseudoconvexity and disprisonment of a given such
class on the one hand, and geodesic connectedness by members of that class on
the other.
