  We explore the relation among volume, curvature and properness of a
$m$-dimensional isometric immersion in a Riemannian manifold. We show that,
when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq
p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded
geometry, properness is equivalent to the finiteness of the volume of extrinsic
balls. We also relate the total absolute curvature of a surface isometrically
immersed in a Riemannian manifold with its properness. Finally, we relate the
curvature and the topology of a complete and non-compact $2$-Riemannian
manifold $M$ with non-positive Gaussian curvature and finite topology, using
the study of the focal points of the transverse Jacobi fields to a geodesic ray
in $M$ . In particular, we have explored the relation between the minimal focal
distance of a geodesic ray and the total curvature of an end containing that
geodesic ray.
