  Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an
ambient Riemannian manifold $N^n$ with a pole and with radial sectional
curvatures bounded from above by the corresponding radial sectional curvatures
of a radially symmetric space $M^n_w$, we determine a set of conditions on the
extrinsic curvatures of $P$ that guarantees that the immersion is proper and
that $P$ has finite topology, in the line of the paper "On Submanifolds With
Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009),
authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient manifold is
a radially symmetric space, it is shown an inequality between the (extrinsic)
volume growth of a complete and minimal submanifold and its number of ends
which generalizes the classical inequality stated in Anderson's paper "The
compactification of a minimal submanifold by the Gauss Map", (Preprint IEHS,
1984), for complete and minimal submanifolds in $\erre^n$. We obtain as a
corollary the corresponding inequality between the (extrinsic) volume growth
and the number of ends of a complete and minimal submanifold in the Hyperbolic
space together with Bernstein type results for such submanifolds in Euclidean
and Hyperbolic spaces, in the vein of the work due to A. Kasue and K. Sugahara
"Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space
forms", (Osaka J. Math. 24,1987).
