  We study, from the extrinsic point of view, the structure at infinity of open
submanifolds isometrically immersed in the real space forms of constant
sectional curvature $\kappa \leq 0$. We shall use the decay of the second
fundamental form of the the so-called tamed immersions to obtain a description
at infinity of the submanifold in the line of the structural results in the
papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P.
Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and
W. Tuschmann. We shall obtain too an estimation from below of the number of its
ends in terms of the volume growth of a special class of extrinsic domains, the
extrinsic balls.
