COMPLETE CONICAL TYPE END IMMERSED MANIFOLDS IN R-N

We introduce and describe the topology of a family of complete immerse d manifolds in R-N, having a nice behaviour at infinity, which we call conical type end manifolds. Our main result states that a complete, n on compact immersed manifold in R-N, whose lim sup of the norm of the second fundamental form times the intrinsic distance of the manifold t o a fixed point is strictly less than 1, as the distance goes to infin ity, is a conical type end manifold. In particular, it follows that th e manifold has finite topology and is properly immersed in R-N.