Complete Conical Type End Immersed Manifolds in R N

We introduce and describe the topology of a family of complete immersed manifolds in R N ; having a nice behaviour at innnity, which we call conical type end manifolds. Our main result states that a complete, non compact immersed manifold in R N ; whose lim sup of the norm of the second fundamental form times the intrinsic distance of the manifold to a xed point is strictly less than 1, as the distance goes to innnity, is a conical type end manifold. In particular, it follows that the manifold has nite topology and is properly immersed in R N :