Homotopy Groups, Focal Points and Totally Geodesic Immersions

Abstract

In this paper we consider on a complete Riemannian manifold $M$ an immersed totally geodesic hypersurface \Si existing together with an immersed submanifold $N$ without focal points. No curvature condition is needed. We obtained several connectedness results relating the topologies of $M$ and \Si which depend on the codimension of $N$.Comment: 24 pages. To appear in Indiana University Mathematical Journa