Homotopy Groups, Focal Points and Totally Geodesic Immersions

In this paper we consider on a complete Riemannian manifold $M$ an immersed totally geodesic hypersurface $\text{Si}$ 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 $\text{Si}$ which depend on the codimension of $N$.