Homotopy Groups, Focal Points and Totally Geodesic Immersions
  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 $N$. 
S\'ergio Mendon\c{c}a and Heudson Mirandola
math.DG