Homotopy Groups, Focal Points and Totally Geodesic Immersions

TL;DR

This paper explores the topological relationships between a complete Riemannian manifold, a totally geodesic hypersurface, and a submanifold without focal points, without requiring curvature conditions.

Contribution

It establishes new connectedness results linking the topologies of the manifold, hypersurface, and submanifold based on codimension, without curvature restrictions.

Findings

Connectedness results relating $M$, $ ext{Si}$, and $N$

Topological implications depending on codimension

No curvature conditions needed for results

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 $N$.