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

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 an immersed totally geodesic hypersurface existing together with an immersed submanifold without focal points. No curvature condition is needed. We obtained several connectedness results relating the topologies of and which depend on the codimension of .
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