On the Cut Locus of Submanifolds of a Finsler Manifold

TL;DR

This paper studies the structure and properties of the cut locus of submanifolds within Finsler manifolds, extending classical results to a more general Finsler setting and considering geodesic loops.

Contribution

It generalizes Klingenberg's lemma for closed geodesics to the context of N-geodesic loops in reversible Finsler manifolds, expanding understanding of cut loci.

Findings

Extended the deformation and characterization of the cut locus in Finsler manifolds.

Generalized Klingenberg's lemma for N-geodesic loops.

Provided new insights into the structure of geodesics in Finsler geometry.

Abstract

In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and the second author (\emph{Algebraic and Geometric Topology}, 2023). Given a submanifold $N$, we consider an $N$-geodesic loop as an $N$-geodesic starting and ending in $N$, possibly at different points. This class of geodesics were studied by Omori (\emph{Journal of Differential Geometry}, 1968). We obtain a generalization of Klingenberg's lemma for closed geodesics (\emph{Annals of Mathematics}, 1959) for $N$-geodesic loops in the reversible Finsler setting.