On the Cut Locus of Submanifolds of a Finsler Manifold

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.

arXiv comment37 pages, 8 figures. Published in the Journal of Geometric Analysis

References in corpus (5)

• Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds
• A connection between cut locus, Thom space and Morse-Bott functions
• A comparison theorem for Finsler submanifolds and its applications
• On conjugate points and geodesic loops in a complete Riemannian manifold
• Cut Locus of Submanifolds: A Geometric and Topological Viewpoint

Cited by in corpus (2)

• Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric
• On the Focal Locus of Submanifolds of a Finsler Manifold