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 Prasad (Algebr Geom Topol 23(9):4185–4233, 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 (J Differ Geom 2:233–252, 1968). We obtain a generalization of Klingenberg’s lemma for closed geodesics (Klingenberg in: Ann Math 2(69):654–666, 1959). for N-geodesic loops in the reversible Finsler setting.