On the Focal Locus of Submanifold of a Finsler Manifold

Abstract

In this article, we investigate the focal locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. The main goal is to show that the associated normal exponential map is \emph{regular} in the sense of F.W. Warner (\textit{Am. J. of Math.}, 87, 1965). This leads to the proof of the fact that the normal exponential is non-injective near tangent focal points. As an application, following R.L. Bishop's work (\textit{Proc. Amer. Math. Soc.}, 65, 1977), we express the tangent cut locus as a closure of a certain set of points, called separating tangent cut points. This strengthens the results from the present authors' previous work (\textit{J. Geom. Anal.}, 34, 2024).