On the Focal Locus of Submanifolds of a Finsler Manifold

TL;DR

This paper studies the focal locus of submanifolds in Finsler manifolds, proving regularity of the normal exponential map and analyzing its properties to better understand the structure of the cut locus.

Contribution

It extends Warner's ideas to Finsler geometry, showing the normal exponential map's regularity and analyzing the focal locus and cut locus structure.

Findings

Normal exponential map is regular in Finsler manifolds.

Focal time maps are smooth on an open dense subset.

Tangent cut locus characterized as closure of separating tangent cut points.

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). As a consequence, we show that the normal exponential is non-injective near any tangent focal point. Extending the ideas of Warner, we study the connected components of the regular focal locus. This allows us to identify an open and dense subset, on which the focal time maps are smooth, provided they are finite. We explicitly compute the derivative at a point of differentiability. As an application of the local form of the normal exponential map, following R.L. Bishop's work (\textit{Proc. Amer. Math. Soc.}, 65, 1977), we express the tangent cut locus as the closure of a certain set of points, called the separating tangent cut points. This strengthens the results from the present authors' previous work (\textit{J. Geom. Anal.}, 34, 2024).