On the Focal Locus of Submanifold of a Finsler Manifold

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 regular in the sense of F.W. Warner (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 (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 (J. Geom. Anal., 34, 2024).