Let Ω be a domain in a smooth complete Finsler manifold, and let G be the largest open subset of Ω such that for every x in G there is a unique closest point from ∂Ω to x (measured in the Finsler metric). We prove that the distance function from ∂Ω is in C k,α loc (G ∪ ∂Ω), k ≥ 2 and 0 < α ≤ 1, if ∂Ω is in C k,α .

We study non-reversible Finsler metrics with constant flag curvature 1 on S and show that the geodesic flow of every such metric is conjugate to that of one of Katok’s examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with c...

This note considers cooperative irreducible systems of ordinary differential equations admitting a C1 first integral with positive gradient. We prove that all forward or backward nonwandering points are equilibria. We obtain also some results on the global phase portrait of such systems. The main tool of proof is a canonically defined Finsler structure with respect to which the derivative skew-...

In Finsler geometry, the notion of parallel for Finsler tensor fields is defined, already. In this paper, however, for vector fields on a base manifold, the author studies the notion of parallel. In the last section, the obtained results are shown as compare corresponding properties to them of Riemannian geometry. Introduction The author gave the definition (1.1) of the parallel displacement al...

The present paper is a continuation of the foregoing paper [16]. The main aim is to establish an intrinsic investigation of the conformal change of the most important special Finsler spaces. Necessary and sufficient conditions for such special Finsler manifolds to be invariant under a conformal change are obtained. Moreover, the conformal change of Chern and Hashiguchi connections, as well as t...

In this paper we define vertical and horizontal Laplace type operators for functions on the total space of a complex Finsler bundle (E, L). We also define the ′′ v-Laplacian for (p, q, r, s)-forms with compact support on E and we get the local expression of this Laplacian explicitly in terms of vertical covariant derivatives with respect to the Chern-Finsler linear connection of (E, L).

Asking that the metric of a complex Finsler space should be strong convex, Abate and Patrizio ([1]) associate to the real tangent bundle a real Finsler metric for which they analyze the relation between Cartan (real) connection of the obtained space and the real image of Chern-Finsler complex connection. Following the same ideas, in the present paper we shall deal with the more general case of ...

We find the abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric defined by a seminorm two-dimensional subspace of algebra generating algebra. In terms structure constants and Minkowski support function unit ball which defines quasimetric, we establish criterion for strict abnormality these extremals.