Geodesic fields for Pontryagin type C⁰-Finsler manifolds

Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil.

^** Corresponding author: rfukuoka@uem.br

Received: 20 July 2020
Accepted: 7 February 2022

Abstract

Let M be a differentiable manifold, T_xM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ T_xM} be its tangent bundle. A C⁰-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : T_xM → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C⁰-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T^*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C⁰-Finsler structure where ℰ is a locally Lipschitz vector field.