Geodesic fields for Pontryagin type C⁰-Finsler manifolds

Abstract

Let M be a differentiable manifold, T x M be its tangent space at x ε M and TM={(x,y);x ε M; y ε T x M} be its tangent bundle. A C 0 -Finsler structure is a continuous function F:TM → [0, ∞ ) such that F(x, · ):T x M → [0, ∞ ) is an asymmetric norm. In this work we introduce the Pontryagin type C 0 -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 C 0 -Finsler structure where E is a locally Lipschitz vector field.