Abstract
Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ TxM} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : TxM → [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.
Highlights
Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and Tx∗M be its cotangent space at x
In this work we study Pontryagin type C0-Finsler structures on differentiable manifolds
We apply the Pontryagin’s maximum principle (PMP) for the time-optimal problem, and as the result, we obtain the extended geodesic field E on T ∗M \0, which is a generalization of the geodesic spray of Finsler geometry
Summary
Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and Tx∗M be its cotangent space at x. It is clear that there is not a standard way to study C0-sub-Finsler structures with “horizontal smoothness” outside the sub-Riemannian geometry and the left invariant C0-sub-Finsler structures on Lie groups This is a natural situation because the geometrical structures that can be studied with PMP can vary and even different problems on the same manifold needs its own formulation. A relevant question is what happens if we drop the horizontal smoothness of a C0-Finsler structure In this case the theory is much less developed due to the lack of a model theory (like Riemannian geometry) and due to the lack of tools in order to study variational problems in detail. The authors would like to thank the referee for his/her remarks and suggestions, which helped us to improve the original version of this work
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