A large family of projectively equivalent $C^0$-Finsler manifolds

Fukuoka, Ryuichi

doi:10.2748/tmj/1601085624

Cited by 3 publications

(2 citation statements)

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“…In [20,21], the first author of this work and his collaborators used the term C 0 -Finsler structure for the second definition of Finsler structure. The term C 0 -Finsler structure given in Definition 2.3 coincides with the third definition of Finsler structure above.…”

Section: Preliminariesmentioning

confidence: 99%

“…Given x ∈ R 2 and a vector y in the tangent space of x, there exist infinitely many geodesics γ : (− , ) → R 2 with constant speed such that γ(0) = x and γ (0) = y. On the other hand, we have a large family of projectively equivalent C 0 -Finsler manifolds on R 2 such that for every pair of points, there exists a unique minimizing path connecting them: All minimizing paths are line segments parallel to the vectors (0, 1), ( √ 3/2, 1/2) or (− √ 3/2, 1/2) or else a concatenation of two of these line segments (see [21]). Therefore if y isn't parallel to one of these three vectors, then there isn't any geodesic satisfying γ(0) = x and γ (0) = y.…”

Section: Introductionmentioning

confidence: 99%

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Geodesic fields for Pontryagin type C⁰-Finsler manifolds

Rodrigues

Fukuoka

2022

ESAIM: COCV

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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.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%