论文标题
Pontryagin类型$ C^0 $ -FINSLER歧管
Geodesic fields for Pontryagin type $C^0$-Finsler manifolds
论文作者
论文摘要
令$ m $为一个可区分的流形，$ t_xm $是其切线空间，$ x \ in m $，$ tm = \ {（x，y）; x \ in m; y \ in t_xm \} $是其切线束。 A $ C^0 $ -FINSLER结构是连续函数$ f：tm \ rightarrow \ mathbb [0，\ infty）$，以至于$ f（x，\ cdot）：t_xm \ rightarrow [0，\ infty）$是不对称的规范。在这项工作中，我们介绍了Pontryagin型$ C^0 $ -FINSLER结构，这些结构满足了Pontryagin最大程度的最低原理要求最小化路径问题的最低要求。我们在缝隙cotangent Bundle $ t^\ ast m \ backslash 0 $ of $（m，f）$上定义了扩展的大地测量字段$ \ MATHCAL E $，这是Finsler几何形状的大地测量喷雾剂的概括。我们研究了$ \ Mathcal E $是本地Lipschitz向量领域的情况。我们展示了一些示例，这些示例比$ tm $上的类似结构更自然地以$ \ Mathcal e $表示。最后，我们证明，独立的Finsler结构的最大值是Pontryagin型$ C^0 $ -FINSLER结构，其中$ \ Mathcal E $是本地Lipschitz Vector Field。
Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $x\in M$ and $TM=\{(x,y);x\in M;y \in T_xM\}$ be its tangent bundle. A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow \mathbb [0,\infty)$ such that $F(x,\cdot): T_xM \rightarrow [0,\infty)$ 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 $\mathcal E$ on the slit cotangent bundle $T^\ast M\backslash 0$ of $(M,F)$, which is a generalization of the geodesic spray of Finsler geometry. We study the case where $\mathcal E$ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by $\mathcal E$ 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 $\mathcal E$ is a locally Lipschitz vector field.
