强凸 $C^0$-芬斯勒结构的磨光子光滑化

中文摘要 AI 辅助

设 $M$ 为光滑流形，$TM$ 为其切丛。$M$ 的 $C^0$-芬斯勒结构是连续函数 $F: TM \to [0, \infty)$，其在每个切空间上的限制为非对称范数。$F$ 的磨光子光滑化是一族芬斯勒结构 $F_\varepsilon: TM \to [0, \infty)$，以 $\varepsilon > 0$ 为参数，通过标准磨光子构造，满足当 $\varepsilon \to 0$ 时，$F_\varepsilon$ 在紧子集上一致收敛到 $F$。本研究构造了 $F$ 的两种磨光子光滑化：第一种假设 $F: TM \to [0, \infty)$ 是强凸的，即其在每个切空间上的限制为强凸函数；第二种假设 $M=G$ 是李群，且 $F:TG \to [0, \infty)$ 是强凸且左不变的，此时 $F_\varepsilon$ 也为左不变的。在两种情形下，均证明若 $F$ 是芬斯勒结构，则 $(M, F_\varepsilon)$ 的基本张量、Chern联络、Cartan联络、Hashiguchi联络、Berwald联络及旗曲率等对象，当 $\varepsilon \to 0$ 时在紧子集上一致收敛到 $(M, F)$ 的对应对象。

英文摘要

Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F: TM \to [0, \infty)$ such that restricted to each tangent space is an asymmetric norm. A mollifier smoothing of $F$ is a family of Finsler structures $F_\varepsilon: TM \to [0, \infty)$, parameterized by $\varepsilon > 0$ and constructed using the standard mollifier, such that $F_\varepsilon \to F$ uniformly on compact subsets as $\varepsilon \to 0$. In this work, we construct two mollifier smoothings of $F$. In the first, we assume that $F: TM \to [0, \infty)$ is strongly convex, that is, that $F$ restricted to each tangent space is strongly convex. In the second, we assume that $M=G$ is a Lie group and that $F:TG \to [0, \infty)$ is strongly convex and left-invariant, and in this case, $F_\varepsilon$ is also left-invariant. In both cases, we prove that if $F$ is a Finsler structure, objects such as the fundamental tensor, the Chern, Cartan, Hashiguchi and Berwald connections, and the flag curvature of $(M, F_\varepsilon)$ converge uniformly on compact subsets to the corresponding objects of $(M, F)$ as $\varepsilon \to 0$.

发表机构

• State University of Maringá（马里兰加州立大学）
• Federal University of Mato Grosso（马托格罗索联邦大学）

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