AI 中文总结
本文提出Wasserstein空间的全局微分演算框架，证明底流形的Levi-Civita联络可提升为兼容扩展Otto度量的无挠联络，重新研究J. Lott的相关工作并得到部分不同结论。
AI 中文摘要
我们在闭黎曼流形上的$L^2$-Wasserstein空间上建立了基于柱函数导数而非标准逐点方法的全局微分演算框架，在此框架中定义了若干基础几何工具。特别地，我们证明了底流形上的Levi-Civita联络可提升为与‘扩展’Otto度量兼容的唯一无挠联络，对应的黎曼张量恰好是底黎曼张量的提升，这表明在该框架下，经典梯度形式的修正项并非内蕴曲率项，而是源于向依赖测度的梯度分布的投影。这使我们能用全局纯微分方法重新研究J. Lott在《Comm. Math. Phys., 2007, 277(2):423-437》中的光滑计算，得到了部分不同的结论。
英文摘要
We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the 'extended' Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient formalism are not intrinsic curvature terms, but arise from the projection onto the measure-dependent gradient distribution. This allows us to revisit with a global and purely differential approach the smooth computations by J. Lott, Comm. Math. Phys., 277(2):423-437, 2007, reaching partially different conclusions.
Comments21 pages, 2 figures
