发表机构
Instituto de Matemática e Estatística - USP; Universidade Estadual de Campinas; Max-Planck-Institute for Mathematics in the Sciences（圣保罗大学数学与统计研究所; 坎皮纳斯州立大学; 马克斯·普朗克科学促进研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明圆上的Wasserstein空间非平坦，其截面曲率由Fourier系数上的辛形式控制且无界，纠正了一维底流形导致平坦性的错误预期，并利用Lott的全局标架给出度量和联络的闭式表达。
AI 中文摘要
我们证明圆上的Wasserstein空间不是平坦的，这与实数直线和区间上的Wasserstein空间的平坦性形成对比。其截面曲率由Fourier系数上的一个辛形式控制，因此一个平面恰好当它对该形式是迷向的时才是平坦的，而在由第$n$个Fourier模态张成的平面上，曲率以$3n^2$为上界，且等号恰好出现在均匀测度处。因此曲率是无界的。我们在Otto的形式Riemann结构内计算该曲率，并直接从$W_2$的定义加以确认，这也表明$P_2(S^1)$中均匀测度的任何邻域都不是CAT(0)的。该结果纠正了广泛流传的预期，即只要底流形是一维的，Wasserstein空间就是平坦的。计算在由$M$的Laplacian特征函数诱导的$P^\infty(M)$切丛的全局标架中进行，该标架的存在性由Lott观察到。我们证明它是每个切空间的Schauder基，并在此标架中写出Otto度量及其Levi-Civita联络。在圆上，这以闭式形式给出$P^\infty(S^1)$的度量和联络系数，作为密度Fourier系数的有限组合。最后一节将该标架应用于动力学中的计算，将扩张圆映射的push-forward作用在其不变测度处的导数读作Fourier频率上的加权移位。
英文摘要
We show that the Wasserstein space of the circle is not flat, in contrast with the flatness of the Wasserstein spaces of the real line and of an interval. Its sectional curvature is governed by a symplectic form on the Fourier coefficients, so that a plane is flat exactly when it is isotropic for that form, and on the plane spanned by the $n$-th Fourier modes the curvature is bounded above by $3n^2$, with equality exactly at the uniform measure. It is therefore unbounded. We compute it inside the formal Riemannian structure of Otto and confirm it directly from the definition of $W_2$, which also shows that no neighbourhood of the uniform measure in $P_2(S^1)$ is CAT(0). The result corrects the widely stated expectation that a Wasserstein space is flat whenever its base manifold is one-dimensional. The computation is carried out in the global frame of the tangent bundle of $P^\infty(M)$ induced by the eigenfunctions of the Laplacian of $M$, whose existence was observed by Lott. We prove that it is a Schauder basis of every tangent space and write Otto's metric and its Levi-Civita connection in it. On the circle this gives the metric and connection coefficients of $P^\infty(S^1)$ in closed form, as finite combinations of the Fourier coefficients of the density. A final section puts the frame to work on a computation from dynamics, reading the derivative of the push-forward action of an expanding circle map at its invariant measure as a weighted shift on the Fourier frequencies.
CommentsThis preprint encompasses the previous preprints: arXiv:2406.05268 and arXiv:2504.11559
