黎曼流形上的期望签名及其几何意义
Expected Signature on a Riemannian Manifold and Its Geometric Implications
AI总结：
本文在紧致黎曼流形上利用布朗桥与布朗环的期望签名渐近重构距离函数与曲率，并推导签名动力学的内在偏微分方程。
摘要：
在紧致黎曼流形$M$上，我们证明了黎曼距离函数$d(x,y)$可以从从$x$到$y$的布朗桥的期望签名的适当渐近行为中显式重构。此外，通过考察基于$x\in M$的布朗环的第四层期望签名的渐近展开，可以显式重构$M$在$x$处的内在（里奇曲率）和外在（第二基本形式）曲率性质。作为独立关注点，我们还从Eells-Elworthy-Malliavin水平提升的角度推导了$M$上期望布朗签名动力学的内在偏微分方程。
英文摘要：
On a compact Riemannian manifold $M,$ we show that the Riemannian distance function $d(x,y)$ can be explicitly reconstructed from suitable asymptotics of the expected signature of Brownian bridge from $x$ to $y$. In addition, by looking into the asymptotic expansion of the fourth level expected signature of the Brownian loop based at $x\in M$, one can explicitly reconstruct both intrinsic (Ricci curvature) and extrinsic (second fundamental form) curvature properties of $M$ at $x$. As independent interest, we also derive the intrinsic PDE for the expected Brownian signature dynamics on $M$ from the perspective of the Eells-Elworthy-Malliavin horizontal lifting.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
