Finsler manifolds

Definitions and mathematical statement

For $N\ge1$ and $1\le p\le\infty$, let $d_p(x,y)=\|x-y\|_p$ on ${\mathbb{R}}^N$, with the usual maximum norm when $p=\infty$. Coarse Ricci curvature requires not only a metric but a family of probability measures $(m_x)$, for example the transition laws of a random walk. For measures of finite first moment, set \[ W_1(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int d(u,v)\,\mathrm d\pi(u,v), \qquad \kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)}\quad(x\ne y), \] where $\Pi(\mu,\nu)$ is the set of couplings. Thus the flat observation $\kappa(x,y)=0$ means $W_1(m_x,m_y)=d_p(x,y)$. A customary flat-space choice is a fixed finite-first-moment law translated with the base point, $m_x=(z\mapsto x+z)_*\nu$; the extracted source does not state its choice of laws or scale.

Ollivier's Problem B asks whether this flat $L^p$-norm observation yields useful results for Finsler manifolds, whose tangent spaces carry varying Finsler norms or gauges and whose distance is obtained by integrating them along curves. This is a research programme, not a specified universal theorem: the record does not identify a desired inequality, invariant, classification, or other conclusion. Nor does it prescribe a reference measure, transition laws and scale on a Finsler manifold, regularity/convexity assumptions, or treatment of a possibly nonreversible distance. Consequently a unique precise yes/no assertion beyond the stated programme cannot be extracted without adding mathematical goals.

The supplied research summary reports partial progress through displacement-convexity approaches of Ohta and Ohta--Sturm, while distinguishing that work from a comprehensive theory based on this specific coarse-Ricci observation. It supplies no theorem asserting that the requested programme has been completed.

Short English statement

Can the vanishing of coarse Ricci curvature in normed ${\mathbb{R}}^N$ lead to useful theorems for Finsler manifolds? The source leaves the Finsler random walk and the desired mathematical conclusions unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $4000002$, exact source code catalogue entry, "Finsler manifolds."
• Yann Ollivier, Discrete Ricci curvature: Open problems (2008), Problem B; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf. Recorded access 2026-07-29; pdf-text extraction.
• Supplied research summary mentions Ohta and Ohta--Sturm on displacement convexity in Finsler manifolds. It provides no complete publication references or precise results.