Overview
This research establishes a fundamental equivalence between the geometric property of nonnegative sectional curvature in a Riemannian manifold and the analytical property of matrix displacement convexity of its entropy tensor. Furthermore, it details consequences of this equivalence, specifically intrinsic dimensional strengthenings of established inequalities, and notes a characterization derived from the Wasserstein contraction inequality.
Research Context
The study operates within the framework of Riemannian geometry and optimal transport theory, focusing on properties related to sectional curvature and various inequalities concerning entropy and heat flow. Riemannian manifolds provide the geometric setting, while concepts like the entropy tensor, HWI inequality, and evolution variational inequality are central to the analytical and functional aspects investigated.
Approach
The methodology involves demonstrating a biconditional relationship between sectional curvature and matrix displacement convexity. Subsequent to this, the research applies the assumption of nonnegative sectional curvature to derive specific strengthenings of known inequalities and to establish a characterization.
Findings
- The sectional curvature of a Riemannian manifold is nonnegative if, and only if, its entropy tensor is matrix displacement convex. This establishes a direct equivalence between a geometric curvature condition and a convexity property of a tensor.
- Under the condition of nonnegative sectional curvature, the research obtains intrinsic dimensional strengthenings of the HWI inequality.
- Similarly, under the condition of nonnegative sectional curvature, intrinsic dimensional strengthenings of the evolution variational inequality are obtained.
- Intrinsic dimensional strengthenings of the corresponding Wasserstein contraction along heat flows are also derived under the assumption of nonnegative sectional curvature.
- The intrinsic dimensional Wasserstein contraction inequality specifically characterizes nonnegative sectional curvature. This indicates that the presence of this particular contraction property is not merely a consequence but a definitive indicator of nonnegative sectional curvature.
Why This Matters
The established equivalence provides a new analytical perspective on the geometric concept of sectional curvature by linking it to matrix displacement convexity. The derived strengthenings of fundamental inequalities, such as HWI and evolution variational inequalities, offer more precise bounds or insights in spaces with nonnegative sectional curvature. The characterization of nonnegative sectional curvature through Wasserstein contraction provides a powerful tool for identifying this geometric property via an optimal transport mechanism.