Sectional curvature and matrix displacement convexity

Gautam Aishwarya; Liran Rotem; Yair Shenfeld

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arXiv · 2509.23399

Sectional curvature and matrix displacement convexity

Gautam Aishwarya·Liran Rotem·Yair Shenfeld

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Abstract

We show that the sectional curvature of a Riemannian manifold is nonnegative if, and only if, the entropy tensor is matrix displacement convex. As an application, under the assumption of nonnegative sectional curvature, we obtain intrinsic dimensional strengthenings of the HWI inequality, the evolution variational inequality, and the corresponding Wasserstein contraction along heat flows. Moreover, we show that the intrinsic dimensional Wasserstein contraction inequality in fact characterizes nonnegative sectional curvature.

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Gautam Aishwarya, Liran Rotem, Yair Shenfeld. 2025-09-27. Sectional curvature and matrix displacement convexity. https://arxiv.org/abs/2509.23399

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