Sectional curvature and matrix displacement convexity
Gautam Aishwarya, Liran Rotem, Yair Shenfeld · arXiv (Cornell University) · 2025
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.