Abstract:On a smooth Riemannian manifold, Aishwarya, Rotem and Shenfeld characterised nonnegative sectional curvature as the matrix displacement convexity of an entropy tensor, the Lagrangian, matrix-valued refinement of Shenfeld's entropy matrix. In order to extend the entropy tensor to a finite-dimensional Alexandrov space of curvature bounded below, we construct a parallel trivialisation satisfying both the cocycle property and the second variation formula. The construction is strongly inspired by Petrunin's synthetic parallel transport. The entropy tensor defined is taken in block-diagonal form; on smooth manifolds the resulting convexity property still characterises nonnegative sectional curvature exactly. We show that the smooth equivalence persists synthetically: an Alexandrov space has nonnegative curvature if and only if its entropy tensor is matrix displacement convex.
From: Jordan Serres [view email] [via CCSD proxy]
[v1]
Wed, 24 Jun 2026 08:54:55 UTC (46 KB)
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