Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below

TL;DR

This paper proves that in certain metric measure spaces with Ricci curvature bounds, almost every point has a Euclidean space as a weak tangent, revealing local Euclidean structure in these spaces.

Contribution

It establishes the existence of Euclidean weak tangents at almost every point in infinitesimally Hilbertian $CD^*(K,N)$-spaces using iterated tangents and splitting theorem techniques.

Findings

Almost every point has a Euclidean weak tangent.

Weak tangents are obtained via dilations and convergence in Gromov-Hausdorff topology.

The approach relies on properties of $CD^*(K,N)$-spaces and splitting theorems.

Abstract

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian $CD^*(0,N)$-spaces.