Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian

TL;DR

This paper establishes that metric spaces with tangent spaces splitting off a line are universally infinitesimally Hilbertian, linking their local geometry to their analytic structure and applying this to RCD and Alexandrov spaces.

Contribution

It provides the first general criterion connecting tangent space splitting properties to universal infinitesimal Hilbertianity, with applications to RCD and Alexandrov spaces.

Findings

Finite dimensional RCD-spaces are universally infinitesimally Hilbertian.

Alexandrov spaces are universally infinitesimally Hilbertian.

Established an isometric embedding of tangent modules.

Abstract

We show that a metric space $X$ that, at every point, has a Gromov-Hausdorff tangent with the splitting property (i.e. every geodesic line splits off a factor $\mathbb{R}$), is universally infinitesimally Hilbertian (i.e. $W^{1,2}(X,\mu)$ is a Hilbert space for every measure $\mu$). This connects the infinitesimal geometry of $X$ to its analytic properties and is, to our knowledge, the first general criterion guaranteeing universal infinitesimal Hilbertianity. Using it we establish universal infinitesimal Hilbertianity of finite dimensional RCD-spaces. We moreover show that (possibly infinite dimensional) Alexandrov spaces are universally infinitesimally Hilbertian and construct an isometric embedding of tangent modules.