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arXiv · 2508.05483

Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian

Jesús Núñez-Zimbrón·Enrico Pasqualetto·Elefterios Soultanis
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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.

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Jesús Núñez-Zimbrón, Enrico Pasqualetto, Elefterios Soultanis. 2025-08-07. Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian. https://arxiv.org/abs/2508.05483

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