Riemann curvature tensor on RCD spaces and possible applications

Abstract

We show that on every RCD spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor.Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an RCD space is Alexandrov if and only if the sectional curvature -defined in terms of such abstract Riemann tensor -is bounded from below. RésuméNous montrons que sur chaque espace RCD il est possible d'introduire, par une approche distributionnelle, un tenseur de courbure de Riemann.Puisque après les travaux de Petrunin et de Zhang-Zhu nous savons que les espaces d'Alexandrov de dimension finie sont des espaces RCD, notre construction s'applique en particulier au cadre d'Alexandrov. Nous conjecturons qu'un espace RCD est Alexandrov si et seulement si la courbure sectionnelle -définie en termes de ce tenseur de Riemann abstrait -est bornée par dessous.

Curvature tensor of smoothable Alexandrov spaces

“…The problem of introducing Ricci tensor was studied in far more general settings [7,9,11,24]. Curvature tensor for RCD spaces was defined by Nicola Gigli [6]; it works for a more general class of spaces, but this approach does not see the curvature of singularities. It is expected that our definitions agree on the regular locus.…”