Comptes Rendus
Functional analysis/Differential geometry
Riemann curvature tensor on RCD spaces and possible applications
Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 613-619.

We show that, on every RCD space, 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.

Nous montrons que, sur chaque espace RCD, il est possible d'introduire, par une approche distributionnelle, un tenseur de courbure de Riemann.

Puisque, d'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 en dessous.

Published online:
DOI: 10.1016/j.crma.2019.06.003

Nicola Gigli 1

1 SISSA, Trieste, Italy
     author = {Nicola Gigli},
     title = {Riemann curvature tensor on $ \mathsf{RCD}$ spaces and possible applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {613--619},
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     year = {2019},
     doi = {10.1016/j.crma.2019.06.003},
     language = {en},
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Nicola Gigli. Riemann curvature tensor on $ \mathsf{RCD}$ spaces and possible applications. Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 613-619. doi : 10.1016/j.crma.2019.06.003.

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