Riemann curvature tensor on $ \mathsf{RCD}$ spaces and possible applications

Nicola Gigli

doi:10.1016/j.crma.2019.06.003

Functional analysis/Differential geometry

Riemann curvature tensor on

RCD

spaces and possible applications
[Tenseur de courbure de Riemann sur les espaces

RCD

et applications possibles]

Présenté par : the Editorial Board

Nicola Gigli ¹

¹ SISSA, Trieste, Italy

Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 613-619.

Résumé
Abstract

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.

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.

Reçu le : 2019-02-06
Accepté le : 2019-06-07
Publié le : 2019-07-01

DOI : 10.1016/j.crma.2019.06.003

Affiliations des auteurs :

Nicola Gigli ¹

¹ SISSA, Trieste, Italy

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     author = {Nicola Gigli},
     title = {Riemann curvature tensor on $ \mathsf{RCD}$ spaces and possible applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {613--619},
     publisher = {Elsevier},
     volume = {357},
     number = {7},
     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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.003/

Bibliographie
Cité par

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