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Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

Published online by Cambridge University Press:  20 November 2018

Philippe Delanoë  and
François Rouvière
Philippe Delanoë
Affiliation:
Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice Cedex 2, e-mail: Philippe.Delanoe@unice.fr, Francois.Rouviere@unice.fr
François Rouvière
Affiliation:
Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice Cedex 2, e-mail: Philippe.Delanoe@unice.fr, Francois.Rouviere@unice.fr
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Abstract

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The squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

Keywords

53C3553C2153C2649N60symmetric spacesrank onepositive curvaturealmost-positive c-curvature

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 4 , 01 August 2013 , pp. 757 - 767
Copyright
Copyright © Canadian Mathematical Society 2013

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