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# Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

Published:2012-07-16
Printed: Aug 2013
• Philippe Delanoë,
Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice Cedex 2
• François Rouvière,
Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice Cedex 2
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## Abstract

The squared distance curvature is a kind of two-point curvature the sign of which turned out 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, 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: symmetric spaces, rank one, positive curvature, almost-positive $c$-curvature
 MSC Classifications: 53C35 - Symmetric spaces [See also 32M15, 57T15] 53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C26 - Hyper-Kahler and quaternionic Kahler geometry, special'' geometry 49N60 - Regularity of solutions