Positively curved Riemannian locally symmetric spaces are positively squared distance curved

Philippe Delanoë 1, * François Rouvière 2
* Corresponding author
1 Géométrie et Analyse
JAD - Laboratoire Jean Alexandre Dieudonné
2 Géométrie et Analyse, Membre Emérite
JAD - Laboratoire Jean Alexandre Dieudonné
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.
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Philippe Delanoë, François Rouvière. Positively curved Riemannian locally symmetric spaces are positively squared distance curved. Canadian Journal of Mathematics, University of Toronto Press, 2013, 65 (4), pp.757-767. ⟨10.4153/CJM-2012-015-1⟩. ⟨hal-00667519v2⟩

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