Positively curved Riemannian locally symmetric spaces are positively squared distance curved (Q2841817)

A new notion of curvature, the squared distance curvature, was recently introduced in the paper by \textit{X.-N. Ma} et al. [Arch. Ration. Mech. Anal. 177, No. 2, 151--183 (2005; Zbl 1072.49035)]. In the introduction, the authors discuss the various developments of that curvature variously named \textit{square-distance curvature} by the authors, \textit{cost-curvature}, \textit{\(c\)-curvature} or \textit{MTW tensor} (up to a constant factor) by others. This notion is instrumental in showing the smoothness of the solution of Monge's problem with smooth data.NEWLINENEWLINEThe authors mention that several positivity results have been established by direct and indirect means for the square-distance curvature namely for the simply connected compact rank one symmetric spaces (except the Cayley plane), for the sphere and for the complex and quaternionic projective spaces.NEWLINENEWLINEIn their paper, the authors provide a direct proof for positively Riemannian locally symmetric spaces, essentially a generalization of all the previous results.NEWLINENEWLINEThey proceed in two main steps: They prove that the case of the constant curvature spheres imply the general result and then handle that case.NEWLINENEWLINEIn the conclusion, the authors recall a more general definition of the \(c\)-curvature, namely the extended \(c\)-curvature introduced in [\textit{A. Figalli} et al., Am. J. Math. 134, No. 1, 109--139 (2012; Zbl 1241.53031)]. They point out that their result also holds for the extended \(c\)-curvature.