François Rouvière

Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

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.

SYMMETRIC SPACES AND THE KASHIWARA-VERGNE METHOD

Introduction.- Notation.- The Kashiwara-Vergne method for Lie groups.- Convolution on homogeneous spaces.- The role of e-functions.- e-functions and the Campbell Hausdorff formula.- Bibliography.

e-Functions and the Campbell-Hausdorff Formula

The goal of this chapter is to adapt the Kashiwara-Vergne method to an arbitrary symmetric space. We show that it admits an e-function, which can be constructed by means of the Campbell-Hausdorff formula. We prove several properties of this function which reflect, by Chap. 3, on invariant analysis on the space. The results extend to line bundles.

Convolution on Homogeneous Spaces

In this short chapter we define convolution products on a general homogeneous space G/H. No symmetric space structure is assumed here.We give some examples, the most important being the application to invariant differential operators. Then we extend the definitions to line bundles over homogeneous spaces.

The Role of e-Functions

Functions and distributions are transferred between a symmetric space and its tangent space at the origin by means of the exponential mapping. We introduce the concept of e-function, which appears when transferring convolution products of invariant functions or distributions. Admitting the existence of this function (proved in Chap. 4), we study its links with invariant differential operators, mean value operators, spherical functions and some integral formulas, with special consideration of the classical case of Riemannian symmetric spaces of the noncompact type; we give an explicit e-function for rank one spaces. The results extend to line bundles.