Strongly exponential symmetric spaces

TL;DR

This paper investigates the exponential map in symmetric spaces, providing conditions for it to be a diffeomorphism and characterizing exponential solvable symmetric spaces through geometric properties, with implications for quantization.

Contribution

It generalizes the Dixmier-Saito theorem to symmetric spaces and offers a new geometric characterization of exponential solvable symmetric spaces.

Findings

Exponential map is a diffeomorphism under specific conditions.

Strongly exponential symmetric spaces have unique double triangles for all triangles.

Results are motivated by applications in Weinstein's quantization program.

Abstract

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spaces as those spaces for which every triangle admits a unique double triangle. This work is motivated by Weinstein's quantization by groupoids program applied to symmetric spaces.