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.

Some corrections and typos. To appear in International Mathematics Research Notices