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Given a metric space   $X$  , one defines its Wasserstein space   ${\mathscr {W}_2}(X)$   as a set of sufficiently decaying probability measures on   $X$   endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of   ${\mathscr {W}_2}(X)$   when   $X$   is a simply connected, non-positively curved metric spaces by considering its isometry group. When   $X$   is Euclidean, the second-named author proved that this isometry group is larger than the isometry group of   $X$  . In contrast, we prove here a rigidity result: when   $X$   is negatively curved, any isometry of   ${\mathscr {W}_2}(X)$   comes from an isometry of   $X$  .

A Geometric Study of Wasserstein Spaces: Isometric Rigidity in Negative Curvature