Isometric Rigidity of compact Wasserstein spaces

Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved useful in describing several geometrical properties of $X.$ In this paper we focus on the study of isometries of $\mathbb{P}_p(X)$ for $p \in (1,\infty)$ under the assumption that there is some characterization of optimal maps between measures, the so called Good transport behaviour $GTB_p$. Our first result states that the set of Dirac deltas is invariant under isometries of the Wasserstein space. Additionally we obtain that the isometry groups of the base Riemannian manifold $M$ coincides with the one of the Wasserstein space $\mathbb{P}_p(M)$ under assumptions on the manifold; namely, for $p=2$ that the sectional curvature is strictly positive and for general $p\in (1,\infty)$ that $M$ is a Compact Rank One Symmetric Space.