  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.
