  We study the Wasserstein space (with quadratic cost) of Euclidean spaces as
an intrinsic metric space. In particular we compute their isometry groups.
Surprisingly, in the case of the line, there exists a (unique) "exotic"
isometric flow. This contrasts with the case of higher-dimensional Euclidean
spaces, where all isometries of the Wasserstein space preserve the shape of
measures. We also study the curvature and various ranks of these spaces.
