Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space W2(Sn,‖⋅‖) over the sphere endowed with the distance inherited from the norm of Rn+1. We prove that W2(Sn,‖⋅‖) is isometrically rigid, meaning that its isometry group is isomorphic to that of (Sn,‖⋅‖). This is in striking contrast to the non-rigidity of its ambient space W2(Rn+1,‖⋅‖) but in line with the rigidity of the geodesic space W2(Sn,∢). One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in W2(Sn,‖⋅‖). A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of Wp(S1,‖⋅‖) for 1≤p<2.