A geometric study of Wasserstein spaces: Euclidean spaces

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.

💡 Research Summary

The paper investigates the intrinsic geometry of Wasserstein spaces (W_{2}(X)) equipped with the quadratic cost, focusing on Euclidean base spaces (X=\mathbb{R}^{n}). After recalling basic definitions—probability measures with finite second moment, the Wasserstein distance defined via optimal couplings, and the fact that (W_{2}(X)) is itself a Polish, geodesic metric space—the author studies three central geometric aspects: isometry groups, curvature, and rank.

A key methodological tool is the explicit description of optimal transport in one dimension: the optimal plan is the monotone (non‑decreasing) rearrangement, which yields closed‑form formulas for the distance and for constant‑speed geodesics in terms of inverse distribution functions. This explicitness allows the author to characterize Dirac measures and convex combinations of two Dirac masses purely by metric properties, a step crucial for later rigidity arguments.

The notion of “shape‑preserving” isometries is introduced: an isometry (\Phi) of (W_{2}(X)) preserves shape if for every (\mu) there exists an isometry (\varphi) of the base space such that (\Phi(\mu)=\varphi_{#}\mu). An isometry that fails this condition is called “exotic”.

One‑dimensional case ((n=1)).
The main result (Theorem 1.1) states that \