Rigidity of Wasserstein spaces over Riemannian manifolds

David Lenze

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arXiv · 2606.21986

Rigidity of Wasserstein spaces over Riemannian manifolds

David Lenze

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Abstract

We show that L2 Wasserstein spaces over Riemannian manifolds are isometrically rigid if and only if their underlying Riemannian manifolds do not admit a Euclidean de Rham factor. We further show that, unless the manifold is isometric to the real line, every isometry of the Wasserstein space is shape-preserving in the sense of Kloeckner. Finally, we demonstrate that two such Wasserstein spaces are isometric if and only if their underlying Riemannian manifolds are isometric.

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David Lenze. 2026-06-20. Rigidity of Wasserstein spaces over Riemannian manifolds. https://arxiv.org/abs/2606.21986

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