Isometric rigidity of $L^2$-spaces with manifold targets

Abstract

We describe the isometry group of $L^2(\Omega ,M)$ for Riemannian manifolds $M$ of dimension at least two with irreducible universal cover. We establish a rigidity result for the isometries of these spaces: any isometry arises from an automorphism of $\Omega$ and a family of isometries of $M$, distinguishing these spaces from the classical $L^2(\Omega )$. Additionally, we prove that these spaces lack irreducible factors and that two such spaces are isometric if and only if the underlying manifolds are.

25 pages. To appear in Trans. Amer. Math. Soc.; revised based on referee's comments