Isometric rigidity of $L^2$-spaces with manifold targets
David Lenze
TL;DR
The paper establishes a sharp rigidity description of the isometry group of $L^2(\Omega,M)$ for complete Riemannian manifolds $M$ of dimension at least two with irreducible universal cover: every isometry arises from a measure-preserving automorphism of $\Omega$ and a square-integrable field of isometries of $M$, yielding $\mathrm{Isom}(L^2(\Omega,M)) = L^2(\Omega,\mathrm{Isom}(M)) \rtimes \mathrm{Aut}(\Omega)$. It proves that $L^2(\Omega,M)$ has no irreducible factors when $\Omega$ is atomless, and that isometry to $L^2(\Omega,N)$ forces $M$ and $N$ to be isometric; further, it shows a rigidity phenomenon via a decomposition into “local” pieces $L^2(A,M)$ and $L^2(A^c,M)$ in atomless settings. The core method combines Alexandrov angle theory in $L^2(\Omega,X)$, a precise affine-map analysis on products and in $L^2$-spaces, and an isometric localization argument to extract a global pair $(\varphi,\rho)$. The results illuminate how geometric properties of the target manifold $M$ strongly constrain the structure of isometries in function-valued metric spaces and connect to de Rham-type decompositions in metric geometry. The irreducibility assumption is crucial, as counterexamples appear in reducible settings, highlighting the delicate balance between base measure dynamics and target geometry.
Abstract
We describe the isometry group of $L^2(Ω, M)$ 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 $Ω$ and a family of isometries of $M$, distinguishing these spaces from the classical $L^2(Ω)$. Additionally, we prove that these spaces lack irreducible factors and that two such spaces are isometric if and only if the underlying manifolds are.