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

TL;DR

This paper characterizes the isometry group of $L^2$ spaces with manifold targets, revealing a rigidity property that links isometries to automorphisms of the domain and isometries of the target manifold.

Contribution

It provides a detailed description of the isometry group for $L^2( ext{domain}, M)$ spaces where $M$ has an irreducible universal cover, highlighting their rigidity and uniqueness properties.

Findings

Isometry group described explicitly for these $L^2$ spaces.

Any isometry is induced by domain automorphism and target isometries.

Spaces are uniquely determined by the underlying manifold.

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.