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

We describe the isometry group of $L^2(\mathrm{\Omega },M)$ 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 $\mathrm{\Omega }$ and a family of isometries of $M$, distinguishing these spaces from the classical $L^2(\mathrm{\Omega })$. Additionally, we prove that these spaces lack irreducible factors and that two such spaces are isometric if and only if the underlying manifolds are.