Isometric embeddings of Teichmuller spaces are covering constructions

Let h : Sigma' -> Sigma be a branched covering of topological surfaces. By pulling back complex structures, h induces a holomorphic isometric embedding of Teichmuller spaces T(Sigma) hooked right arrow T(Sigma'). We show that for dim T(Sigma) >= 2, all isometric embeddings arise from branched coverings. This generalizes a theorem of Royden [16]. As a consequence, we obtain that totally geodesic submanifolds of T(Sigma), which are isometric to some Teichmuller space, are covering constructions. Another consequence is the classification of local isometric embeddings of moduli spaces of Riemann surfaces. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.