Isometric embeddings of Teichm\"uller spaces are covering constructions

Abstract

Pulling back complex structures along a branched covering induces a holomorphic isometric embedding of Teichm\"uller spaces. We show that for dimension at least $2$, all isometric embeddings arise from branched coverings. This generalizes a theorem of Royden. As a consequence we obtain that totally geodesic submanifolds of Teichm\"uller space, which are isometric to some Teichm\"uller space, are covering constructions. Another consequence is the classification of locally isometric embeddings of moduli spaces of Riemann surfaces.