Isometric embeddings of Teichmüller spaces are covering constructions

Abstract

Pulling back complex structures along a branched covering induces a holomorphic isometric embedding of Teichmüller 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 Teichmüller space, which are isometric to some Teichmüller space, are covering constructions. Another consequence is the classification of locally isometric embeddings of moduli spaces of Riemann surfaces.

17 pages