  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.
