Totally geodesic discs in bounded symmetric domains

Abstract

In this paper, we characterize C2-smooth totally geodesic isometric embeddings f: Ω → Ω ′ between bounded symmetric domains Ω and Ω ′ which extend C1-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if Ω is irreducible, there exist totally geodesic bounded symmetric subdomains Ω 1 and Ω 2 of Ω ′ such that f= (f1, f2) maps into Ω 1× Ω 2⊂ Ω where f1 is holomorphic and f2 is anti-holomorphic totally geodesic isometric embeddings. If rank (Ω ′) < 2 rank (Ω) , then either f or f¯ is a standard holomorphic embedding.