Complete totally geodesic subsets of the complex hyperbolic plane: an elementary classification

The non-trivial complete totally geodesic submanifolds of the complex hyperbolic plane \(\mathbb {H}_\mathbb {C}^2\) ${\mathbb{H}}_{\mathbb{C}}^2$ are the complex geodesics and the real planes. We present two new proofs for this fact. One is a short proof based on an algebraic formula for the Riemann curvature tensor due to S. Anan \('\) $_{}^{\prime }$ in and C. Grossi and resembles the traditional proof using Lie theory. The other is purely elementary and geometric, relying on the structures in \(\mathbb {H}_\mathbb {C}^2\) ${\mathbb{H}}_{\mathbb{C}}^2$ instead of general theories. In this second approach, we prove a slightly stronger result: the only non-trivial complete totally geodesic subsets of \(\mathbb {H}_\mathbb {C}^2\) ${\mathbb{H}}_{\mathbb{C}}^2$ are the complex geodesics and the real planes without assuming that the subsets are submanifolds a priori. This second proof is also intriguing for only making use of elementary geometric constructions.