Totally geodesic submanifolds of a complex flag

Abstract

This work presents a classification of the maximal totally geodesic submanifolds of the complex manifold M = SU(n+2)/S(U(1) x U(1) x U(n)) \cong F_{1,2}(\mathbb{C}^{n+2}), n\geq 2, equipped with its standard homogeneous Riemannian metric. Using the natural fibration S^2 \to M \to G_2(\mathbb{C}^{n+2}) and curvature computations derived from O'Neill's formulas, we identify curvature-invariant subspaces of the tangent representation. As a consequence, we obtain a complete description of the maximal totally geodesic submanifolds of M. These include twistor spaces of lower-dimensional complex Grassmannians, real flag manifolds, products of complex projective spaces, and twistor spaces of quaternionic projective spaces.