Second homotopy and invariant geometry of flag manifolds

Abstract

We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $2$-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold. We characterize when the generators with the same invariant geometry are in the same homotopy class. This is done by exploring the action of Weyl group on the irreducible components of isotropy representation of the flag manifold.