Second Homotopy Group 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, generalizing a result in Burstall and Rawnsley (Springer Lect. Notes Math. 2(84):1424, 1990). This illustrates how “rubber-band” topology can, in the presence of symmetry, single out very rigid objects. We characterize when these 2-spheres in the same homotopy class have the same geometry for all invariant metrics. This is done by exploring the action of Weyl group of the flag manifold, generalizing results of Patrao and San Martin (Indag. Math. 26:547–579, 2015) and de Siebenthal (Math. Helvetici 44(1):1–3, 1969). This illustrates how some aspects of “continuum” invariant geometry can, in the presence of symmetry, be reduced to the study of discrete objects. We remark that the topology singling out very rigid objects and the study of a continuum object being reduced to discrete ones is a characteristic of situations with a lot of symmetry and, thus, are recurring themes in Lie theory.