Second homotopy and invariant geometry of flag manifolds

TL;DR

This paper explicitly computes the second homotopy group generators of flag manifolds using the Hopf fibration, revealing their geometric properties and homotopy classifications through Weyl group actions.

Contribution

It provides a detailed geometric and algebraic analysis of second homotopy groups of flag manifolds, including explicit generator constructions and homotopy class characterizations.

Findings

Second homotopy group generators are explicitly constructed using Hopf fibration.

These generators are totally geodesic surfaces under any invariant metric.

Homotopy classes are characterized by Weyl group actions on isotropy components.

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.