Equigeodesics on Generalized Flag Manifolds with Two Isotropy Summands

Abstract

In this paper we study homogeneous curves in generalized flag manifolds with two isotropy summands with the additional property that such curves are geodesics with respect to each invariant metric on the flag manifold. These curves are called equigeodesics. We give an algebraic characterization for such curves and we exhibit families of equigeodesics in several flag manifolds of classical and exceptional Lie groups.

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

“…We discuss an algebraic approach in order to study equigeodesics on SU (n)-flags and flag manifolds with isotropy summands. These results are proved in [6] and [11].…”

“…The study of equigeodesics in generalized flag manifolds started in [6] with the description of equigeodesics on SU (n)-flags. All results in this section are proved in [6] and [11].…”

Equigeodesics on Generalized Flag Manifolds with b 2 (G / K) = 1

“…Let Π = {α 1 , α 2 , α 3 , α 4 , α 5 , α 6 } be a system of simple roots for E 6 such that the highest root is given by μ = α 1 + 2α 2 + 2α 3 + 3α 4 …”

“…In Sect. 4 we give the results about structural equigeodesic vectors on generalized flag manifolds associated to the exceptional Lie groups F 4 , E 6 and E 7 with three isotropy summands.…”

Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

“…Several authors have contributed to the realm of equigeodesics within flag manifolds. For instance, in [13], and [21], the authors focus on the study of equigeodesics on generalized flag manifolds with two, and four isotropy summands, respectively, in [17] are investigated equigeodesics on flag manifolds with G 2 -type t-roots, and, in [20], the authors examine the existence and properties of equigeodesics in flag manifolds where the second Betti number b 2 (G/K) = 1. For other homogeneous spaces, it is noteworthy to mention the works of Statha [18], which includes a characterization of algebraic equigeodesics on some homogeneous spaces, such as Stiefel manifolds, generalized Wallach spaces, and some spheres, and Xu and Tan [22], who have extended the concept of homogeneous equigeodesics to the context of homogeneous Finsler spaces, expanding the scope of this field of study.…”