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

Abstract

Abstract. In this paper we provide a characterization of structural equigeodesics on generalized flag manifolds with second Betti number b2(G/K) = 1, and give examples of structural equigeodesics on generalized flag manifolds of the exceptional Lie groups F4, E6 and E7 with three isotropy summands.

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

“…It is important to note that this characterization depends on determining whether a vector is equigeodesic on the corresponding generalized flag manifold. For a comprehensive understanding of equigeodesic vectors on generalized flag manifolds, we refer to [6,17,20,21].…”

“…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.…”

Riemannian M-spaces with homogeneous geodesics

“…. , k − 1) in (32), there exist X ∈ m iq , Y ∈ m i q+1 eigenvectors of Λ such that [X, Y ] = 0. If we had that λ iq = λ i q+1 , then Proposition 5 implies that [X, Y ] ⊂ m iq ⊕ m i q+1 , which contradicts (34), hence λ iq = λ i q+1 , (q = 1, .…”

“…Other interesting results about g.o. spaces can be found in [3], [8], [11], [16], [19], [23], [25], [26], [27], [28] and [32]. Finally, the notion of homogeneous geodesics can be extended to geodesics which are orbits of a product of two exponential factors (cf.…”