Equigeodesics on generalized flag manifolds with $G_2$-type $t$-roots

TL;DR

This paper investigates equigeodesics, which are homogeneous curves that are geodesics with respect to all invariant metrics, on generalized flag manifolds with $G_2$-type $t$-roots, extending known results for the exceptional full flag manifold.

Contribution

It characterizes structural equigeodesics and identifies subspaces containing vectors that are structural equigeodesic in flag manifolds with $G_2$-type $t$-roots.

Findings

Characterization of structural equigeodesics in these flag manifolds

List of subspaces with structural equigeodesic vectors for each manifold

Extension of results from $G_2/T$ to more general flag manifolds

Abstract

We study homogeneous curves in generalized flag manifolds $G/K$ with $G_2$-type $t$-roots, which are geodesics with respect to each $G$-invariant metric on $G/K$. These curves are called equigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with $t$-roots. Also, these spaces are a generalization of the exceptional full flag manifold $G_2/T$. We give a characterization for structural equigeodesics for flag manifolds with $G_2$-type $t$-roots, and we give for each such flag manifold, a list of subspaces in which the vectors are structural equigeodesic vectors.