Equigeodesics on generalized flag manifolds with ${\rm G}_2$-type $\mathfrak{t}$-roots

Abstract

We study homogeneous curves in generalized flag manifolds $G/K$ with ${\rm G}_2$-type $\mathfrak{t}$-roots, which are geodesics with respect to each $G$-invariant metric on $G/K$. These curves are calledequigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with $\mathfrak{t}$-roots. Also, these spaces are a generalization of the exceptional full flag manifold ${\rm G}_2/T$. We give a characterization for structural equigeodesics for flag manifolds with ${\rm G}_2$-type $\mathfrak{t}$-roots, and we give for each such flag manifold, a list of subspaces in which the vectors are structural equigeodesic vectors.