Riemannian Geometry of $G_2$-type Real Flag Manifolds

Abstract: In this paper, we investigate homogeneous Riemannian geometry on real flag manifolds of the split real form of $\mathfrak{g}_2$. We characterize the metrics that are invariant under the action of a maximal compact subgroup of $G_2.$ Our exploration encompasses the analysis of g.o. metrics and equigeodesics on the $\mathfrak{g}_2$-type flag manifolds. Additionally, we explore the Ricci flow for the case where the isotropy representation has no equivalent summands, employing techniques from the qualitative theory of dynamical systems.