Riemannian flag manifolds with homogeneous geodesics

A geodesic in a Riemannian homogeneous manifold $(M=G/K, g)$ is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group $G$. We investigate $G$-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when $M=G/K$ is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group $G$. We use an important invariant of a flag manifold $M=G/K$, its $T$-root system, to give a simple necessary condition that $M$ admits a non-standard $G$-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds $M=G/K$ of a simple Lie group $G$, only the manifold $\operatorname{Com}(\mathbb{R}^{2\ell +2}) = SO(2\ell +1)/U(\ell )$ of complex structures in $\mathbb{R}^{2\ell + 2}$, and the complex projective space $\mathbb{C} P^{2\ell -1}= Sp(\ell )/U(1) \cdot Sp(\ell -1)$ admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only $G$-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra $\mathfrak{g}$ of $G$). According to F. Podestà and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.