Compact Riemannian Manifolds with Homogeneous Geodesics

A homogeneous Riemannian space $(M=G\mathrm{/}H,g)$ is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group $G$. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric $g$ with homogeneous geodesics on a homogeneous space of a compact Lie group $G$. We give a classification of compact simply connected GO-spaces $(M = G/H,g)$ofpositiveEulercharacteristic\mathrm{.}Ifthegroup$G$ is simple and the metric $g$ does not come from a bi-invariant metric of $G$, then $M$ is one of the flag manifolds $M_1=SO(2n+1)\mathrm{/}U(n)$ or $M_2=Sp(n)\mathrm{/}U(1)\cdot Sp(n-1)$ and $g$ is any invariant metric on $M$ which depends on two real parameters. In both cases, there exists unique (up to a scaling) symmetric metric $g_0$ such that $(M,g_0)$ is the symmetric space $M=SO(2n+2)\mathrm{/}U(n+1)$ or, respectively, $\mathbb{C}{P}^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces.