Compact Riemannian Manifolds with Homogeneous Geodesics

Abstract

A homogeneous Riemannian space $(M=G/H,g)$ 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 $(M=G/H,g)$ of positive Euler characteristic. If the group $G$ is simple and the metric $g$ does not come from a bi-invariant metric of $G$, then $M$ is one of the flag manifolds $M_1=SO(2n+1)/U(n)$ or $M_2=Sp(n)/U(1)\cdot Sp(n-1)$ 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 $M=SO(2n+2)/U(n+1)$ or, respectively, $\mathbb{C}{P}^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces.

Cited by in corpus (13)

• On the structure of geodesic orbit Riemannian spaces
• Geodesic Orbit Riemannian Structures on ${\mathbf{R}}^n$
• On left-invariant Einstein Riemannian metrics that are not geodesic orbit
• Geodesic orbit Riemannian spaces with two isotropy summands. I
• Killing vector fields of constant length on compact homogeneous Riemannian manifolds
• On Lagrangian and Hamiltonian systems with homogeneous trajectories
• Compact geodesic orbit spaces with a simple isotropy group
• Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds
• Geodesic flows on Riemannian g.o. spaces
• Spectral properties of Killing vector fields of constant length
• Equigeodesics on flag manifolds
• Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups
• Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds