Compact Riemannian Manifolds with Homogeneous Geodesics

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) . 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, CP2n-1. The manifolds M-1, M-2 are weakly symmetric spaces.