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 M1 = SO(2n + 1)/U(n) or M2 = 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 g0 such that (M,g0) is the symmetric space M = SO(2n + 2)/U(n + 1) or, respectively, CP 2n 1 . The manifolds M1, M2 are weakly symmetric spaces.
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