TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 8, August 2007, Pages 3769–3789 S 0002-9947(07)04277-8 Article electronically published on March 20, 2007 RIEMANNIAN FLAG MANIFOLDS WITH HOMOGENEOUS GEODESICS DMITRI ALEKSEEVSKY AND ANDREAS ARVANITOYEORGOS Abstract. 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 compo- nents. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2+2 )= SO(2 + 1)/U () of complex structures in R 2+2 , and the complex projective space CP 2−1 = Sp()/U (1) · Sp( − 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 associ- ated to the negative of the Killing form of the Lie algebra g of G). According to F. Podest`a and G.Thorbergsson (2003), these manifolds are the only non- Hermitian symmetric flag manifolds with coisotropic action of the stabilizer. 1. Introduction A Riemannian manifold (M,g) is called homogeneous if it admits a transitive connected Lie group G of isometries. Then M can be viewed as a coset space G/K with a G-invariant metric, where K is the isotropy subgroup of some point in M . A geodesic γ (t) through the origin o = eK is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G, i.e., (1) γ (t) = (exp tX) · o, where X is a non-zero vector in the Lie algebra g of G. Homogeneous geodesics were originally studied quite a long time ago by several authors such as R. Hermann, B. Kostant, and E.B. Vinberg to name a few. In particular, in [Kos1] and [Vin] a simple algebraic condition was found so that the orbit (1) is a geodesic. Homogeneous geodesics in a Lie group were studied by V.V. Kajzer in [Kaj] where he proved that a Lie group G with a left-invariant metric has at least one Received by the editors June 23, 2005. 2000 Mathematics Subject Classification. Primary 53C22, 53C30; Secondary 14M15. Key words and phrases. Homogeneous Riemannian manifolds, flag manifolds, homogeneous geodesics, g.o. spaces, coisotropic actions. The first author was supported by Grant Luverhulme trust, EM/9/2005/0069. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 3769 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use