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Journal Article

# Riemannian Flag Manifolds with Homogeneous Geodesics

Dmitri Alekseevsky and Andreas Arvanitoyeorgos
Transactions of the American Mathematical Society
Vol. 359, No. 8 (Aug., 2007), pp. 3769-3789
Stable URL: http://www.jstor.org/stable/20161752
Page Count: 21

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## 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 components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold $\text{Com}(ℝ^{2l+2}=SO(2l+1)/U(l)$ of complex structures in $ℝ^{2l+2}$, and the complex projective space $ℂP^{2l-1}=Sp(l)/U(1)\cdot Sp(l-1)$ 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 associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podestà and G. Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

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