Riemannian flag manifolds with homogeneous geodesics |
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| Authors: | Dmitri Alekseevsky Andreas Arvanitoyeorgos |
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| Affiliation: | School of Mathematics and Maxwell Institute for Mathematical Studies, Edinburgh University, Edinburgh EH9 3JZ, United Kingdom ; Department of Mathematics, University of Patras, GR-26500 Patras, Greece |
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| Abstract: | A geodesic in a Riemannian homogeneous manifold  is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group  . We investigate  -invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when  is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group  . We use an important invariant of a flag manifold  , its  -root system, to give a simple necessary condition that  admits a non-standard  -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  of a simple Lie group  , only the manifold  of complex structures in  , and the complex projective space  admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only  -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  of  ). 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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| Keywords: | Homogeneous Riemannian manifolds flag manifolds homogeneous geodesics g.o. spaces coisotropic actions |
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