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Transactions of the American Mathematical Society

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Riemannian flag manifolds with homogeneous geodesics


Authors: Dmitri Alekseevsky and Andreas Arvanitoyeorgos
Journal: Trans. Amer. Math. Soc. 359 (2007), 3769-3789
MSC (2000): Primary 53C22, 53C30; Secondary 14M15
DOI: https://doi.org/10.1090/S0002-9947-07-04277-8
Published electronically: March 20, 2007
MathSciNet review: 2302514
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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 $ \operatorname{Com}(\mathbb{R}^{2\ell +2}) = SO(2\ell +1)/U(\ell )$ of complex structures in $ \mathbb{R}^{2\ell + 2}$, and the complex projective space $ \mathbb{C} P^{2\ell -1}= Sp(\ell )/U(1) \cdot Sp(\ell -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 $ \mathfrak{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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Additional Information

Dmitri Alekseevsky
Affiliation: School of Mathematics and Maxwell Institute for Mathematical Studies, Edinburgh University, Edinburgh EH9 3JZ, United Kingdom
Email: D.Aleksee@ed.ac.uk

Andreas Arvanitoyeorgos
Affiliation: Department of Mathematics, University of Patras, GR-26500 Patras, Greece
Email: arvanito@math.upatras.gr

DOI: https://doi.org/10.1090/S0002-9947-07-04277-8
Keywords: Homogeneous Riemannian manifolds, flag manifolds, homogeneous geodesics, g.o. spaces, coisotropic actions
Received by editor(s): June 23, 2005
Published electronically: March 20, 2007
Additional Notes: The first author was supported by Grant Luverhulme trust, EM/9/2005/0069.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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