Riemannian flag manifolds with homogeneous geodesics
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- by Dmitri Alekseevsky and Andreas Arvanitoyeorgos PDF
- Trans. Amer. Math. Soc. 359 (2007), 3769-3789 Request permission
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.References
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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
- MR Author ID: 226278
- ORCID: 0000-0002-6622-7975
- Email: D.Aleksee@ed.ac.uk
- Andreas Arvanitoyeorgos
- Affiliation: Department of Mathematics, University of Patras, GR-26500 Patras, Greece
- MR Author ID: 307519
- Email: arvanito@math.upatras.gr
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2302514