Metrics with homogeneous geodesics on flag manifolds

Dimitri V. Alekseevsky; Andreas Arvanitoyeorgos

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 189-199
  • ISSN: 0010-2628

Abstract

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A geodesic of a homogeneous Riemannian manifold ( M = G / K , g ) is called homogeneous if it is an orbit of an one-parameter subgroup of G . In the case when M = G / H is a naturally reductive space, that is the G -invariant metric g is defined by some non degenerate biinvariant symmetric bilinear form B , all geodesics of M are homogeneous. We consider the case when M = G / K is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group G , and we give a simple necessary condition that M admits a non-naturally reductive invariant metric with homogeneous geodesics. Using this, we enumerate flag manifolds of a classical Lie group G which may admit a non-naturally reductive G -invariant metric with homogeneous geodesics.

How to cite

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Alekseevsky, Dimitri V., and Arvanitoyeorgos, Andreas. "Metrics with homogeneous geodesics on flag manifolds." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 189-199. <http://eudml.org/doc/248966>.

@article{Alekseevsky2002,
abstract = {A geodesic of a homogeneous Riemannian manifold $(M=G/K, g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$. In the case when $M=G/H$ is a naturally reductive space, that is the $G$-invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$, all geodesics of $M$ are homogeneous. We consider the case when $M=G/K$ is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group $G$, and we give a simple necessary condition that $M$ admits a non-naturally reductive invariant metric with homogeneous geodesics. Using this, we enumerate flag manifolds of a classical Lie group $G$ which may admit a non-naturally reductive $G$-invariant metric with homogeneous geodesics.},
author = {Alekseevsky, Dimitri V., Arvanitoyeorgos, Andreas},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {homogeneous Riemannian spaces; homogeneous geodesics; flag manifolds; homogeneous Riemannian spaces; homogeneous geodesics; flag manifolds},
language = {eng},
number = {2},
pages = {189-199},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Metrics with homogeneous geodesics on flag manifolds},
url = {http://eudml.org/doc/248966},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Alekseevsky, Dimitri V.
AU - Arvanitoyeorgos, Andreas
TI - Metrics with homogeneous geodesics on flag manifolds
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 189
EP - 199
AB - A geodesic of a homogeneous Riemannian manifold $(M=G/K, g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$. In the case when $M=G/H$ is a naturally reductive space, that is the $G$-invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$, all geodesics of $M$ are homogeneous. We consider the case when $M=G/K$ is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group $G$, and we give a simple necessary condition that $M$ admits a non-naturally reductive invariant metric with homogeneous geodesics. Using this, we enumerate flag manifolds of a classical Lie group $G$ which may admit a non-naturally reductive $G$-invariant metric with homogeneous geodesics.
LA - eng
KW - homogeneous Riemannian spaces; homogeneous geodesics; flag manifolds; homogeneous Riemannian spaces; homogeneous geodesics; flag manifolds
UR - http://eudml.org/doc/248966
ER -

References

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  9. Kowalski O., Ž. Nikčević S., Vlášek Z., Homogeneous geodesics in homogeneous Riemannian manifolds - examples, in: Geometry and Topology of Submanifolds, X (Beijing/Berlin, 1999), pp.104-112, World Sci. Publishing, River Edge, NJ, 2000. MR1801906
  10. Kowalski O., Vanhecke L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B (7) 5 (1991), 189-246. (1991) Zbl0731.53046MR1110676
  11. Kowalski O., Szenthe J., On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), 209-214; Erratum: 84 (2001), 331-332. (2000) Zbl0980.53061MR1772203
  12. Vinberg E.B., Invariant linear connections in a homogeneous manifold, Trudy MMO 9 (1960), 191-210. (1960) MR0176418

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