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Homogeneous Einstein metrics on $ G_2/T$

Authors: Andreas Arvanitoyeorgos, Ioannis Chrysikos and Yusuke Sakane
Journal: Proc. Amer. Math. Soc. 141 (2013), 2485-2499
MSC (2010): Primary 53C25; Secondary 53C30
Published electronically: March 12, 2013
MathSciNet review: 3043029
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Abstract: We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $ M=G_2/T$. By computing a Gröbner basis for a system of polynomials on six variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus $ G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits a non-Kähler and not normal homogeneous Einstein metric.

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  • 1. D. V. Alekseevsky and A. M. Perelomov: Invariant Kähler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (1986) 171-182. MR 868557 (88c:53049)
  • 2. A. Arvanitoyeorgos: New invariant Einstein metrics on generalized flag manifolds, Trans. Amer. Math. Soc. 337 (1993) 981-995. MR 1097162 (93h:53043)
  • 3. A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on generalized flag manifolds with two isotropy summands, J. Aust. Math. Soc. 90 (2011) 237-251. MR 2821781
  • 4. A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on generalized flag manifolds with four isotropy summands, Ann. Glob. Anal. Geom. 37 (2010) 185-219. MR 2578265 (2011d:53083)
  • 5. C. Böhm and M. Kerr: Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc. 358 (2006) 1455-1468. MR 2186982 (2006g:53056)
  • 6. M. Borderman, M. Forger and H. Römer: Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models, Comm. Math. Phys. 102 (1986) 605-617. MR 824094 (87c:53096)
  • 7. A. Borel and F. Hirzebruch: Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958) 458-538. MR 0102800 (21:1586)
  • 8. N. Bourbaki: Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • 9. D. Cox, J. Little and D. O'Shea: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1189133 (93j:13031)
  • 10. E. C. F. Dos Santos and C. J. C. Negreiros: Einstein metrics on flag manifolds, Revista Della, Unión Mathemática Argentina, 47 (2006) 77-84. MR 2301378 (2008b:53061)
  • 11. S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. MR 514561 (80k:53081)
  • 12. M. Kimura: Homogeneous Einstein metrics on certain Kähler C-spaces, Adv. Stud. Pure Math. 18-I, Academic Press, Boston, MA, 1990, 303-320. MR 1145261 (93b:53039)
  • 13. J. L. Koszul: Sur la forme hermitienne canonique des espaces homogènes complexes, Can. J. Math. 7 (1955) 562-576. MR 0077879 (17:1109a)
  • 14. Yu. G. Nikonorov, E. D. Rodionov and V. V. Slavskiĭ: Geometry of homogeneous Riemannian manifolds, J. Math. Sciences 146 (2007) 6313-6390. MR 2568572 (2011a:53083)
  • 15. J-S. Park and Y. Sakane: Invariant Einstein metrics on certain homogeneous spaces, Tokyo J. Math. 20 (1997) 51-61. MR 1451858 (98c:53058)
  • 16. Y. Sakane: Homogeneous Einstein metrics on flag manifolds, Lobachevskii J. Math. 4 (1999) 71-87. MR 1743146 (2000m:53069)
  • 17. M. Takeuchi: Homogeneous Kähler submanifolds in complex projective spaces, Japan. J. Math. 4 (1978) 171-219. MR 528871 (80i:32059)
  • 18. P. Tauvel and R. W. T. Yu: Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2146652 (2006c:17001)
  • 19. M. Wang and W. Ziller: On normal homogeneous Einstein manifolds, Ann. Scient. Éc. Norm. Sup. 18 (1985) 563-633. MR 839687 (87k:53113)
  • 20. M. Wang and W. Ziller: Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84 (1986) 177-194. MR 830044 (87e:53081)

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Additional Information

Andreas Arvanitoyeorgos
Affiliation: Department of Mathematics, University of Patras, GR-26500 Rion, Greece

Ioannis Chrysikos
Affiliation: Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 611 37 Brno, Czech Republic

Yusuke Sakane
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan

Keywords: Homogeneous Einstein metric, full flag manifold, exceptional Lie group $G_2$
Received by editor(s): October 14, 2011
Published electronically: March 12, 2013
Additional Notes: The third author was supported by Grant-in-Aid for Scientific Research (C) 21540080
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society

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