Homogeneous Einstein metrics on $G_2/T$
HTML articles powered by AMS MathViewer
- by Andreas Arvanitoyeorgos, Ioannis Chrysikos and Yusuke Sakane PDF
- Proc. Amer. Math. Soc. 141 (2013), 2485-2499 Request permission
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.References
- D. V. Alekseevskiĭ and A. M. Perelomov, Invariant Kähler-Einstein metrics on compact homogeneous spaces, Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 1–16, 96 (Russian). MR 868557
- Andreas Arvanitoyeorgos, New invariant Einstein metrics on generalized flag manifolds, Trans. Amer. Math. Soc. 337 (1993), no. 2, 981–995. MR 1097162, DOI 10.1090/S0002-9947-1993-1097162-3
- Andreas Arvanitoyeorgos and Ioannis Chrysikos, Invariant Einstein metrics on generalized flag manifolds with two isotropy summands, J. Aust. Math. Soc. 90 (2011), no. 2, 237–251. MR 2821781, DOI 10.1017/S1446788711001303
- Andreas Arvanitoyeorgos and Ioannis Chrysikos, Invariant Einstein metrics on flag manifolds with four isotropy summands, Ann. Global Anal. Geom. 37 (2010), no. 2, 185–219. MR 2578265, DOI 10.1007/s10455-009-9183-7
- Christoph Böhm and Megan M. Kerr, Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1455–1468. MR 2186982, DOI 10.1090/S0002-9947-05-04096-1
- M. Bordemann, M. Forger, and H. Römer, Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models, Comm. Math. Phys. 102 (1986), no. 4, 605–617. MR 824094, DOI 10.1007/BF01221650
- A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2
- Evandro C. F. dos Santos and Caio J. C. Negreiros, Einstein metrics on flag manifolds, Rev. Un. Mat. Argentina 47 (2006), no. 2, 77–84 (2007). MR 2301378
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Masahiro Kimura, Homogeneous Einstein metrics on certain Kähler $C$-spaces, Recent topics in differential and analytic geometry, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 303–320. MR 1145261, DOI 10.2969/aspm/01810303
- J. L. Koszul, Sur la forme hermitienne canonique des espaces homogènes complexes, Canadian J. Math. 7 (1955), 562–576 (French). MR 77879, DOI 10.4153/CJM-1955-061-3
- Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskiĭ, Geometry of homogeneous Riemannian manifolds, Sovrem. Mat. Prilozh. 37, Geometriya (2006), 101–178 (Russian); English transl., J. Math. Sci. (N.Y.) 146 (2007), no. 6, 6313–6390. MR 2568572, DOI 10.1007/s10958-007-0472-z
- Joon-Sik Park and Yusuke Sakane, Invariant Einstein metrics on certain homogeneous spaces, Tokyo J. Math. 20 (1997), no. 1, 51–61. MR 1451858, DOI 10.3836/tjm/1270042398
- Y. Sakane, Homogeneous Einstein metrics on flag manifolds, Lobachevskii J. Math. 4 (1999), 71–87. Towards 100 years after Sophus Lie (Kazan, 1998). MR 1743146
- Masaru Takeuchi, Homogeneous Kähler submanifolds in complex projective spaces, Japan. J. Math. (N.S.) 4 (1978), no. 1, 171–219. MR 528871, DOI 10.4099/math1924.4.171
- Patrice Tauvel and Rupert W. T. Yu, Lie algebras and algebraic groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2146652, DOI 10.1007/b139060
- McKenzie Y. Wang and Wolfgang Ziller, On normal homogeneous Einstein manifolds, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 563–633. MR 839687, DOI 10.24033/asens.1497
- McKenzie Y. Wang and Wolfgang Ziller, Existence and nonexistence of homogeneous Einstein metrics, Invent. Math. 84 (1986), no. 1, 177–194. MR 830044, DOI 10.1007/BF01388738
Additional Information
- Andreas Arvanitoyeorgos
- Affiliation: Department of Mathematics, University of Patras, GR-26500 Rion, Greece
- MR Author ID: 307519
- Email: arvanito@math.upatras.gr
- Ioannis Chrysikos
- Affiliation: Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 611 37 Brno, Czech Republic
- Email: xrysikos@master.math.upatras.gr
- Yusuke Sakane
- Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
- Email: sakane@math.sci.osaka-u.ac.jp
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 2485-2499
- MSC (2010): Primary 53C25; Secondary 53C30
- DOI: https://doi.org/10.1090/S0002-9939-2013-11682-5
- MathSciNet review: 3043029