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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11682-5
PII: S 0002-9939(2013)11682-5
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