New homogeneous Einstein metrics on SO(7)/T

Abstract

The authors compute non-zero structure constants of the full flag manifold M = SO(7)/T with nine isotropy summands, then construct the Einstein equations. With the help of computer they get all the forty-eight positive solutions (up to a scale ) for SO(7)/T, up to isometry there are only five G-invariant Einstein metrics, of which one is Kähler Einstein metric and four are non-Kähler Einstein metrics.

Keywords

Generalized flag manifold / Einstein metric / Ricci tensor / Isotropy representation / Isometry

References

[1] Arvanitoyeorgos A.. New invariant Einstein metrics on generalizd flag manifolds. Trans. Amer. Math. Soc., 1993, 337(2): 981-995

[2] Sakane Y.. Homogeneous Einstein metrics on flag manifold. Lobachevskii J. Math., 1999, 4: 71-87

[3] Dos Santos E. C. F., Negreiros C. J. C.. Einstein metrics on flag manifolds, Revista Della. Uni´on Mathem´atica Aregetina, 2006, 47(2): 77-84

[4] Alekseevsky D. V., Perelomov A. M.. Invariant K¨ahler-Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl., 1986, 20(3): 171-182

[5] Siebenthal J.. Sur certains modules dans une algébre de Lie semisimple. Comment. Math. Helv., 1964, 44(1): 1-44

[6] Alekseevsky D. V.. Homogeneous Einstein metrics. Differential Geometry and Its Applications (Brno Conference Proceedings), 1987 1-21

[7] Chrysikos I.. Flag manifolds, symmetric t-triples and Einstein metrics. Differential Geom. Appl., 2012, 30(6): 642-659

[8] Wang Y., Ziller W.. Existence and non-excistence of homogeneous Einstein metrics. Invent. Math., 1986, 84: 177-194

[9] Graev M. M.. On the number of invariant Eistein metrics on a compact homogeneous space, Newton polytopes and contraction of Lie algebras. Int. J. Geom. Methods Mod. Phys., 2006, 3(5–6): 1047-1075

[10] Arvanitoyeorgos A., Chrysikos I., Sakane Y.. Homogeneous Einstein metrics on G₂/T. Proc. Amer. Math. Soc., 2013, 14(7): 2485-2499

[11] Park J. S., Sakane Y.. Invariant Einstein metrics on certain homogeneous space. Tokyo J. Math., 1997, 20(1): 51-61

[12] Varadarajan V. S.. Lie Group, Lie Algebra and Their Representations, Springer-Verlag, New York, Berlin, 1984, Tokyo: Heidelberg

[13] Onishchik A. L., Vinberg E. B.. Lie Groups and Lie Algebras III, Structure of Lie Groups and Lie Algebras, 1994, Berlin, Heideberg: Springer-Verlag