Einstein metrics on compact Lie groups which are not naturally reductive

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• 作者：Andreas Arvanitoyeorgos (1) arvanito@math.upatras.gr
  Kunihiko Mori (2)
  Yusuke Sakane (3) sakane@math.sci.osaka-u.ac.jp
• 关键词：Einstein metrics – ; Homogeneous spaces – ; Naturally reductive metrics – ; K盲hler C ; spaces
• 刊名：Geometriae Dedicata
• 出版年：2012
• 出版时间：October 2012
• 年：2012
• 卷：160
• 期：1
• 页码：261-285
• 全文大小：310.5 KB
• 参考文献：1. Arvanitoyeorgos A., Chrysikos I.: Motion of charged particles and homogeneous geodesics in K盲hler C-spaces with two isotropy summands. Tokyo J. Math. 32(2), 487–500 (2009)
  2. Besse A.L.: Einstein Manifolds. Springer, Berlin (1987)
  3. B枚hm C.: Homogeneous Einstein metrics and simplicial complexes. J. Differ. Geom. 67, 79–165 (2004)
  4. B枚hm C., Kerr M.M.: Low-dimensional homogeneous Einstein manifolds. Trans. Am. Math. Soc. 358(4), 1455–1468 (2006)
  5. B枚hm C., Wang M., Ziller W.: A variational approach for homogeneous Einstein metrics. GAFA 14, 681–733 (2004)
  6. Borel A., Hirzebruch F.: Characteristic classes and homogeneous spaces I. Am. J. Math. 80, 458–538 (1958)
  7. Burstall F.E., Rawnsley J.H.: Twistor Theory for Riemannian Symmetric Spaces. Lecture Notes in Mathematics, vol. 1424. Springer, Heidelberg (1990)
  8. Dickinson W., Kerr M.M.: The geometry of compact homogeneous spaces with two isotropy summands. Ann. Global Anal. Geom. 34(4), 329–350 (2008)
  9. D’Atri, J.E., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Am. Math. Soc. 18 (No. 215), Rhode Island (1979)
  10. Jensen G.: The scalar curvature of left-invariant Riemannian metrics. Indiana Univ. Math. J. 20, 1125–1144 (1971)
  11. Kimura M.: Homogeneous Einstein metrics on certain K盲hler C-spaces. Recent Top. Differ. Anal. Geom. Adv. Stud. Pure Math. 18-I, 303–320 (1990)
  12. Mori, K.: Left Invariant Einstein Metrics on SU(N) that are Not Naturally Reductive. Master Thesis (in Japanese) Osaka University 1994, English translation Osaka University RPM 96—10 (preprint series) (1996)
  13. Nikonorov Y.G., Rodionov E.D., Slavskii V.V.: Geometry of homogeneous Riemannian manifolds. J. Math. Sci. 146(6), 6313–6390 (2007)
  14. Wang M., Ziller W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)
  15. Wolf J.A.: Spaces of Constant Curvature. Publish or Perish, Wilmington (1984)
• 作者单位：1. Department of Mathematics, University of Patras, 26500 Rion, Greece2. Saibi-Heisei Junior & Senior High School, 5-6-3, K奴k艒-d艒ri, Matsuyama, Ehime, 791-0054 Japan3. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-043 Japan
• ISSN：1572-9168

文摘

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D’Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E ₆, E ₇, and E ₈.