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Transactions of the American Mathematical Society

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Some New Homogeneous Einstein Metrics
on Symmetric Spaces


Author: Megan M. Kerr
Journal: Trans. Amer. Math. Soc. 348 (1996), 153-171
MSC (1991): Primary 53C25; Secondary 53C30, 53C35
DOI: https://doi.org/10.1090/S0002-9947-96-01512-7
MathSciNet review: 1327258
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Abstract | References | Similar Articles | Additional Information

Abstract: We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank$(M)\!> 1$, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup $G$ of Isom$(M)$ acting transitively on $M$ we find all $G$-homogeneous (non-symmetric) Einstein metrics on $M$.


References [Enhancements On Off] (What's this?)

  • [A] A. Arvanitoyeorgos, New Invariant Einstein Metrics on Generalized Flag Manifolds, Transactions of the Amer. Math. Soc., 337, (1993), 981--995. MR 93k:53043
  • [Ber] M. Berger, Quelques formules de variation pour une structure Riemannienne, Ann. Sci. Ec. Norm. Super. 3, $4^e$ serie, (1970), 285--294.MR 43:3969
  • [Bes] A. Besse, Einstein Manifolds, Springer, 1987. MR 88f:53087
  • [DA-Z] J. E. D'Atri and W. Ziller, Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups, Memoirs of the Amer. Math. Soc., 18, 215 (1979).MR 80i:53023
  • [H] D. Hilbert, Die Grundlagen der Physik, Nachr. Akad. Wiss. Gött., (1915), 395--407.
  • [Je1] G. Jensen, The Scalar Curvature of Left Invariant Riemannian Metrics, Indiana Univ. Math. J., 20, (1970/71), 1125--1144. MR 44:6914
  • [Je2] G. Jensen, Einstein Metrics on Principal Fibre Bundles, J. Diff. Geom., 8, (1973), 599--614.MR 50:5694
  • [K] M. Kimura, Homogeneous Einstein Metrics on Certain Kähler $C$-Spaces, Adv. Studies in Pure Math, 18-I, (1990), 303--320. MR 93b:53039
  • [M] S. Murakami, Exceptional Simple Lie Groups and Related Topics in Recent Differential Geometry, Diff. Geom. and Topol. Proceedings, (Tianjin, 1986--87), vol. 1369, Springer, 1989. MR 90g:22009
  • [O1] A. L. Oniscik, Inclusion Relations Among Transitive Compact Transformation Groups, Am. Math. Soc. Transl., 50, (1966), 5--58. MR 27:3740
  • [O2] A. L. Oniscik, Transitive Compact Transformation Groups, Am. Math. Soc. Transl., 55, (1966), 153--194. MR 27:5868
  • [O3] A. L. Oniscik, Lie Groups Transitive on Grassmann and Stiefel Manifolds, Math. USSR, Sb., 12, (1970), 405--427. MR 43:414
  • [S] A. Shchetinin, On a class of compact homogeneous spaces I, Ann. Global Anal. Geom., 2, (1988), 119--140. MR 90d:57049
  • [T] E. Tsukada, Transitive actions of compact connected Lie groups on symmetric spaces, Sci. Rep. Niigata Univ., 15, (1978), 1--13.MR 57:13997
  • [W-Z] M. Y. Wang and W. Ziller, Existence and Nonexistence of Homogeneous Einstein Metrics, Invent. Math. 84, (1986), 177--194.MR 87e:53081
  • [W] A. J. Wolf, Spaces of Constant Curvature, $5^{th}$ ed., Publish or Perish, Inc., 1984.
  • [Z] W. Ziller, Homogeneous Einstein Metrics on Spheres and Projective Spaces, Math. Ann. 259, (1982), 351--358. MR 84h:53062

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Additional Information

Megan M. Kerr
Affiliation: Department of Mathematics, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, Pennsylvania 19104-6395
Address at time of publication: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Email: megan@math.upenn.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01512-7
Received by editor(s): August 29, 1994
Article copyright: © Copyright 1996 American Mathematical Society

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