Some New Homogeneous Einstein Metrics on Symmetric Spaces
HTML articles powered by AMS MathViewer
- by Megan M. Kerr PDF
- Trans. Amer. Math. Soc. 348 (1996), 153-171 Request permission
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
- V. A. Aleksandrov, An interconnection between the problem of unique determination of a domain in $\textbf {R}^n$ and the problem of reconstruction of a locally Euclidean metric, Sibirsk. Mat. Zh. 33 (1992), no. 4, 206–211, 224 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 4, 732–736 (1993). MR 1185450, DOI 10.1007/BF00971139
- Marcel Berger, Quelques formules de variation pour une structure riemannienne, Ann. Sci. École Norm. Sup. (4) 3 (1970), 285–294 (French). MR 278238, DOI 10.24033/asens.1194
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
- J. E. D’Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), no. 215, iii+72. MR 519928, DOI 10.1090/memo/0215
- D. Hilbert, Die Grundlagen der Physik, Nachr. Akad. Wiss. Gött., (1915), 395–407.
- Gary R. Jensen, The scalar curvature of left-invariant Riemannian metrics, Indiana Univ. Math. J. 20 (1970/71), 1125–1144. MR 289726, DOI 10.1512/iumj.1971.20.20104
- Gary R. Jensen, Einstein metrics on principal fibre bundles, J. Differential Geometry 8 (1973), 599–614. MR 353209
- 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
- Shingo Murakami, Exceptional simple Lie groups and related topics in recent differential geometry, Differential geometry and topology (Tianjin, 1986–87) Lecture Notes in Math., vol. 1369, Springer, Berlin, 1989, pp. 183–221. MR 1001187, DOI 10.1007/BFb0087534
- A. L. Oniščik, Inclusion relations between transitive compact transformation groups, Trudy Moskov. Mat. Obšč. 11 (1962), 199–242 (Russian). MR 0153779
- A. L. Oniščik, Transitive compact transformation groups, Mat. Sb. (N.S.) 60 (102) (1963), 447–485 (Russian). MR 0155935
- A. L. Oniščik, Lie groups that are transitive on Grassmann and Stiefel manifolds, Mat. Sb. (N.S.) 83 (125) (1970), 407–428 (Russian). MR 0274651
- Alexander Shchetinin, On a class of compact homogeneous spaces. I, Ann. Global Anal. Geom. 6 (1988), no. 2, 119–140. MR 982761, DOI 10.1007/BF00133035
- Etsuo Tsukada, Transitive actions of compact connected Lie groups on symmetric spaces, Sci. Rep. Niigata Univ. Ser. A 15 (1978), 1–13. MR 474351
- 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
- A. J. Wolf, Spaces of Constant Curvature, $5^{th}$ ed., Publish or Perish, Inc., 1984.
- W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann. 259 (1982), no. 3, 351–358. MR 661203, DOI 10.1007/BF01456947
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
- Received by editor(s): August 29, 1994
- © Copyright 1996 American Mathematical Society
- 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