Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Low-dimensional homogeneous Einstein manifolds

Author(s): Christoph Böhm; Megan M. Kerr
Journal: Trans. Amer. Math. Soc. 358 (2006), 1455-1468.
MSC (2000): Primary 53C30; Secondary 53C25
Posted: November 18, 2005
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We show that compact, simply connected homogeneous spaces up to dimension $ 11$ admit homogeneous Einstein metrics.

References:

[Ad]
J. F. Adams: Lectures on Exceptional Lie Groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, (1996) MR 1428422 (98b:22001)

[ADF]
D. V. Alekseevsky, I. Dotti and C. Ferraris: Homogeneous Ricci positive 5-manifolds, Pacific J. Math. 175, 1-12, (1996) MR 1419469 (97k:53049)

[AlKi]
D. V. Alekseevsky, B. N. Kimel'fel'd: Structure of homogeneous Riemann spaces with zero Ricci curvature, Func. Anal. Appl. 9, 97-102, (1975) MR 0402650 (53:6466)

[Ar]
A. Arvanitoyeorgos: New invariant Einstein metrics on generalized flag manifolds, Trans. Amer. Math. Soc. 337, 981-995, (1993) MR 1097162 (93h:53043)

[AlWa]
S. Aloff, N. Wallach: An infinite family of distinct $ 7$-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81, 93-97, (1975) MR 0370624 (51:6851)

[BaHs]
A. Back, W.-Y. Hsiang: Equivariant geometry and Kervaire spheres, Transactions Amer. Math. Soc. 304, 207-227 (1987) MR 0906813 (88m:53073)

[Bes]
A. L. Besse: Einstein Manifolds, Springer-Verlag, (1987) MR 0867684 (88f:53087)

[Bo]
S. Bochner: Curvature and Betti numbers, Ann. of Math. 49, 379-390, (1948) MR 0025238 (9:618d)

[Bö]
C. Böhm: Homogeneous Einstein metrics and simplicial complexes, J. Diff. Geom. 67, 79-165, (2004) MR 2153482

[BWZ]
C. Böhm, M. Wang, W. Ziller: A variational approach for homogeneous Einstein metrics, GAFA 14, 681-733, (2004) MR 2084976 (2005g:53074)

[BoGa]
C. P. Boyer, K. Galicki: On Sasakian-Einstein geometry, Internat. J. Math. 11, 873-909, (2000) MR 1792957 (2001k:53081)

[BGMR]
C. P. Boyer, K. Galicki, B. M. Mann, E. G. Rees: Compact $ 3$-Sasakian $ 7$-manifolds with arbitrary second Betti number, Invent. Math. 131, 321-344, (1998) MR 1608567 (99b:53066)

[CDF]
L. Castellani, R. D'Auria, P. Fré: $ SU(3)\times SU(2) \times U(1)$ from $ D=11$ supergravity, Nuclear Physics 239 B, 610-652, (1984) MR 0749348 (86e:83052a)

[CRW]
L. Castellani, L. J. Romans and N. P. Warner: A classification of compactifying solutions for $ d=11$ supergravity, Nuclear Physics 241 B, 429-262, (1984) MR 0757859 (86b:83039)

[DN]
J. E. D'Atri, N. K. Nickerson: Divergence-preserving geodesic symmetry, J. Diff. Geom. 9, 251-262, (1979)

[DFN]
R. D'Auria, P. Fré, P. van Nieuwenhuisen: $ N=2$ matter coupled supergravity from compactification on a coset $ G/H$ possessing an additional Killing vector, Phys. Lett. B 136, 347-353, (1984) MR 0735731 (85d:83080)

[DK1]
W. Dickinson, M. M. Kerr: The geometry of compact homogeneous spaces with two isotropy summands, preprint (2002)

[DK2]
W. Dickinson, M. M. Kerr: Einstein metrics on compact homogeneous spaces, preprint (2002)

[Dy1]
E. B. Dynkin: Maximal subalgebras of the classical groups, Transl. Am. Math. Soc. Series 2, Vol. 6, 245-378, (1957) MR 0049903 (14:244d)

[Dy2]
E. B. Dynkin: Semisimple subalgebras of semisimple Lie algebras, Transl. Am. Math. Soc. Series 2, Vol. 6, 111-244, (1957) MR 0047629 (13:904c)

[He]
J. Heber: Noncompact homogeneous Einstein spaces, Invent. Math. 133, 279-352, (1998) MR 1632782 (99d:53046)

[Hi]
D. Hilbert: Die Grundlagen der Physik, Nachr. Akad. Wiss. Gött., 395-407, (1915)

[Je1]
G. Jensen: Homogeneous Einstein spaces of dimension 4, J. Diff. Geom. 3, 309-349, (1969) MR 0261487 (41:6100)

[Je2]
G. Jensen: The scalar curvature of left invariant Riemannian metrics, Indiana U. Math. J. 20, 1125-1144, (1971) MR 0289726 (44:6914)

[Je3]
G. Jensen: Einstein metrics on principal fibre bundles, J. Diff. Geom. 8, 599-614 (1973) MR 0353209 (50:5694)

[Jo]
D. Joyce: Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press (2000) MR 1787733 (2001k:53093)

[Ke]
M. M. Kerr: Some new homogeneous Einstein metrics on symmetric spaces, Trans. Amer. Math. Soc. 348, 153-172, (1996) MR 1327258 (96f:53064)

[Ki]
M. Kimura: Homogeneous Einstein Metrics On Certain Kähler $ C$-Spaces, Adv. Stud. Pure Math., 18-I, Recent topics in differential and analytic geometry, 303-320, (1990) MR 1145261 (93b:53039)

[Kl]
S. Klaus: Einfach-zusammenhängende Kompakte Homogene Räume bis zur Dimension Neun, Diplomarbeit, Johannes Gutenberg Universität (1988)

[Ko]
S. Kobayashi: Topology of positively pinched Kähler manifolds, Tohoku Math. J. 15, 121-139, (1963) MR 0154235 (27:4185)

[Kosz]
J. L. Koszul: Sur la forme hermitienne canonique des espaces homogènes complexes, Can. J. of Math. 7, 562-576, (1955) MR 0077879 (17:1109a)

[KS1]
M. Kreck, S. Stolz: A diffeomorphism classification of $ 7$-dimensional homogeneous Einstein manifolds with $ {\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1)$-symmetry, Ann. of Math. 127, 373-388, (1988) MR 0932303 (89c:57042)

[KS2]
M. Kreck, S. Stolz: Some nondiffeomorphic homeomorphic homogeneous $ 7$-manifolds with positive sectional curvature, J. Diff. Geom. 33, 465-486, (1991); correction, J. Diff. Geom. 49, 203-204, (1998) MR 1094466 (92d:53043)

[Kr]
B. Kruggel: Quotienten halbeinfacher, kompakter Liescher Gruppen; Eine Klassifikation bis zur Dimension Zwölf, Diplomarbeit, Heinrich-Heine-Universität Düsseldorf (1993)

[LeWa]
C. LeBrun, M. Y. Wang: Surveys in differential geometry VI: essays on Einstein manifolds, eds. C. LeBrun, M. Y. Wang, International Press, (1999) MR 1798603 (2001f:53003)

[Mi]
J. Milnor: Construction of Universal Bundles, II, Ann. of Math. 63, 430-436, (1956) MR 0077932 (17:1120a)

[NiRo]
Yu. G. Nikonorov and E. D. Rodionov: Compact 6-dimensional homogeneous Einstein manifolds, Dokl. Math. RAS 336, 599-601, (1999) MR 1721419 (2000g:53054)

[Ni1]
Yu. G. Nikonorov: On one class of homogeneous compact Einstein manifolds, Siberian Math. J. 41, No. 1, 168-172, (2000) MR 1756487 (2001i:53075)

[Ni2]
Yu. G. Nikonorov: Compact homogeneous Einstein 7-manifolds, Geom. Dedicata 109, 7-30, (2004) MR 2113184

[On]
A. L. Onishchik: Topology of Transitive Transformation Groups, Barth, (1994)

[PP]
D. Page, C. Pope: New squashed solutions of $ d=11$ supergravity, Phys. Lett. 147B, 55-60, (1984) MR 0768319 (86h:83062)

[Sa]
Y. Sakane: Homogeneous Einstein Metrics on flag manifolds, Lobach. Jour. of Math. 4, 71-87, (1999) MR 1743146 (2000m:53069)

[Tia]
G. Tian: Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130, 1-37, (1997) MR 1471884 (99e:53065)

[Wa1]
M. Wang: Some examples of homogeneous Einstein manifolds in dimension seven, Duke Math. J. 49, 23-28, (1982) MR 0650366 (83k:53069)

[Wa2]
M. Wang: Einstein metrics and quaternionic Kähler manifolds, Math. Z. 210, no. 2, 305-325 (1992) MR 1166528 (93h:53046)

[WZ1]
M. Wang, W. Ziller: On normal homogeneous Einstein manifolds, Ann. Scient. Éc. Norm. Sup., $ 4^e$ série t. 18, 563-633, (1985) MR 0839687 (87k:53113)

[WZ2]
M. Wang, W. Ziller: Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84, 177-194, (1986) MR 0830044 (87e:53081)

[WZ3]
M. Wang, W. Ziller: Einstein metrics on principal torus bundles, J. Diff. Geom. 31, 215-248, (1990) MR 1030671 (91f:53041)

[Wo]
J. A. Wolf: The geometry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120, 59-148 (1968); correction, Acta Math. 152, 141-142 (1984). MR 0223501 (36:6549)

[Yau]
S. T. Yau: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31, 339-411, (1978) MR 0480350 (81d:53045)

[Zi]
W. Ziller: Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann. 259, 351-358, (1982) MR 0661203 (84h:53062)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C30, 53C25

Retrieve articles in all Journals with MSC (2000): 53C30, 53C25

Additional Information:

Christoph Böhm
Affiliation: Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: cboehm@math.uni-muenster.de

Megan M. Kerr
Affiliation: Department of Mathematics, Wellesley College, 106 Central St., Wellesley, Massachusetts 02481
Email: mkerr@wellesley.edu

DOI: 10.1090/S0002-9947-05-04096-1
PII: S 0002-9947(05)04096-1
Received by editor(s): December 17, 2003
Posted: November 18, 2005
Additional Notes: The second author was partially supported by the Radcliffe Institute for Advanced Study and by the Clare Boothe Luce Foundation.
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google