Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$S^1$-quotients of quaternion-Kähler manifolds


Author: Fiammetta Battaglia
Journal: Proc. Amer. Math. Soc. 124 (1996), 2185-2192
MSC (1991): Primary 53C25; Secondary 58F05
DOI: https://doi.org/10.1090/S0002-9939-96-03208-X
MathSciNet review: 1307492
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of symplectic reduction has been generalized to manifolds endowed with other structures, in particular to quaternion-Kähler manifolds, namely Riemannian manifolds with holonomy in $Sp(n)Sp(1)$. In this work we prove that the only complete quaternion-Kähler manifold with positive scalar curvature obtainable as a quaternion-Kähler quotient by a circle action is the complex Grassmannian $Gr_2(\mathbb {C} ^n)$.


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

  • 1. D.V. Alekseevskii: Riemannian manifolds with exceptional holonomy groups , Functional Anal. Appl. 2 (1968), 97--105; Compact quaternion spaces, Functional Anal. Appl. 2 (1968), 106--114. MR 37:6868; MR 37:6869
  • 2. D.V. Alekseevskii: Classification of quaternionic spaces with transitive solvable group of motions, Math. USSR--Izv. 9 (1975), 297--339. MR 53:6465
  • 3. F. Battaglia: Tesi di Dottorato, Università di Firenze, 1993.
  • 4. M. Berger: Sur les groupes d'holonomie des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279--330. MR 18:149a
  • 5. A. Besse: Einstein Manifold, Springer-Verlag, 1987. MR 88f:53087
  • 6. B. de Wit, A. van Proetyen: Special geometry, cubic polynomials and homogeneous space. Commun. Math. Phys. 149, (1992), 307-333. MR 94a:53079
  • 7. K. Galicki: A generalization of the momentum mapping construction for quaternionic Kähler manifolds, Commun. Math. Phys. 108 (1987) 117--138. MR 88f:53088
  • 8. K. Galicki: Multi-centre metrics with negative comological constant. Class. Quantum Grav. 8, (1991), 1529-1543. MR 92i:53040
  • 9. K. Galicki, H.B. Lawson: Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988), 1--21. MR 89m:53075
  • 10. T. Gocho, H. Nakajima: Einstein-Hermitian connections on hyperkähler quotients, J. Math. Soc. Japan, 44 (1992), 43--51. MR 92k:53079
  • 11. N.J. Hitchin, A. Karlhede, U. Lindström, M. Ro\v{c}ek: Hyperkähler metrics and supersimmetry. Commun. Math. Phys. 108 (1988), 535-589. MR 88g:53048
  • 12. D. Joyce: The hypercomplex quotient and the quaternionic quotient, Math. Ann. 290 (1991), 323--340. MR 92f:53052
  • 13. F.C. Kirwan: Cohomology of Quotients in Algebraic Geometry, Princeton University Press, 1984. MR 86i:58050
  • 14. S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, Vol. I and II, Wiley, New York, 1963. MR 27:2945
  • 15. B. Kostant: Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Amer. Math. Soc. 80 (1955), 528--542. MR 18:930a
  • 16. C.R. LeBrun: On the topology of quaternionic manifolds, Twistor Newsletter, 32 (1991), 6--7.
  • 17. C.R. LeBrun: On complete quaternionic-Kähler manifolds. Duke Math. J. 63, (1991), 723-743. MR 92i:53042
  • 18. C.R. LeBrun, S.M. Salamon : Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118, (1994), 109-132. CMP 94:16
  • 19. A. Lichnerowicz: Espaces homogenes Kähler iens, Colloque International de Geometrie Differentielle, Strasbourg, 1953, 171--184. MR 16:519c
  • 20. J. Marsden, A. Weinstein: Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121--130. MR 53:6633
  • 21. T. Nitta: Yang-Mills connections on quaternionic Kähler quotients, Proc. Japan Acad., 66 (1990), 245--247. MR 92g:53022
  • 22. Y.S. Poon, S. Salamon: Eight-dimensional quaternionic Kähler manifolds with positive scalar curvature, J. Diff. Geometry 33 (1991), 363--378. MR 92b:53071
  • 23. S.M. Salamon: Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143--171. MR 83k:53054
  • 24. S.M. Salamon: Differential geometry of quaternionic manifolds, Ann. scient. Éc. Norm. Sup. 19 (1986), 31--55. MR 87m:53079
  • 25. S.M. Salamon: Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics 201, Longman Scientific, 1989. MR 90g:53058
  • 26. A. Swann: Hyperkähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421--450. MR 92c:53030
  • 27. J.A. Wolf: Complex homogeneous contact structures and quaternionic symmetric spaces, J. Math. Mech. 14 (1965), 1033--1047. MR 32:3020

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C25, 58F05

Retrieve articles in all journals with MSC (1991): 53C25, 58F05


Additional Information

Fiammetta Battaglia
Affiliation: Dipartimento di Matematica Applicata G. Sansone via S. Marta 3 50139 Firenze Italy.
Email: fiamma@ingfi1.ing.unifi.it

DOI: https://doi.org/10.1090/S0002-9939-96-03208-X
Keywords: Quaternion-K\"ahler manifolds, moment map, reduction
Received by editor(s): April 5, 1994
Received by editor(s) in revised form: December 16, 1994
Communicated by: Christopher Croke
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society