Skip to main content
SpringerLink
Log in

HyperKähler and quaternionic Kähler geometry

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Alekseevskiî, D.V.: Riemannian manifolds with exceptional holonomy groups Funkts. Anal. Prilozh.2(2), 1–10 (1968) [Engl. transl.: Funct. Anal. Appl.2, 97–105 (1968)]

    Google Scholar 

  2. Alekseevskiî, D.V.: Compact quaternion spaces. Funkts. Anal. Prilozh.2(2), 11–20 (1968) [Engl. transl.: Funct. Anal. Appl.2, 106–114 (1968)]

    Google Scholar 

  3. Alekseevskiî, D.V.: Quaternion Riemannian spaces with transitive reductive or solvable group of motions. Funkts. Anal. Prilozh.4 (4), 68–69 (1970) [Engl. transl.: Funct. Anal. Appl.4, 321–322 (1970)]

    Google Scholar 

  4. Alekseevskiî, D.V.: Classification of quaternionic spaces with a transitive solvable group of motions. Izv. Akad. Nauk SSSR Ser. Mat.9 (2), 315–362 (1975) [Engl. transl.: Math. USSR, Izv.9, 297–339 (1975)]

    Google Scholar 

  5. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond., Ser. A362, 425–461 (1978)

    Google Scholar 

  6. Atiyah, M.F., Drinfeld, D.G., Hitchin, N.J., Manin, Y.I.: Construction of instantons. Phys. Lett. A65, 185–187 (1978)

    Google Scholar 

  7. Atiyah, M.F., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles. New Jersey: Princeton University Press 1988

    Google Scholar 

  8. Bagger, J., Witten, E.: Matter couplings inn=2 supergravity. Nucl. Phys. B222, 1–10 (1983)

    Google Scholar 

  9. Besse, A.L.: Einstein manifolds. Erg. Math. Grenzgeb. Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  10. Brinkman, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann.94, 119–145 (1925)

    Google Scholar 

  11. Bryant, R.L., Salamon, S.M.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J.58, 829–850 (1989)

    Google Scholar 

  12. Calabi, E.: Métriques kählŕiennes et fibrés holomorphes. Ann. Sci. Éc. Norm. Supér., IV. Sér.12, 269–294 (1979)

    Google Scholar 

  13. Calabi, E.: Isometric families of Kähler structures. In: The Chern Symposium, 1979, Hsiang, W.-Y., et al. (eds.), pp. 23–39. Springer 1980

  14. Cordero, L.A., Fernández, M., Gray, A.: Variétés symplectiques sans structures kählériennes. C.R. Acad. Sci., Paris, Sér. I301, 217–218 (1985)

    Google Scholar 

  15. Cordero, L.A., Fernández, M., de Léon, M.: Examples of compact non-Kähler almost Kähler manifolds. Proc. Am. Math. Soc.95, 280–286 (1985)

    Google Scholar 

  16. Cordero, L.A., Fernández, M., Gray, A.: Symplectic manifolds with no Kähler structure. Topology25, 375–380 (1986)

    Google Scholar 

  17. Fernández, M., Gray, A.: The Iwasawa manifold. In: Differential geometry, Peñíscola 1985, Proceedings, Naveira, A., Ferrandez, A., Mascaro, F. (eds.) (Lect. Notes Math., vol. 1209). Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  18. Galicki, K.: Quaternionic Kähler and hyperKähler non-linear σ-models. Nucl. Phys. B271, 402–416 (1986)

    Google Scholar 

  19. Galicki, K.: A generalization of the momentum mapping construction for quaternionic Kähler manifolds. Commun. Math. Phys.108, 117–138 (1987)

    Google Scholar 

  20. Galicki, K., Lawson, H.B.: Quaternionic reduction and quaternionic orbifolds. Math. Ann.282, 1–21 (1988)

    Google Scholar 

  21. Gibbons, G.W., Pope, C.N.: The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys.66, 267–290 (1979)

    Google Scholar 

  22. Gray, A.: A note on manifolds whose holonomy group is a subgroup ofSp(n)·Sp(1). Mich. Math. J.16, 125–128 (1969)

    Google Scholar 

  23. Gudmundsson, S.: On the geometry of harmonic morphisms. Preprint

  24. Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press 1984

    Google Scholar 

  25. Hitchin, N.J.: Metrics on moduli spaces. In: Lefschetz Centennial Conference. Part I: Proceedings on algebraic geometry, Mexico City, 1984. Sundararaman, D. (ed.) (Contemp. Math., vol. 58, pp. 157–178). AMS, Providence, R.I. (1986)

    Google Scholar 

  26. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc.55, 59–126 (1987)

    Google Scholar 

  27. Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: HyperKähler metrics and supersymmetry. Commun. Math. Phys.108, 535–589 (1987)

    Google Scholar 

  28. Ishihara, S.: Quaternion Kählerian manifolds. J. Differ. Geom.9, 483–500 (1974)

    Google Scholar 

  29. Jensen, G.: Einstein metrics in principal fibre bundles. J. Differ. Geom.8, 599–614 (1973)

    Google Scholar 

  30. Kobayashi, S.: Remarks on complex contact manifolds. Proc. Am. Math. Soc.10, 164–167 (1959)

    Google Scholar 

  31. Kobayashi, S., Nomizu, K.: Foundations of differential geometry. 2 volumes. New York: Wiley 1963

    Google Scholar 

  32. Kronheimer, P.B.: The construction of ALE spaces as hyperKähler quotients. J. Differ. Geom.29, 665–683 (1989)

    Google Scholar 

  33. Kronheimer, P.B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom.29, 685–697 (1989)

    Google Scholar 

  34. Kronheimer, P.B.: Instantons and the geometry of the nilpotent variety. J. Differ. Geom.32, 473–490 (1990)

    Google Scholar 

  35. Lang, S.: Algebra. Second edition. Reading, Mass.: Addison-Wesley 1984

    Google Scholar 

  36. LeBrun, C.: Spaces of complex null geodesics in complex-Riemannian geometry. Trans. Am. Math. Soc.278, 209–231 (1983)

    Google Scholar 

  37. McDuff, D.: Examples of simply-connected symplectic non-Kählerian manifolds. J. Differ. Geom.20, 267–277 (1984)

    Google Scholar 

  38. Maciocia, A.: Metrics on the moduli spaces of instantons over Euclidean 4-space. Commun. Math. Phys.135, 467–482 (1991)

    Google Scholar 

  39. Marchiafava, S., Romani, G.: Sui fibrati construttura quaternionale generalizzata. Ann. Mat. Pura Appl., IV. Ser.107, 131–157 (1976)

    Google Scholar 

  40. Marchiafava, S.: Su alouna sottovarietà che ha interesse considerare i una varietà Kahleriana quaternionale. Preprint

  41. Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121–130 (1974)

    Google Scholar 

  42. Obata, M.: Affine connections on manifolds with almost complex, quaternion or Hermititian structure. Jap. J. Math.26, 43–73 (1956)

    Google Scholar 

  43. O'Neill, B.: The fundamental equations of a submersion. Mich. Math. J.13, 459–469 (1966)

    Google Scholar 

  44. Page, D.N., Pope, C.N.: Einstein metrics on quaternionic line bundles. Classical Quantum Gravity3, 249–259 (1986)

    Google Scholar 

  45. Salamon, S.M.: Quaternionic Kähler manifolds. Invent. Math.67, 143–171 (1982)

    Google Scholar 

  46. Salamon, S.M.: Differential geometry of quaternionic manifolds. Ann. Sci. Éc. Norm. Supér., IV. Sér.19, 31–55 (1986)

    Google Scholar 

  47. Swann, A.F.: Aspects symplectiques de la géométrie quaternionique. C.R. Acad. Sci., Paris, Sér. I308, 225–228 (1989)

    Google Scholar 

  48. Swann, A.F.: Some quaternionically equivalent Einstein metrics. Twistor Newsl.30, 18–20 (1990)

    Google Scholar 

  49. Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc.55, 467–468 (1976)

    Google Scholar 

  50. Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech.14, 1033–1047 (1965)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK-5230, Odense 14, Denmark

    Andrew Swann

Authors
  1. Andrew Swann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Swann, A. HyperKähler and quaternionic Kähler geometry. Math. Ann. 289, 421–450 (1991). https://doi.org/10.1007/BF01446581

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01446581

Search

Navigation