The Work of Peter B. Kronheimer

  • Simon Donaldson

Abstract

Kronheimer’s work in this period was mainly concerned with the subject of “hyperkahler geometry”. Recall that a Kahler manifold is a Riemannian manifold with a complex structure I on its tangent spaces which is preserved by the parallel transport of the Levi-Civita connection: equivalently, it is a manifold whose holonomy group reduces to the unitary group. A hyperkahler manifold is a Riemannian manifold with a triple of complex structures I, J, K, satisfying the algebraic identities of the quaternions, each of which yields a Kahler structure. Equivalently, it is a manifold whose holonomy group reduces to the compact symplectic group. A hyperkahler structure gives a natural kind of “quaternionic manifold” and they are also very interesting from the point of view of Riemannian geometry: in any dimension they have vanishing Ricci tensor, and in dimension 4 they can be characterised, locally, as metrics which are simultaneously Ricci-flat and “self-dual”.

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. B. Kronheimer: The construction of ALE spaces as hyperkahler quotients. J. Diff. Geom. 29, 665–685 (1989)MATHMathSciNetGoogle Scholar
  2. 2.
    P. B. Kronheimer: A Torelli-type theorem for gravitational instantons. J. Diff. Geom. 29, 685–699 (1989)MATHMathSciNetGoogle Scholar
  3. 3.
    P. B. Kronheimer: lnstantons and the geometry of the nilpotent variety. J. Diff. Geom. 32, 473–490 (1990)MATHMathSciNetGoogle Scholar
  4. 4.
    P. B. Kronheimer: A Hyper Kahlerian structure on co-adjoint orbits of a semi-simple complex Lie group. J. Math. Soc. (London) 42, 193–208 (1990)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Simon Donaldson

There are no affiliations available

Personalised recommendations