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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

Modulus Space Riemannian Manifold Natural Kind Twistor Space Holonomy Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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