We consider a hyperkähler 4-manifold M. Using local holomorphic Darboux coordinates with respect to a compatible complex structure I on M, we find local necessary and sufficient conditions for a real smooth vector field X on M to be quaternionic Killing. We then apply this result to the case of a hyperkähler manifold M admitting n commuting quaternionic Killing fields, X1,..., Xn, the first n-1 of which are further assumed to be triholomorphic and quaternionically linearly independent pointwise. We then have two cases: if the self-dual part of DX n vanishes, we get back the Hitchin-Karlhede-Lindström-Roĉek result, and if the self-dual part of DXn is non-zero, we obtain a partial generalization of the Boyer and Finley equation.
|School:||State University of New York at Stony Brook|
|School Location:||United States -- New York|
|Source:||DAI-B 73/12(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics|
|Keywords:||Differential geometry, Hyperkahler, Quaternionic killing fields, Symmetry|
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