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Hypercomplex manifolds and the space of framings. (English) Zbl 0905.53023

Huggett, S. A. (ed.) et al., The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 9-30 (1998).
A. Ashtekar, T. Jacobson and L. Smolin [Commun. Math. Phys. 115, 631-648 (1988; Zbl 0642.53079)] have shown how the Einstein equations may be reduced to equations for a positive-definite self-dual spacetime (a 4-dimensional hyperkähler manifold) which are equivalent to Nahm’s equations for a triple of volume-preserving vector fields on a 3-manifold. In the present paper, the author seeks to relax the volume-preserving condition and recast the result in a general setting.
Contents include: introduction; framings; \(SU(2)\)-invariance; hypercomplex manifolds; twistor spaces and isomonodromic deformations; holonomy and hypergeometric functions.
For the entire collection see [Zbl 0890.00046].

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
32L25 Twistor theory, double fibrations (complex-analytic aspects)

Citations:

Zbl 0642.53079
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