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Complex paraconformal manifolds - their differential geometry and twistor theory. (English) Zbl 0728.53005

The authors propose the notion of a complex paraconformal \(\{\) q,p\(\}\) structure on a complex manifold which involves the factorization of its holomorphic tangent bundle as a tensor product of a pair of bundles of ranks q and p for q,p\(\geq 2\). Such manifolds include as special cases 4- dimensional conformal manifolds having spin-structure, as well as complexified quaternionic, quaternionic Kähler and hyper-Kähler manifolds. The differential-geometric properties of complex paraconformal manifolds are discussed and formally they are very similar to those of 4- dimensional manifolds having a conformal structure. Examples are cited to show that the theory has a rich twistor structure which include applications to Penrose’s notion of a non-linear graviton. Contents: an introduction, paraconformal structures, the flat model, real structures, twistor spaces for paraconformal structures, local twistors and quaternionic curves, quaternionic Kähler metrics and Einstein scales, the Penrose transform, twistor description of Einstein scales and compatible metrics, and an appendix.

MSC:

53A30 Conformal differential geometry (MSC2010)
32L25 Twistor theory, double fibrations (complex-analytic aspects)
53C56 Other complex differential geometry
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