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Almost Hermitian symmetric manifolds. I: Local twistor theory. (English) Zbl 0724.53019
Roughly speaking, an almost Hermitian symmetric (AHS) manifold is a complex manifold whose holomorphic tangent bundle admits a structure group isomorphic to that of the tangent bundle of a Hermitian symmetric space. This paper contains a theory of the general class of AHS structures, which include conformal, projective, quaternionic and paraconformal structures. The author studies here the particular case of almost Lagrangian structures. Then a local twistor theory for AMS manifolds is developed in order to obtain local structural invariants and to show that the vanishing of these invariants is equivalent with the flatness of AHS, i.e. locally isomorphic to a Hermitian symmetric space. The main results are proved by means of Spencer cohomology. [Part II, see the following review.]
Reviewer: C.-L.Bejan (Iaşi)

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Citations:
Zbl 0724.53020
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