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Recent advances in the theory of holonomy. (English) Zbl 1014.53029

Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque. 266, 351-374, Exp. No. 861 (2000).
This survey expository paper provides a valuable account of recent advances in the study of holonomy. Here the reader will find the historical remarks, information on the holonomy group, \(H\)-structures and torsion, criteria for holonomy in the torsion-free case, classification, pseudo-Riemannian irreducible holonomy in \(\mathbb R^n\), local existence, explicit construction, ‘classical’ non-metric irreducible holonomies, exotic conformal holonomies, exotic symplectic holonomies, twistor methods, exterior differential systems, Poisson construction, Kähler manifolds, special Kähler manifolds, hyper-Kähler manifolds, quaternion Kähler manifolds, \(G_2\) and \(\text{Spin}(7)\) manifolds. Both the Riemannian holonomy (de Rham splitting theorem, non-closed holonomy, compact manifolds with exceptional holonomy, fundamental 3-form) and irreducible case of torsion-free non-metric connections (twistor constructions, Merkulov’s generalization, Poisson constructions, algebraic classification, two leftover cases) are considered in detail. Extensive references are added.
For the entire collection see [Zbl 0939.00019].

MSC:

53C29 Issues of holonomy in differential geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58A15 Exterior differential systems (Cartan theory)
53C10 \(G\)-structures
53C28 Twistor methods in differential geometry
53A17 Differential geometric aspects in kinematics
53B05 Linear and affine connections
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