Bryant, Robert 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]. Reviewer: Serguey M.Pokas (Odessa) Cited in 8 Documents 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 Keywords:Riemannian holonomy; exterior differential systems; torsion-free non-metric connections; holonomy group; \(H\)-structures; pseudo-Riemannian irreducible holonomies in \(\mathbb R^n\); exotic conformal holonomies; ‘classical’ non-metric irreducible holonomi PDFBibTeX XMLCite \textit{R. Bryant}, in: Séminaire Bourbaki. Volume 1998/99. Exposés 850--864. Paris: Société Mathé\-ma\-ti\-que de France. 351--374, Exp. No. 861 (2000; Zbl 1014.53029) Full Text: arXiv Numdam EuDML