Recent Advances in the Theory of Holonomy

Abstract

This article is a report on the status of the problem of classifying the irriducibly acting subgroups of GL(n,R) that can appear as the holonomy of a torsion-free affine connection. In particular, it contains an account of the completion of the classification of these groups by Chi, Merkulov, and Schwachhofer as well as of the exterior differential systems analysis that shows that all of these groups do, in fact, occur. Some discussion of the results of Joyce on the existence of compact examples with holonomy G_2 or Spin(7) is also included.

24 pages, plain tex with amssym.tex and amssym.def. To appear in Asterisque. This is the text of a report to the Seminaire Bourbaki in June 1999. Amended to include the new exotic symplectic example of Spin(6,H) in GL(32,R)

Cited by in corpus (10)

• On the geometry of Riemannian manifolds with density
• Holonomy of supermanifolds
• Irreducible complex skew-Berger algebras
• Notes on exterior differential systems
• Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2
• Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems
• Tangent Lie algebra of a diffeomorphism group and application to holonomy theory
• The uniqueness in the de Rham-Wu decomposition
• Holonomy groups of stable vector bundles
• Legendrian Submanifold Path Geometry