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Mathematics > Differential Geometry

Title: 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.
Comments: 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)
Subjects: Differential Geometry (math.DG)
MSC classes: 53C10 (Primary), 53B05 (Secondary)
Journal reference: Seminaire Bourbaki, Volume 1998/99, Asterisque 266 (2000), 351-374
Cite as: arXiv:math/9910059 [math.DG]
  (or arXiv:math/9910059v2 [math.DG] for this version)

Submission history

From: Robert L. Bryant [view email]
[v1] Mon, 11 Oct 1999 19:58:44 GMT (24kb)
[v2] Tue, 12 Oct 1999 15:38:25 GMT (24kb)