Publications [#9575] of Robert Bryant

Papers Published

1. Metrics with exceptional holonomy, Ann. of Math. (2), vol. 126 no. 3 (1987), pp. 525–576 [MR89b:53084]
   (last updated on 2004/04/06)

   Author's Comments:
   This is an exposition, with complete details, of the existence and generality of metrics with holonomy G₂ or Spin(7). In the last section, I also construct explicit examples (which, however, are not complete).

   There has been a great deal of progress on the holonomy problem in the intervening years. One source for further information is my 1999 article Recent advances in the theory of holonomy.

   As for errata and addenda to the article itself, I am only aware of two: First, McLean has pointed out that SO^*(2p) does not satisfy Berger's criteria and so should never have appeared on the list of possible holonomies in the first place. Second, on page 537, I make a couple of remarks about the ideal I in two special cases that are either misleading or wrong. I say that, in the Sp(n)Sp(1) case, the closure of the fundamental 4-form does not imply 1-flatness when n>1, but, in fact, the only cases where the 4-form is closed but the structure is not 1-flat happen when n=2. For all n>2, it's OK. I also say that the ideal I is not involutive when the group is Sp(n) and n>1. However, this is false. It is involutive for all n.