Math @ Duke

Publications [#243397] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, Metrics with exceptional holonomy,
Ann. of Math. (2), vol. 126 no. 3
(1987),
pp. 525576 [MR89b:53084]
(last updated on 2017/12/17)
Author's Comments: This is an exposition, with complete details, of the
existence and
generality of metrics with holonomy G_{2} 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
4form does not imply 1flatness when n>1, but, in fact,
the only cases
where the 4form is closed but the structure is not 1flat
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.


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