Abstract |
This article consists of loosely related
remarks about the geometry of G2-structures on
7-manifolds, some of which are based on unpublished
joint work with two other people: F. Reese Harvey and
Steven Altschuler.
After some preliminary background information about
the group G2 and its representation theory, a
set of techniques is introduced for calculating the
differential invariants of G2-structures and
the rest of the article is applications of these
results. Some of the results that may be of interest
are as follows:
First, a formula is derived for the scalar curvature
and Ricci curvature of a G2-structure in terms of
its torsion and covariant derivatives with respect to
the `natural connection' (as opposed to the Levi-Civita
connection) associated to a G2-structure. When
the fundamental 3-form of the G2-structure is
closed, this formula implies, in particular, that
the scalar curvature of the underlying metric is nonpositive
and vanishes if and only if the structure is torsion-free.
These formulae are also used to generalize a recent
result of Cleyton and Ivanov [3]
about the nonexistence of
closed Einstein G2-structures (other than the
Ricci-flat ones) on compact 7-manifolds to a nonexistence
result for closed G2-structures whose Ricci tensor
is too tightly pinched.
Second, some discussion is given of the geometry of
the first and second order invariants of G2-structures
in terms of the representation theory of G2.
Third, some formulae are derived for closed solutions
of the Laplacian flow that specify how various related
quantities, such as the torsion and the metric, evolve
with the flow. These may be useful in studying convergence
or long-time existence for given initial data.
Some of this work was subsumed in the work of
Hitchin [12] and Joyce [14].
I am making it available now mainly because of interest expressed
by others in seeing these results written up since they
do not seem to have all made it into the literature.
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