 
Summary: DEFORMATIONS IN G2 MANIFOLDS
SELMAN AKBULUT AND SEMA SALUR
Abstract. Here we study the deformations of associative submanifolds
inside a G2 manifold M7
with a calibration 3form . A choice of 2plane
field on M (which always exists) splits the tangent bundle of M as a direct
sum of a 3dimensional associate bundle and a complex 4plane bundle
T M = E V, and this helps us to relate the deformations to Seiberg
Witten type equations. Here all the surveyed results as well as the new ones
about G2 manifolds are proved by using only the cross product operation
(equivalently ). We feel that mixing various different local identifications
of the rich G2 geometry (e.g. cross product, representation theory and the
algebra of octonions) makes the study of G2 manifolds looks harder then it
is (e.g. the proof of McLean's theorem [M]). We believe the approach here
makes things easier and keeps the presentation elementary. This paper is
essentially selfcontained.
1. G2 manifolds
We first review the basic results about G2 manifolds, along the way we give
a selfcontained proof of the McLean's theorem and its generalization [M],
[AS1]. A G2 manifold (M, , ) with an oriented 2plane field gives various
