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DEFORMATIONS IN G2 MANIFOLDS SELMAN AKBULUT AND SEMA SALUR
 

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 3-form . A choice of 2-plane
field on M (which always exists) splits the tangent bundle of M as a direct
sum of a 3-dimensional associate bundle and a complex 4-plane 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 self-contained.
1. G2 manifolds
We first review the basic results about G2 manifolds, along the way we give
a self-contained proof of the McLean's theorem and its generalization [M],
[AS1]. A G2 manifold (M, , ) with an oriented 2-plane field gives various

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics