2 SELMAN AKBULUT AND SEMA SALUR Alternatively

Here we study the deformations of associative submanifolds inside a G 2 manifold M 7 with a calibration 3-form ϕ. A choice of 2-plane field Λ on M (which always exits) 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 G 2 manifolds are proved by using only the cross product operation (equivalently ϕ). We feel that mixing various different local identifications of the rich G 2 geometry (e.g. cross product, representation theory and the algebra of octonions) makes the study of G 2 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. We first review the basic results about G 2 manifolds, along the way we give a self contained proof of the McLean's theorem and its generalization [M], [AS1]. A G 2 manifold (M, ϕ, Λ) with an oriented 2-plane field gives various complex structures on some of subbundles of T (M). This imposes interesting structures on the deformation theory of its associative submanifolds. By using this we relate them to the Seiberg-Witten type equations.