Two-forms on four-manifolds and elliptic equations

Let V be a four-dimensional real vector space with a fixed orientation. Then the wedge product can be viewed, up to a positive factor, as a canonical quadratic form of signature (3, 3) on the six-dimensional space ΛV . This gives a homomorphism from the identity component of the general linear group GL(V ) to the conformal group of the indefinite form, which is a local isomorphism. The significance of this is that geometrical structures on V can be expressed in terms of the six-dimensional space ΛV with its quadratic form. Now if X is an oriented 4-manifold we can apply this idea to the cotangent spaces of X . Many important differential geometric structures on X can fruitfully be expressed in terms of the bundle of 2-forms, with its quadratic wedge product and exterior derivative. We recall some examples