0 Some Special Geometry in Dimension Six

We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution. Definitions Suppose M is a 6-dimensional connected oriented smooth manifold and H is a rank 4 smooth subbundle of its tangent bundle TM . Let Q denote the quotient bundle TM/H . There is a homomorphism of vector bundles L : H ∧ H → Q induced by Lie bracket:– L(ξ, η) = [ξ, η] modH for ξ, η ∈ Γ(H). Regard L as a tensor L ∈ Γ(ΛH⊗Q). Then L∧L ∈ Γ(ΛH⊗ 2Q) may be regarded as a quadratic form on Q defined up to scale. We shall say that (M,H) is non-degenerate if and only if L∧L is non-degenerate as such a quadratic form. Since Q has rank two, there are only two cases:– • (M,H) is elliptic ⇐⇒ L∧L is definite; • (M,H) is hyperbolic ⇐⇒ L∧L is indefinite. An elliptic example may be obtained by taking a 3-dimensional complex contact manifold and forgetting its complex structure. A hyperbolic example may be obtained by taking the product of two 3-dimensional real contact manifolds. These two examples will be referred to as the ‘flat’ models. The motivations for our investigation are discussed in the end of this article. Acknowledgements We are pleased to acknowledge several useful conversations with Gerd Schmalz, Jan Slovák, and Peter Vassiliou. Senior Research Fellow of the Australian Research Council. This research was undertaken whilst the second author was visiting the Erwin Schrödinger International Institute for Mathematical Physics. Its support is gratefully acknowledged.