The G 2 sphere of a 4-manifold

Abstract

We present a construction of a canonical G 2 structure on the unit sphere tangent bundle S M of any given orientable Riemannian 4-manifold M . Such structure is never geometric or 1-flat, but seems full of other possibilities. We start by the study of the most basic properties of our construction. The structure is co-calibrated if, and only if, M is an Einstein manifold. The fibres are always associative. In fact, the associated 3-form φ results from a linear combination of three other volume 3-forms, one of which is the volume of the fibres. We also give new examples of co-calibrated structures on well known spaces. We hope this contributes both to the knowledge of special geometries and to the study of 4-manifolds.

Weighted Metrics on Tangent Sphere Bundles

“…It is today well established that any oriented Riemannian 4-manifold M gives rise to a canonical G 2 structure on S 1 M. This was discovered in [7,9,10] partly recurring to twistor methods; so we call it the G 2 -twistor bundle of M. Indeed, the pull-back of the volume form coupled with each point u ∈ S 1 M, say a 3-form α, induces a quaternionic structure which is reproduced twice in horizontal and vertical parts of T u S 1 M. Then the Cayley-Dickson process gives the desired G 2 = Aut O-structure over S 1 M. Some properties of the so called gwistor space have been discovered, namely that it is cocalibrated if and only if the 4-manifold is Einstein. The first variation of that structure, which may yield interesting features, is by choosing both any metric connection (i.e.…”

On the characteristic connection of gwistor space

“…The former was discovered in [5,6] and we shall start here by recalling how it was obtained. Often we abbreviate the name G 2 -twistor for gwistor, as suggested in [3].…”

“…There is also a natural map θ : T T M → T T M (2) which is a π * ∇ L-C -parallel endomorphism of T T M identifying H isometrically with the vertical bundle π * T M = ker dπ and defined as 0 on the vertical side. It was introduced in [3,5,6]. Then we define the horizontal vector field θ U.…”

Hermitian and quaternionic Hermitian structures on tangent bundles

“…This picture has led to the construction in [3] of G 2 -structures on the 7-manifold which is the unit sphere tangent bundle of M.…”

“…[2,12]) or of the sphere tangent bundle (cf. [3]), where a similar canonical section ξ was defined. In sum, it follows from the theory that X ∈ H D ⇔ (π * D) X ξ = 0.…”