On the G2 bundle of a Riemannian 4-manifold

Abstract

We study the natural G 2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M , as it was discovered in [3]. A name is proposed for the space. We work in the context of metric connections, or so called geometry with torsion, and describe the components of the torsion of the connection which imply certain equations of the G 2 structure. This article is devoted to finding the G 2 -torsion tensors which classify our structure according to the theory in [10].

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.…”

“…The G 2 -twistor structure verifies dφ = −(dµ) 2 − 2µ ∧ α 1 . The simply-connected germ of such a gwistor, say SM = R 4 × S 3 , was described in [3] with canonical coordinates, inducing a trivial framing. Recall from (12) Notice both metric connections preserve the Riemannian splitting.…”

“…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].Given M as before, the G 2 -twistor space of M consists of a natural G 2 structure on the S 3 -bundle over M of unit tangent vectors SM = { ∈ T M : = 1}exclusively induced by the metric = · · and orientation. We shall describe the characteristic connection ∇ c of SM in the case where M is an Einstein manifold.…”