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].

We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle E −→ M , over a Riemannian manifold M , when E is endowed with a metric connection. The tangent bundle of E admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of E; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type G 2 manifolds.

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.

We prove a Theorem on homotheties between two given tangent sphere bundles S r M of a Riemannian manifold M, g of dim ≥ 3, assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I G and symplectic structure ω G on the manifold T M , generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds T M and S r M .

Natural metric structures on the tangent bundle and tangent sphere bundles S r M of a Riemannian manifold M with radius function r enclose many important unsolved problems. Admitting metric connections on M with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of T M to the almost Hermitian category. Our purpose is the study of the natural contact structure on S r M and the G 2 -twistor space of any oriented Riemannian 4-manifold.