On vector bundle manifolds with spherically symmetric metrics

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 , 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 manifolds.