  We consider the problem of preserving positive Ricci curvature along
connected sums. In this context, based on earlier work by Perelman, Burdick
introduced the notion of core metrics and showed that the connected sum of
manifolds with core metrics admits a Riemannian metric of positive Ricci
curvature. He subsequently showed that core metrics exist on compact rank one
symmetric spaces, certain sphere bundles and manifolds obtained as boundaries
of certain plumbings. In this article, we show that core metrics can be lifted
along general fibre bundles, including sphere bundles and projective bundles.
Our techniques also apply to spaces that decompose as the union of two disc
bundles such as the Wu manifold. As application we show that all classes in the
torsion-free oriented bordism ring can be represented by connected manifolds of
positive Ricci curvature.
