Metric transformations under collapsing of Riemannian manifolds

Let (M,g) be a Riemannian manifold with an isometric action of the Lie group G. Let g_G be a left invariant metric on G. Consider the diagonal G action on the product $M\times G$ with the metric g+g_G. In this paper we calculate the formula for the metric h on the quotient space $(M\times G)\mathrm{/}G$; the map from g to h is the metric transformation. In particular when g is the hyperbolic metric on H^2 and G=S^1, the transformed metric h is Hamilton’s cigar soliton metric studied in the Ricci flow.