  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 $(M \times G) / G$; 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.
