Metric transformations under collapsing of Riemannian manifolds

Gromov-Hausdorff convergence is an important tool in comparison Riemannian geometry. Given a sequence of Riemannian manifolds of dimension n with Ricci curvature bounded from below, Gromov’s precompactness theorem says that a subsequence will converge in the pointed Gromov-Hausdorff topology to a length space [G-99, Section 5A]. If the sequence has bounded sectional curvature, then the limit will be an Alexandrov space [BGP-92]. The general idea in using Gromov-Hausdorff convergence is to construct sequences and to inspect their limits closely (see [P-97], for example). A sequence is called collapsing if its Gromov-Hausdorff limit has a lower Hausdorff dimension. It is known that interesting phenomena occur when a sequence of Riemannian manifolds collapses [CG-86], [CG-90], [Fu-90]. In this paper we combine the ideas of isometric group actions and collapsing of Riemannian manifolds to explore metric transformations. More precisely, for several Riemannian manifolds (M, g) with isometric group actions G, we construct a sequence of Riemannian manifolds (M̃i, g̃i) of higher dimension which collapse to the original manifold M with a new metric h. G also acts on (M,h) by isometries. The map g 7→ h is the metric transformation in the title of this paper. When g is the hyperbolic metric on H and G = S, the transformed metric h is Hamilton’s cigar soliton metric studied in Ricci flow [H-88], [ H-95] (see also [W-91, p. 315]). When g is the exploding soliton (see the appendix for its definition), the transformed metric h is the 2-sphere. When g is the standard metric on S and G = S, the transformed metrics h are the Berger metrics on S [CE-75]. In section 2 we construct examples of sequences (M̃i, g̃i) and show that the collapsing limit (M,h) of (M̃i, g̃i) is a quotient of a Riemannian manifold (M̃, g̃) by a Lie group G̃. In section 3 we calculate the transformed metric h for the examples in section 2. In section 4 we compute the metrics on quotients (M̃, g̃)/G̃ for other choices of (M̃, g̃) and G̃ which give metric transformations.