论文信息 - Metric transformations under collapsing of Riemannian manifolds

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

B. Chow | P. Lu | David Glickenstein

[1] B. Chow,et al. The Ricci flow on surfaces , 2004 .

[2] J. R. David. String Theory and Black Holes , 1999, hep-th/9911003.

[3] H. Verlinde,et al. String Propagation in a Black Hole Geometry , 1992 .

[4] J. Cheeger,et al. Collapsing riemannian manifolds while keeping their curvature bounded. I. , 1986 .

[5] J. Milnor. Curvatures of left invariant metrics on lie groups , 1976 .

[6] J. Cheeger. Some examples of manifolds of nonnegative curvature , 1973 .

[7] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[8] P. Petersen. Convergence Theorems in Riemannian Geometry , 1997 .

[9] R. Hamilton,et al. The formations of singularities in the Ricci Flow , 1993 .

[10] J. Isenberg,et al. Ricci flow of locally homogeneous geometries on closed manifolds , 1992 .

[11] K. Fukaya. Hausdorff Convergence of Riemannian Manifolds and Its Applications , 1990 .

[12] Richard S. Hamilton,et al. The Ricci flow on surfaces , 1986 .

[13] J. Cheeger,et al. Comparison theorems in Riemannian geometry , 1975 .