Comparison Geometry

This is an up to date work on a branch of Riemannian geometry called Comparison Geometry.

Karsten Grove (Edited by), Peter Petersen (Edited by)

9780521089456, Cambridge University Press

Paperback / softback, published 6 November 2008

276 pages
23.4 x 15.6 x 1.5 cm, 0.39 kg

Review of the hardback: '… a beautiful, comprehensive and up-to-date collection of expository experts in the field.' European Mathematical Society

This book documents the focus on a branch of Riemannian geometry called Comparison Geometry. The simple idea of comparing the geometry of an arbitrary Riemannian manifold with the geometries of constant curvature spaces has seen a tremendous evolution of late. This volume is an up-to-date reflection of the recent development regarding spaces with lower (or two-sided) curvature bounds. The content of the volume reflects some of the most exciting activities in comparison geometry during the year and especially of the Mathematical Sciences Research Institute's workshop devoted to the subject. Both survey and research articles are featured. Complete proofs are often provided, and in one case a new unified strategy is presented and new proofs are offered. This volume will be a valuable source for advanced researchers and those who wish to learn about and contribute to this beautiful subject.

1. Scalar curvature and geometrization conjectures for 3-manifolds Michael T. Anderson
2. Injectivity radius estimates and sphere theorems Uwe Abresch and Wolfgang T. Meyer
3. Aspects of Ricci curvature Tobias H. Colding
4. A genealogy of noncompact manifolds of nonnegative curvature: history and logic R. E. Greene
5. Differential geometric aspects of Alexandrov spaces Yukio Otsu
6. Convergence theorems in Riemannian geometry Peter Petersen
7. The comparison geometry of Ricci curvature Shunhui Zhu
8. Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers G. Perelman
9. Collapsing with no proper extremal subsets G. Perelman
10. Example of a complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and with nonunique asymptotic cone G. Perelman
11. Applications of quasigeodesics and gradient curves Anton Petrunin.

Subject Areas: Differential & Riemannian geometry [PBMP]