Collapsing three-manifolds under a lower curvature bound. (English) Zbl 1036.53028

Shioya, Takashi; Yamaguchi, Takao

J. Differ. Geom. 56, No. 1, 1-66 (2000).

In the present paper the authors study the topology of three-dimensional Riemannian manifolds with a uniform lower bound of sectional curvature converging to a metric of lower dimension. The paper consists of two parts and an appendix. In part 1, the authors discuss the collapsing of three-dimensional Riemannian manifolds by assuming the generalized soul theorem. In part 2 they classify all the three-dimensional complete open Alexandrov spaces with nonnegative curvature (the generalized soul theorem). Some examples are given and the rigidity part of the generalized soul theorem (Theorem 9.6) is proved. In the appendix the authors discuss the total curvature, the Gauss-Bonet theorem and the Cohn-Vossen theorem for Alexandrov surfaces, and give a classification of nonnegatively curved Alexandrov surfaces.

Reviewer: Neculai Papaghiuc (Iaşi)

MSC:

53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20	Global Riemannian geometry, including pinching

Keywords:

Gromov-Hausdorff convergence; Alexandrov spaces; topology of 3-manifolds; Seifert fibred spaces; soul