Metric spaces of curvature \(\geq k\). (English) Zbl 1011.57002

Plaut, Conrad

Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 819-898 (2002).

The paper under review is intended to be an introduction to the theory of metric spaces of curvature bounded below and a survey of recent results. The author does not assume that the reader has knowledge of Riemannian geometry, although such knowledge is helpful for understanding the motivation behind certain topics. A good working knowledge of topology and basic real analysis should be sufficient background. The author has limited himself to the subject of spaces of curvature bounded below in the sense of Alexandrov or Berestovskij-Wald, which generalize the notion of bounded sectional curvature in Riemannian manifolds. One of the main themes of this paper is the tremendous influence that a curvature bound can have on topology, at the infinitesimal, local, and global levels, as well as on a global level across whole families of spaces with geometrically defined restrictions. In this regard, modern results in the field remain very close, in spirit, to their earliest ancestor, the Gauss-Bonnet Theorem.
For the entire collection see [Zbl 0977.00029].

MSC:

57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57N16	Geometric structures on manifolds of high or arbitrary dimension

Keywords:

curvature bound