Riemannian comparison constructions. (English) Zbl 0683.53040

Global differential geometry, MAA Stud. Math. 27, 170-222 (1989).
[For the entire collection see Zbl 0683.53001.]
This is a very fascinating exposition of Riemannian comparison theorems and related topics on manifolds of nonnegative sectional curvature. It can be considered as a continuation of the introductory chapter by S. Kobayashi [see the same collection, 140-169 (1989; Zbl 0683.53043)]. A series of important results in this area is selected and presented with detailed proofs in this paper. In addition to this the reader gets a good guideline through the technical details and a comprehensive survey on recent developments in the topic under consideration. For a rough description of the contents some keywords may be sufficient: methods of curvature control, Hadamard-Cartan theorem, growth of the fundamental group, Ricci-diameter bound, Busemann functions, triangle comparison theorems, bound for the number of generators of the fundamental group, critical points of the distance function, cut locus estimates, sphere theorems, complex projective space and its distance spheres. For a good bibliography the reader is referred to work of T. Sakai [Comparison and finiteness theorems in Riemannian geometry, Adv. Stud. Pure Math. 3, 125-181 (1984; Zbl 0578.53028)].
Reviewer: Bernd Wegner


53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry