The comparison geometry of Ricci curvature. (English) Zbl 0896.53036

Zhu, Shunhui

Grove, Karsten (ed.) et al., Comparison geometry. Cambridge: Cambridge University. Math. Sci. Res. Inst. Publ. 30, 221-262 (1997).

The term “comparison geometry” has its origin with the Rauch comparison theorem and its more powerful global version, the Toponogov theorem. The comparison geometry of sectional curvature represents many ingenious applications of these theorems and produced many interesting results by Berger, Klingenberg, Cheeger, Gromoll, Gromov, Grove, and many others. The comparison geometry of Ricci curvature started as isolated attempts to generalize sectional curvature results to Ricci curvature. It turned out that Toponogov’s theorem does not hold for Ricci curvature.
This paper is a survey which covers most of the important comparison theorems concerning the Ricci curvature, including volume element, Laplacian, and mean curvature comparisons. The author gives proofs of the discussed results. The paper is recommended as a very good reference source for graduate students and experts in this area.
For the entire collection see [Zbl 0871.00023].

Reviewer: A.Bucki (Oklahoma City)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry

Keywords:

comparison geometry of Ricci curvature; positive Ricci curvature; volume comparison; Laplacian comparison; mean curvature comparison; survey