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Comparison and finiteness theorems in Riemannian geometry. (English) Zbl 0578.53028

Geometry of geodesics and related topics, Proc. Symp., Tokyo 1982, Adv. Stud. Pure Math. 3, 125-181 (1984).
The author gives here a wide survey on the comparison theorems and finiteness theorems of topological type of Riemannian manifolds having a restriction on the metrical invariants on it. The survey consists of 3 chapters: in the first one, the author treats preparatory relations of the invariants, like various types of estimates of injectivity radius from sectional curvature making full use of calculations on geodesics and Jacobi fields. In chapter 2, he treats first the ordinary pinching problem and next its generalization to symmetric spaces; the former is a well-known sphere theorem and rigidity theorem, the latter is the comparison theorem between symmetric spaces satisfying a certain condition. In the last chapter, the finiteness theorem of Weinstein-Cheeger and that of Gromov are reviewed.
[For the entire collection see Zbl 0535.00016.]

MSC:

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

Citations:

Zbl 0535.00016