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Hausdorff convergence of Riemannian manifolds and its applications. (English) Zbl 0754.53004

Recent topics in differential and analytic geometry, Adv. Stud. Pure Math. 18-I, 143-238 (1990).
[For the entire collection see Zbl 0733.00013.]
Chapter 1 of this survey contains properties of Hausdorff convergence and some theorems about Riemannian manifolds, whose curvatures and diameter are uniformly bounded and whose volumes are uniformly bounded away from 0: precompactness, rigidity, convergence. Chapter 2 deals with collapsing Riemannian manifolds: almost flat (with bounded sectional curvatures and small diameter), fiber bundles. Chapter 3 contains three types of applications to Riemannian manifolds without assumptions on curvatures or diameter: finiteness theorems, pinching theorems, about ends of complete manifolds. The author gives many examples, conjectures and open problems.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citations:

Zbl 0733.00013
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