Fukaya, Kenji 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. Reviewer: V.Yu.Rovenskij (Krasnoyarsk) Cited in 1 ReviewCited in 42 Documents 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 Keywords:diameter; rigidity; collapsing Riemannian manifolds; finiteness theorems; pinching theorems Citations:Zbl 0733.00013 PDFBibTeX XMLCite \textit{K. Fukaya}, Adv. Stud. Pure Math. None, 143--238 (1990; Zbl 0754.53004)