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Comparison Geometry asks: What can we say about a Riemannian manifold if we know a (lower or upper) bound for its curvature, and perhaps something about its topology? Powerful results that allow the exploration of this question were first obtained in the 1950s by Rauch, Alexandrov, Toponogov, and Bishop, with some ideas going back to Hopf, Morse, Schoenberg, Myers, and Synge in the 1930s.
In the last decade the field has witnessed many important advances: first in conjunction with Morse theory and convexity, then with critical point theory for distance functions, and most recently with the Gromov--Hausdorff topology on spaces of Riemannian manifolds, and the geometry of singular spaces. As a result, our understanding of relations between the geometry and topology of Riemannian manifolds has expanded, and no longer consists of small unrelated pieces of scholarship.
This volume, arising from a 1994 MSRI program, is an up-to-date panorama of Comparison Geometry, featuring surveys and new research. Surveys present classical and recent results, and often include complete proofs, in some cases involving a new and unified approach. The historical evolution of the subject is summarized in charts and tables of examples.
This volume will be a valuable source for researchers and graduate
students in Riemannian Geometry.