Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann.
Euclidean geometry -- Transition -- Surfaces from GauB to today -- Riemann's blueprints -- A one page panorama -- Metric geometry and curvature -- Volumes and inequalities on volumes of cycles -- Transition: the next two chapters -- Spectrum of the Laplacian -- Geodesic dynamics -- Best metrics -- From curvatures to topology -- Holonomy groups and Kahler manifolds -- Some other important topics -- The technical chapter
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