Ebook: A Panoramic View of Riemannian Geometry

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.

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Content:
Front Matter....Pages I-XXIII
Old and New Euclidean Geometry and Analysis....Pages 1-99
Transition: The Need for a More General Framework....Pages 101-104
Surfaces from Gau? to Today....Pages 105-142
Riemann’s Blueprints for Architecture in Myriad Dimensions....Pages 143-218
A One Page Panorama....Pages 219-219
Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature....Pages 221-297
Volumes and Inequalities on Volumes of Cycles....Pages 299-368
Transition: The Next Two Chapters....Pages 369-372
Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian....Pages 373-429
Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics....Pages 431-497
What is the Best Riemannian Metric on a Compact Manifold?....Pages 499-541
From Curvature to Topology....Pages 543-635
Holonomy Groups and K?hler Manifolds....Pages 637-657
Some Other Important Topics....Pages 659-691
The Technical Chapter....Pages 693-721
Back Matter....Pages 723-824

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Content:
Front Matter....Pages I-XXIII
Old and New Euclidean Geometry and Analysis....Pages 1-99
Transition: The Need for a More General Framework....Pages 101-104
Surfaces from Gau? to Today....Pages 105-142
Riemann’s Blueprints for Architecture in Myriad Dimensions....Pages 143-218
A One Page Panorama....Pages 219-219
Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature....Pages 221-297
Volumes and Inequalities on Volumes of Cycles....Pages 299-368
Transition: The Next Two Chapters....Pages 369-372
Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian....Pages 373-429
Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics....Pages 431-497
What is the Best Riemannian Metric on a Compact Manifold?....Pages 499-541
From Curvature to Topology....Pages 543-635
Holonomy Groups and K?hler Manifolds....Pages 637-657
Some Other Important Topics....Pages 659-691
The Technical Chapter....Pages 693-721
Back Matter....Pages 723-824
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