A Panoramic View of Riemannian Geometry

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Springer Science & Business Media, Jun 29, 2007 - Mathematics - 824 pages
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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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Contents

Euclidean Geometry
2
and Principal Curvatures
45
Transition
101
with Little Smoothness
133
Riemanns Blueprints
143
A One Page Panorama
219
Volumes and Inequalities on Volumes of Cycles
299
The Next Two Chapters
369
Best Metrics
499
From Curvature to Topology
543
Holonomy Groups and Kahler Manifolds
637
Some Other Important Topics
659
The Technical Chapter
693
References
723
Acknowledgements
789
Subject Index
811

Geodesic Dynamics
431

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Page 747 - Guillemin, The spectrum of positive elliptic operators and periodic ^characteristics. Invent. Math. 29, 1975, 39-79.
Page 727 - The nonsplit case Addendum to " Eta invariants, signature defects of cusps, and values of L-functions" 118 (1983), 131-147 By MF ATIYAH, H.
Page 748 - Les connexions infinitesimales dans un espace fibre differentiable, Colloque de topologie (espaces fibres), Bruxelles, 1950, Georges Thone, Liege, 1951, pp.
Page 746 - PAM Dirac, The quantum theory of the electron, Proc. Roy. Soc. A., 117 (1928;, 616.
Page 750 - A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), 1-21.

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