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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and Principal Curvatures
with Little Smoothness
A One Page Panorama
Volumes and Inequalities on Volumes of Cycles
The Next Two Chapters
From Curvature to Topology
Holonomy Groups and Kahler Manifolds
Some Other Important Topics
The Technical Chapter
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algebraic angles balls Berger boundary called canonical Cartan chapter Cheeger circle compact manifold compact surface complete compute conjecture constant curvature convex coordinates curvature tensor cut locus defined definition denoted derivative diameter diffeomorphism differential dimension dimensional domain eigenfunctions eigenvalues Einstein ellipsoid equation Euclidean space everg example exponential finite flat formula function GauB-Bonnet geodesic flow given Gromov harmonic holonomy hyperbolic hypersurface inequality injectivity radius inner metric integral invariant isometric isoperimetric Kahler manifolds Laplacian length Lie groups look lower bound metric space Moreover negative curvature Note notion orthogonal parallel transport periodic geodesic plane curves positive problem proof prove quadratic form question quotient references result Ricci curvature Riemann Riemannian geometry Riemannian manifold Riemannian metric scalar curvature sectional curvature segment simply connected smooth space forms spectrum structure submanifolds symmetric spaces systole tangent space tangent vectors theorem theory topology torus totally geodesic triangle vanishes vector field volume zero
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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.