A Panoramic View of Riemannian GeometryRiemannian 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. |
Contents
| 2 | |
Transition 101 | 100 |
with Little Smoothness | 133 |
Riemanns Blueprints | 143 |
A One Page Panorama | 219 |
Volumes and Inequalities on Volumes of Cycles 299 | 298 |
The Next Two Chapters | 369 |
Geodesic Dynamics | 432 |
Best Metrics 499 | 500 |
References 723 | 526 |
From Curvature to Topology | 543 |
Holonomy Groups and Kähler Manifolds 637 | 636 |
Some Other Important Topics | 659 |
The Technical Chapter 693 | 692 |
Acknowledgements | 789 |
| 810 | |
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Common terms and phrases
algebraic balls Berger Betti numbers boundary canonical chapter Cheeger compact manifold complete compute conjecture constant curvature convex curvature tensor cut locus defined definition diameter diffeomorphism differential dimension dimensional distance domain eigenfunctions eigenvalues Einstein metrics ellipsoid equation Euclidean space example exponential finite flat formula fundamental group Gauß geodesic flow given Gromov harmonic holonomy group hyperbolic hypersurface implies inequality injectivity radius integral invariant isometry isoperimetric Kähler manifolds Laplacian length Lie group look lower bound Math metric spaces Moreover negative curvature noncompact nonnegative Note notion orthogonal parallel transport periodic geodesics pinching positive curvature problem proof prove quaternionic question quotient Ricci curvature Riemann Riemannian geometry Riemannian manifold Riemannian metric scalar curvature sectional curvature simply connected smooth space forms spectrum spinors structure submanifolds surfaces symmetric spaces systole tangent space technique theorem theory topology torus triangle vanishing vector field volume zero