## 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. |

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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 |

811 | |

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### Common terms and phrases

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

### Popular passages

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.