A Panoramic View of Riemannian Geometry

by Marcel Berger

Overview

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field.

From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." —MATHEMATICAL REVIEWS

Product Details

ISBN-13:	9783642621215
Publisher:	Springer Berlin Heidelberg
Publication date:	11/14/2012
Edition description:	Softcover reprint of the original 1st ed. 2003
Pages:	824
Product dimensions:	6.10(w) x 9.25(h) x 0.07(d)

Table of Contents

0. Vector fields, tensors 1. Tensor Riemannian duality, the connection and the curvature 2. The parallel transport 3. Absolute (Ricci) calculus, commutation formulas 4. Hodge and the Laplacian, Bochners technique 5. Generalizing Gauss-Bonnet, characteristic classes and C. GEOMETRIC MEASURE THEORY AND PSEUDO-HOLOMORPHIC B. HIGHER DIMENSIONS A.THE CASE OF SURFACES IN R3 C. various other bundles 3. Harmonic maps between Riemannian manifolds 4. Low dimensional Riemannian geometry 5. Some generalizations of Riemannian geometry 6. Gromov mm-spaces 7. Submanifolds B. Spinors A. Exterior differential forms (and some others) C. RICCI FLAT KÄHLER AND HYPERKÄHLER MANIFOLDS 6. Kählerian manifolds (Kähler metrics) Chapter XI : SOME OTHER IMPORTANT TOPICS 1. Non compact manifolds 2. Bundles over Riemannian manifolds B. QUATERNIONIC-KÄHLER MANIFOLDS A. G2 AND Spin(7) HIERRACHY : HOLONOMY GROUPS AND KÄHLER MANIFOLDS 1. Definitions and philosophy 2. Examples 3. General structure theorems 4. The classification result 5. The rare cases b. on a given compact manifolds : closures Chapter X : GLOBAL PARALLEL TRANSPORT AND ANOTHER RIEMANNIAN a. collapsing C. THE CASE OF RICCI CURVATURE 12. Compactness, convergence results 13. The set of all Riemannian structures : collapsing B. MORE FINITENESS THEOREMS A. CHEEGERs FINITENESS THEOREM 11. Finiteness results of all Riemannian structures third part : Finiteness, compactness, collapsing and the space D. NEGATIVE VERSUS NONPOSITIVE CURVATURE 10. The negative side : Ricci curvature C. VOLUMES, FUNDAMENTAL GROUP B. QUASI-ISOMETRIES A. INTRODUCTION E. POSSIBLE APPROACHES, LOOKING FOR THE FUTURE 7. Ricci curvature : positive, nonnegative and just below 8. The positive side : scalar curvature 9. The negative side : sectional curvature D. POSITIVITY OF THE CURVATURE OPERATOR C. THE NON-COMPACT CASE B. HOMOLOGY TYPE AND THE FUNDAMENTAL GROUP A. THE KNOWN EXAMPLES 6. The positive side : sectional curvature second part : Curvature of a given sign1. Introduction 2. The positive pinching 3. Pinching around zero 4. The negative pinching 5. Ricci curvature pinching first part : Pinching problems b. hierarchy of curvaturesa. hopfs urge d. the set of constants, ricci flat metrics 18. The Yamabe problem Chapter IX : from curvature to topology 0. Some history and structure of the chapter c. moduli b. uniqueness a. existence b. homogeneous spaces and others 14. Examples from Analysis I : the evolution Ricci flow 15. Examples from Analysis II : the Kähler case 16. The sporadic examples 17. Around existence and uniqueness a. symmetric spaces THIRD PART : EINSTEIN MANIFOLDS 12. Hilberts variational principle and great hopes 13. The examples from the geometric hierachy 10. The case of Min R d/2 when d=4 11. Summing up questions on MinVol and Min(R) d/2 b. the simplicial volume of gromov a. using integral formulas d. cheeger-rong examples 9. Some cases where MinVol > 0 , Min Rd/2 > 0 c. nilmanifolds and the converse : almost flat manifolds b. wallachs type examples a. s1 fibrations and more examples MinDiam = 0 MinVol, MinDiam 5. Definitions 6. The case of surfaces 7. Generalities, compactness, finiteness and equivalence 8.Cases where MinVol = Min R d/2 = 0 and SECOND PART : WHICH METRIC IS THE LESS CURVED : Min R d/2 , FIRST PART: PURE GEOMETRIC FUNCTIONALS 1. Systolic quotients 2. Counting periodic geodesics 3. The embolic volume 4. Diameter/Injectivity riemannian metric on a given compact manifold • 0. Introduction and a possible scheme of attack c. the structure on a given Sd and KPn 19. Inverse problems II : conjugacy of geodesics flows Chapter VIII : the search for distinguished metrics : what is the best b. bott and samelson theorems a. definitions and the need to be careful are closed 14. The case of negative curvature 15. The case of nonpositive curvature 16. Entropies on various space forms 17. From Osserman to Lohkamp 18. Inverse problems I : manifolds all of whose geodesics b. the various notions of

Editorial Reviews

From the reviews:

"In this monumental work, Marcel Berger manages to survey large parts of present day Riemannian geometry. … the book offers a great opportunity to get a first impression of some part of Riemannian geometry, together with hints for further reading." (A.Cap, Monatshefte für Mathematik, Vol. 145 (4), 2005)

"Riemannian geometry has become a vast subject, influencing, famously, the development of general relativity and, more recently, the classification of 3-manifolds by hyperbolic structures … . Marcel Berger’s book is an overview of this enormous subject. … Virtually everything is illustrated with clear and useful diagrams … . This is the sort of book one could dip into or refer to over a period of years." (Peter Giblin, The Mathematical Gazette, March, 2005)

"Marcel Berger’s A Panoramic View of Riemannian Geometry is without doubt the most comprehensive, original and idiosyncratic treatise on differential geometry … . he manages to include the most up-to-date references on even the most classical topics that he presents, and he puts far greater emphasis on applications. … the book concludes with a massive and extremely useful bibliography of 1310 items." (Robert Osserman, SIAM Review, Vol. 47 (1), 2005)

"This book of one of the main contributors to Riemannian geometry has as a first goal to give an overview to most of the living topics of the subject. … A second goal is to show how many intuitive geometric questions lead to Riemannian geometry in a natural way. In my opinion it is this way of explaining … which makes the book highly recommendable to students as well as to experienced geometers. Furthermore, this book is clearly an encyclopedia in Riemannian geometry … ." (F.Manhart, Internationale Mathematische Nachrichten, Issue 197, 2004)

"This book is really a panorama. … the reading creates pleasure for the interested reader. … the book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." (Jürgen Eichhorn, Mathematical Reviews, 2004 h)

"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. … enables the reader to obtain a sweeping panoramic view of almost the entirety of the field." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (3-4), 2003)

From the Publisher