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Geometry of Manifolds (AMS Chelsea Publishing): Richard L. Bishop, Richard J. Crittenden

This volume is a reprint with few corrections of the original work published in 1964. Starting with the notion of differential manifolds, the first six chapters lay a foundation for the study of Riemannian manifolds through specializing the theory of connections on principle bundles and affine connections. The geometry of Riemannian manifolds is emphasized, as opposed to global analysis, so that the theorems of Hopf-Rinow, Hadamard-Cartan, and Cartan's local isometry theorem are included, but no elliptic operator theory. Isometric immersions are treated elegantly and from a global viewpoint. In the final chapter are the more complicated estimates on which much of the research in Riemannian geometry is based: the Morse index theorem, Synge's theorems on closed geodesics, Rauch's comparison theorem, and the original proof of the Bishop volume-comparison theorem (with Myer's Theorem as a corollary). The first edition of this book was the origin of a modern treatment of global Riemannian geometry, using the carefully conceived notation that has withstood the test of time. The primary source material for the book were the papers and course notes of brilliant geometers, including É. Cartan, C. Ehresmann, I. M. Singer, and W. Ambrose. It is tightly organized, uniformly very precise, and amazingly comprehensive for its length.

Table of Contents (by Culin) Preface to the New Printing Preface Ch. 1Manifolds 1.1Introductory Material and Notation1 1.2Definition of a Manifold2 1.3Tangent Space7 1.4Vector Fields13 1.5Submanifolds21 1.6Distributions and Integrability22 Ch. 2Lie Groups 2.1Lie Groups25 2.2Lie Algebras26 2.3Lie Group - Lie Algebra Correspondence28 2.4Homomorphisms29 2.5Exponential Map30 2.6Representations34 Ch. 3Fibre Bundles 3.1Transformation Groups38 3.2Principal Bundles41 3.3Associated Bundles45 3.4Reduction of the Structural Group49 Ch. 4Differential Forms 4.1Introduction53 4.2Classical Notion of Differential Form53 4.3Grassmann Algebras54 4.4Existence of Grassmann Algebras57 4.5Differential Forms62 4.6Exterior Derivative64 4.7Action of Maps68 4.8Frobenius' Theorem70 4.9Vector-Valued Forms and Operations71 4.10Forms on Complex Manifolds72 Ch. 5Connexions 5.1Definitions and First Properties74 5.2Parallel Translation77 5.3Curvature Form and the Structural Equation80 5.4Existence of Connexions and Connexions in Associated Bundles83 5.5Structural Equations for Horizontal Forms84 5.6Holonomy87 Ch. 6Affine Connexions 6.1Definitions89 6.2The Structural Equations of an Affine Connexion99 6.3The Exponential Maps108 6.4Covariant Differentiation and Classical Forms111 Ch. 7Riemannian Manifolds 7.1Definitions and First Properties122 7.2The Bundle of Frames127 7.3Riemannian Connexions129 7.4Examples and Problems132 Ch. 8Geodesics and Complete Riemannian Manifolds 8.1Geodesics145 8.2Complete Riemannian Manifolds152 8.3Continuous Curves158 Ch. 9Riemannian Curvature 9.1Riemannian Curvature161 9.2Computation of the Riemannian Curvature165 9.3Continuity of the Riemannian Curvature166 9.4Rectangles and Jacobi Fields172 9.5Theorems Involving Curvature178 Ch. 10Immersions and the Second Fundamental Form 10.1Definitions185 10.2The Connexions187 10.3Curvature189 10.4The Second Fundamental Form190 10.5Curvature and the Second Fundamental Form192 10.6The Local Gauss Map195 10.7Hessians of Normal Coordinates of N197 10.8A Formulation of the Immersion Problem199 10.9Hypersurfaces207 Ch. 11Second Variation of Arc Length 11.1First and Second Variation of Arc Length213 11.2The Index Form220 11.3Focal Points and Conjugate Points224 11.4The Infinitesimal Deformations226 11.5The Morse Index Theorem233 11.6The Minimum Locus237 11.7Closed Geodesics241 11.8Convex Neighborhoods246 11.9Rauch's Comparison Theorem250 11.10Curvature and Volume253 AppTheorems on Differential Equations258 Bibliography260 Subject Index265

Summary: A Unique Classic

Rating: 5

Differential geometry is one of the most highly developed subjects in all of mathematics. The literature is daunting, both in volume and complexity. The serious student will soon learn that there simply is no single "perfect book" on the subject from which one can learn everything one needs to know. This is doubly true for the student who wants to learn about both Riemannian manifolds and Semi-Riemannian geometry, the language of Einstein's theory of general relativity.

The book by Bishop and Crittenden, long out of print and difficult to find before this recent re-printing emerged, contains a wealth of important and fundamental insights that are simply not to be found in any other differential geometry text. I will describe only one example in detail; many other examples of a similar nature could be cited.

As one studies differential geometry, one quickly learns that there are uncountably many connexions on a typical manifold (M,g). However, most books quickly restrict their attention to the Levi-Civita connexion, the unique connexion that is (1) metrically compatible, and (2) has torsion zero.

While metric compatibility is fairly easy to understand, the notion of torsion zero is far more elusive. Do a quick internet search and you will find scores of hapless students who are begging for help in understanding the GEOMETRIC content of the torsion tensor. Students of general relativity quickly learn that the mathematical expression of Einstein's Equivalence Principle will not hold unless the connexion has zero torsion, and that is sufficient to motivate the condition in GR; however, this still does not explain the tensor's geometric content.

Bishop and Crittenden give a visual interpretation of torsion in terms of geodesic quadrilaterals (see page 97) that will appeal to anyone who is searching for geometric intuition. I have over 150 differential geometry books in my personal library, and Bishop and Crittenden is the only one to provide this intuitive, geometric understanding of the torsion tensor. Richard Bishop continued this trend in his later book, co-authored with Sam Goldberg, where he gives a similar geometric interpretation of the Lie bracket.

If you are a devoted student of geometry, then I suggest you add Bishop and Crittenden to your library, along with Spivak (5 volumes), Kobayashi and Nomizu (2 volumes), Chern, etc., etc. Each of these references contains unique insights not to be found in any of the others. You may only need to refer to this book a few times, but the insights gained will be well worth the meager purchase price.

Summary: excellent but not a first semester textbook

Rating: 4

I recommend this as a supplement for students who have already learned Riemannian Geometry or at least Differential Geometry elsewhere for 1-2 semesters. It is excellent for fibre bundles and viewing a metric as an SO(n) bundle over a manifold. It is also an excellent reference for Bishop's Volume Comparison Theorem which has since been adapted by Gromov in his book "Metric Structures for Riemannian and Non-Riemannian Spaces" and has led to a fundamental change in the study of manifolds with Ricci curvature bounds.

Prerequisite books I read as a grad student were Spivak's Differential Geometry and DoCarmo's "Riemannian Geometry".

Summary: Spectacular geometric insight into differential geometry

Rating: 5

As a differential geometer for the past 30 years, I own 8 introductions to the field, and I have perused a half-dozen others. Bishop & Crittenden's "Geometry of Manifolds" is unparallelled for imparting a strong geometric intuition about Lie derivatives and connections on fiber bundles, which is the key to understanding this field, plus general relativity, Einstein-Cartan theory, and gauge theories in physics. Bishop & Crittenden taught me to see pictures for the main constructions and theorems, though I admit I had to work hard to build my intuition. It is particularly strong in establishing a 1-1 correspondence between geometric images of horizontal planes in fiber bundles and the language of differential forms in fiber bundles. I first read the classic Chevalley's "Lie Groups" which is very strong on geometric intuition about manifolds and Lie groups. I also relied on Kobayashi & Numizu's "Foundations of Differential Geometry" I & II because ity has better algebraic and computational treatment in some areas, and it cover topics not in "Geometry of Manifolds."

If you are really serious about differential geometry, if you prefer to think in terms of geometric visualization when possible and use algebra afterwards, then this book is the greatest -- nothing I have seen comes close. If you prefer to think mainly with abstract algebra, then you might prefer Kobayashi and Nomizu.

Richard J. Petti

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