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Graduate Texts in MathematicsTAKE AIUNG.Introduction toAxiomatic Set Theory. 2nd ed.OXTOBY.Measure and Category. 2nd ed.SCHAEFER.Topological Vector Spaces.HILTONISTAMMBACH.A Course inHomological Algebra. 2nd ed.MACLANE.Categories for the WorkingMathematician.HUGWPPER.Projective Planes.SERRE.A C o m e in Arithmetic.TAKE AIUNG.Axiomatic Set Theory.HUMPHREYS.Introduction to Lie Algebrasand Representation Theory.COHEN.A Course in Simple HomotopyTheory.CONWAY.Functions of One ComplexVariable I. 2nd ed.B'EALS. Advanced Mathematical Analysis.ANDERSON/FWLLER.Rings and Categoriesof Modules. 2nd ed.G O L U B S K YStable/ G Mappings.and Their Singularities.BERBERIAN.Lectures in FunctionalAnalysis and Operator Theory.W m . The Structure of Fields.ROSENBLATT.Random Processes. 2nd ed.HALMos. Measure Theory.HALMos. A Hilbert Space Problem Book.2nd ed.HUSEMOLLER.Fibre Bundles. 3rd ed.HUMPHREYS.Linear Algebraic Groups.B A R N M A CAnK . Algebraic Introductionto Mathematical Logic.GREUB.Linear Algebra. 4th ed.HOLIUIES.Geometric Functional Analysisand Its Applications.HEW /STROMBERG.Real and AbstractAnalysis.MANES.Algebraic Theories.KFLLEY. General Topology.ZARISKI SAMIJEL. Commutative Algebra.V0l.I.ZAR SKJISAMLEL.Commutative Algebra.v01.n.JACOBSON.Lectures in Abstract Algebra I.Basic Concepts.JACOBSON.Lectures in Abstract AlgebraII. Linear Algebra.JACOBSON.Lectures in Abstract AlgebraIII. Theory of Fields and Galois Theory.HIRSCH.Differential Topology.SP IZER.Principles of Random Walk.2nd ed.WERMER.Banach Algebras and SeveralComplex Variables. 2nd ed.KELLEY/NAMIoKAet d. LinearTopological Spaces.MONK.Mathematical Logic.GRAUERT/FRI ZSCHE.Several ComplexVariables.An Invitation to arkov Chains. 2nd ed.APOSTOL.Modular Functions andDichlet Series in Number Theory.2nd ed.SERRE.Linear Representations of FiniteGroups.G W J E R I S O N Rings.of ContinuousFunctions.KENDIG. Elementary Algebraic Geometry.LoiVE. Probability Theory I. 4th ed.LOEVE.Probability Theory II. 4th ed.MOISE.Geometric Topology inDimensions 2 and 3.S m s M r u General.Relativity forMathematicians.L i a r Geometry.GRUENBER WEIR.2nd ed.EDWARDS.Fennat's Last Theorem.KLJNGENBERG.A Course in DifferentialGeometry.HARTSHORNE.Algebraic Geometry.MANIN.A Course in Mathematical Logic.G R A W A T K I N SCombinatorics.withEmphasis on the Theory of Graphs.BROWNJPEARCY.Introduction to OperatorTheory I: Elements of FunctionalAnalysis.MASSEY.Algebraic Topology: AnIntroduction.CROWELLJFOX.Introduction to KnotTheory.K O B L p-adic.Numbers, padicAnalysis, and Zeta-Functions. 2nd ed.LANG.Cyclotomic Fields.ARNOLD.Mathematical Methods inClassical Mechanics. 2nd ed.continued afer index

John M. LeeRiemannian ManifoldsAn Introduction to CurvatureWith 88 IllustrationsSpringer

John M. LeeDepartment of MathematicsUniversity of WashingtonSeattle, W A 981 95-4350USAEditorial BoardS. AxlerDepartment ofMathematicsMichigan State UniversityEast Lansing, M I 48824USAF.W. G e k n gP.R. HalmosDepartment ofMathematicsUniversity of MichiganAnn Arbor, MI 48109USADepartment ofMathematicsSanta Clara UniversitySanta Clara, C A 95053USAMathematics Subject Classification (1991): 53-01, 53C20Library of Congress Cataloging-in-Publication DataLee, John M., 1950Reimannian manifolds : an introduction to curvature I John M. Lee.cm. - (Graduate texts in mathematics ; 176)p.Includes index.ISBN 0-387-98271-X (hardcover : alk. paper)1. Reimannian manifolds. I. Title. 11. Series.QA649.L397 1997516.3'734 21O 1997 Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc., in this publication, even if theformer are not especially identified, is not to be taken as a sign that such names, as understood by theTrade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.ISBN 0-387-98271-X Springer-Verlag New York Berlin Heidelberg SPIN 10630043 (hardcover)ISBN 0-387-98322-8 Springer-Verlag New York Berlin Heidelberg SPIN 10637299 (softcover)

To my family:Pm, Nathan, and Jeremy Weizenbaum

PrefaceThis book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar withtopological and diﬀerentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, itintroduces and demonstrates the uses of all the main technical tools neededfor a careful study of Riemannian manifolds.I have selected a set of topics that can reasonably be covered in ten toﬁfteen weeks, instead of making any attempt to provide an encyclopedictreatment of the subject. The book begins with a careful treatment of themachinery of metrics, connections, and geodesics, without which one cannotclaim to be doing Riemannian geometry. It then introduces the Riemanncurvature tensor, and quickly moves on to submanifold theory in order togive the curvature tensor a concrete quantitative interpretation. From thenon, all eﬀorts are bent toward proving the four most fundamental theoremsrelating curvature and topology: the Gauss–Bonnet theorem (expressingthe total curvature of a surface in terms of its topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositivecurvature), Bonnet’s theorem (giving analogous restrictions on manifoldsof strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature).Many other results and techniques might reasonably claim a place in anintroductory Riemannian geometry course, but could not be included dueto time constraints. In particular, I do not treat the Rauch comparison theorem, the Morse index theorem, Toponogov’s theorem, or their importantapplications such as the sphere theorem, except to mention some of them

viiiPrefacein passing; and I do not touch on the Laplace–Beltrami operator or Hodgetheory, or indeed any of the multitude of deep and exciting applicationsof partial diﬀerential equations to Riemannian geometry. These importanttopics are for other, more advanced courses.The libraries already contain a wealth of superb reference books on Riemannian geometry, which the interested reader can consult for a deepertreatment of the topics introduced here, or can use to explore the moreesoteric aspects of the subject. Some of my favorites are the elegant introduction to comparison theory by Jeﬀ Cheeger and David Ebin [CE75](which has sadly been out of print for a number of years); Manfredo doCarmo’s much more leisurely treatment of the same material and more[dC92]; Barrett O’Neill’s beautifully integrated introduction to pseudoRiemannian and Riemannian geometry [O’N83]; Isaac Chavel’s masterfulrecent introductory text [Cha93], which starts with the foundations of thesubject and quickly takes the reader deep into research territory; MichaelSpivak’s classic tome [Spi79], which can be used as a textbook if plenty oftime is available, or can provide enjoyable bedtime reading; and, of course,the “Encyclopaedia Britannica” of diﬀerential geometry books, Foundations of Diﬀerential Geometry by Kobayashi and Nomizu [KN63]. At theother end of the spectrum, Frank Morgan’s delightful little book [Mor93]touches on most of the important ideas in an intuitive and informal waywith lots of pictures—I enthusiastically recommend it as a prelude to thisbook.It is not my purpose to replace any of these. Instead, it is my hopethat this book will ﬁll a niche in the literature by presenting a selectiveintroduction to the main ideas of the subject in an easily accessible way.The selection is small enough to ﬁt into a single course, but broad enough,I hope, to provide any novice with a ﬁrm foundation from which to pursueresearch or develop applications in Riemannian geometry and other ﬁeldsthat use its tools.This book is written under the assumption that the student alreadyknows the fundamentals of the theory of topological and diﬀerential manifolds, as treated, for example, in [Mas67, chapters 1–5] and [Boo86, chapters1–6]. In particular, the student should be conversant with the fundamentalgroup, covering spaces, the classiﬁcation of compact surfaces, topologicaland smooth manifolds, immersions and submersions, vector ﬁelds and ﬂows,Lie brackets and Lie derivatives, the Frobenius theorem, tensors, diﬀerential forms, Stokes’s theorem, and elementary properties of Lie groups. Onthe other hand, I do not assume any previous acquaintance with Riemannian metrics, or even with the classical theory of curves and surfaces in R3 .(In this subject, anything proved before 1950 can be considered “classical.”) Although at one time it might have been reasonable to expect mostmathematics students to have studied surface theory as undergraduates,few current North American undergraduate math majors see any diﬀeren-

Prefaceixtial geometry. Thus the fundamentals of the geometry of surfaces, includinga proof of the Gauss–Bonnet theorem, are worked out from scratch here.The book begins with a nonrigorous overview of the subject in Chapter1, designed to introduce some of the intuitions underlying the notion ofcurvature and to link them with elementary geometric ideas the studenthas seen before. This is followed in Chapter 2 by a brief review of somebackground material on tensors, manifolds, and vector bundles, includedbecause these are the basic tools used throughout the book and becauseoften they are not covered in quite enough detail in elementary courseson manifolds. Chapter 3 begins the course proper, with deﬁnitions of Riemannian metrics and some of their attendant ﬂora and fauna. The end ofthe chapter describes the constant curvature “model spaces” of Riemanniangeometry, with a great deal of detailed computation. These models form asort of leitmotif throughout the text, and serve as illustrations and testbedsfor the abstract theory as it is developed. Other important classes of examples are developed in the problems at the ends of the chapters, particularlyinvariant metrics on Lie groups and Riemannian submersions.Chapter 4 introduces connections. In order to isolate the important properties of connections that are independent of the metric, as well as to lay thegroundwork for their further study in such arenas as the Chern–Weil theoryof characteristic classes and the Donaldson and Seiberg–Witten theories ofgauge ﬁelds, connections are deﬁned ﬁrst on arbitrary vector bundles. Thishas the further advantage of making it easy to deﬁne the induced connections on tensor bundles. Chapter 5 investigates connections in the contextof Riemannian manifolds, developing the Riemannian connection, its geodesics, the exponential map, and normal coordinates. Chapter 6 continuesthe study of geodesics, focusing on their distance-minimizing properties.First, some elementary ideas from the calculus of variations are introducedto prove that every distance-minimizing curve is a geodesic. Then the Gausslemma is used to prove the (partial) converse—that every geodesic is locally minimizing. Because the Gauss lemma also gives an easy proof thatminimizing curves are geodesics, the calculus-of-variations methods are notstrictly necessary at this point; they are included to facilitate their use laterin comparison theorems.Chapter 7 unveils the ﬁrst fully general deﬁnition of curvature. The curvature tensor is motivated initially by the question of whether all Riemannian metrics are locally equivalent, and by the failure of parallel translationto be path-independent as an obstruction to local equivalence. This leadsnaturally to a qualitative interpretation of curvature as the obstruction toﬂatness (local equivalence to Euclidean space). Chapter 8 departs somewhat from the traditional order of presentation, by investigating submanifold theory immediately after introducing the curvature tensor, so as todeﬁne sectional curvatures and give the curvature a more quantitative geometric interpretation.

xPrefaceThe last three chapters are devoted to the most important elementaryglobal theorems relating geometry to topology. Chapter 9 gives a simplemoving-frames proof of the Gauss–Bonnet theorem, complete with a careful treatment of Hopf’s rotation angle theorem (the Umlaufsatz). Chapter10 is largely of a technical nature, covering Jacobi ﬁelds, conjugate points,the second variation formula, and the index form for later use in comparison theorems. Finally in Chapter 11 comes the dénouement—proofs ofsome of the “big” global theorems illustrating the ways in which curvatureand topology aﬀect each other: the Cartan–Hadamard theorem, Bonnet’stheorem (and its generalization, Myers’s theorem), and Cartan’s characterization of manifolds of constant curvature.The book contains many questions for the reader, which deserve specialmention. They fall into two categories: “exercises,” which are integratedinto the text, and “problems,” grouped at the end of each chapter. Both areessential to a full understanding of the material, but they are of somewhatdiﬀerent character and serve diﬀerent purposes.The exercises include some background material that the student shouldhave seen already in an earlier course, some proofs that ﬁll in the gaps fromthe text, some simple but illuminating examples, and some intermediateresults that are used in the text or the problems. They are, in general,elementary, but they are not optional—indeed, they are integral to thecontinuity of the text. They are chose