(cid:2) (cid:2) (cid:2) (cid:2) Differential Geometry of Manifolds (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Differential Geometry of Manifolds Stephen Lovett AKPeters,Ltd. Natick,Massachusetts (cid:2) (cid:2) (cid:2) (cid:2) CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110714 International Standard Book Number-13: 978-1-4398-6546-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com (cid:2) (cid:2) (cid:2) (cid:2) Contents Preface vii Acknowledgments xiii 1 AnalysisofMultivariableFunctions 1 1.1 Functions from Rn to Rm . . . . . . . . . . . . . . . . . . 1 1.2 Continuity, Limits, and Differentiability . . . . . . . . . . 9 1.3 Differentiation Rules: Functions of Class Cr . . . . . . . . 20 1.4 Inverse and Implicit Function Theorems . . . . . . . . . . 27 2 Coordinates,Frames,andTensorNotation 37 2.1 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . 37 2.2 Moving Frames in Physics . . . . . . . . . . . . . . . . . . 44 2.3 Moving Frames and Matrix Functions . . . . . . . . . . . 53 2.4 Tensor Notation . . . . . . . . . . . . . . . . . . . . . . . 56 3 DifferentiableManifolds 79 3.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 80 3.2 Differentiable Maps between Manifolds . . . . . . . . . . . 94 3.3 Tangent Spaces and Differentials . . . . . . . . . . . . . . 99 3.4 Immersions, Submersions, and Submanifolds . . . . . . . . 113 3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . 122 4 AnalysisonManifolds 125 4.1 Vector Bundles on Manifolds . . . . . . . . . . . . . . . . 126 4.2 Vector Fields on Manifolds . . . . . . . . . . . . . . . . . 135 4.3 Differential Forms . . . . . . . . . . . . . . . . . . . . . . 145 4.4 Integration on Manifolds . . . . . . . . . . . . . . . . . . . 158 4.5 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . 177 v (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) vi Contents 5 IntroductiontoRiemannianGeometry 185 5.1 Riemannian Metrics . . . . . . . . . . . . . . . . . . . . . 186 5.2 Connections and Covariant Differentiation . . . . . . . . . 204 5.3 Vector Fields Along Curves: Geodesics . . . . . . . . . . . 219 5.4 The Curvature Tensor . . . . . . . . . . . . . . . . . . . . 234 6 ApplicationsofManifoldstoPhysics 249 6.1 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 250 6.2 Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . 262 6.3 Geometric Concepts in String Theory . . . . . . . . . . . 269 6.4 A Brief Introduction to General Relativity . . . . . . . . . 278 A PointSetTopology 295 A.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 295 A.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 296 A.3 TopologicalSpaces . . . . . . . . . . . . . . . . . . . . . . 313 A.4 Proof of the Regular Jordan Curve Theorem. . . . . . . . 335 A.5 Simplicial Complexes and Triangulations . . . . . . . . . . 339 A.6 Euler Characteristic . . . . . . . . . . . . . . . . . . . . . 343 B CalculusofVariations 347 B.1 Formulation of Several Problems . . . . . . . . . . . . . . 347 B.2 The Euler-LagrangeEquation . . . . . . . . . . . . . . . . 348 B.3 Several Dependent Variables. . . . . . . . . . . . . . . . . 353 B.4 Isoperimetric Problems and Lagrange Multipliers . . . . . 359 C MultilinearAlgebra 365 C.1 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . 366 C.2 Bilinear and Quadratic Forms . . . . . . . . . . . . . . . . 368 C.3 The Hom Space and the Dual Space . . . . . . . . . . . . 376 C.4 The Tensor Product . . . . . . . . . . . . . . . . . . . . . 381 C.5 Symmetric Product and Alternating Product . . . . . . . 390 C.6 The Wedge Product and Analytic Geometry . . . . . . . . 401 Bibliography 411 Index 415 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Preface Purpose of This Book This book is the second in a pair of books that together are intended to bring the reader through classical differential geometry into the modern formulation of the differential geometry of manifolds, assuming only prior experience in multivariable calculus and linear algebra. The first book in the pair, Differential Geometry of Curves and Surfaces by Banchoff and Lovett[5],introducestheclassicaltheoryofcurvesandsurfaces. Thisbook continuesthedevelopmentofdifferentialgeometrybystudyingmanifolds— the natural generalization of regular curves and surfaces to higher dimen- sions. Though [5] provides many examples of one- and two-dimensional manifolds that lend themselves well to visualization, this book does not rely on [5] and can be read independently. Takenonits own,this bookattempts toprovideanintroductionto dif- ferentiablemanifolds,gearedtowardadvancedundergraduateorbeginning graduate readers and retaining a view toward applications in physics. For readers primarily interested in physics, this book may fill a gap between the geometry typically offered in undergraduate programs and the geom- etry expected in physics graduate programs. For example, some graduate programs in physics first introduce electromagnetism in the context of a manifold. The student who is unaccustomed to the formalism of man- ifolds may be lost in the notation at worst or, at best, be unaware of how to parametrize coordinate patches or how to do explicit calculations of differentials of maps between manifolds. For readers with primarily a mathematicsleaning,this bookgivesaconcreteintroductionto the theory of manifolds at an advanced undergraduate or beginning graduate level. What is Differential Geometry? Differentialgeometrystudiespropertiesofandanalysisoncurves,surfaces, andhigher-dimensionalspacesusingtoolsfromcalculusandlinearalgebra. vii (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) viii Preface Just as the introduction of calculus expands the descriptive and predictive abilities of nearly every field of scientific study, the use of calculus in ge- ometry brings about avenues of inquiry that extend far beyond classical geometry. Though differential geometry does not possess the same restrictions as Euclideangeometryon whattypes of objects it studies, not everyconceiv- able set of points falls within the purview of differential geometry. One of the underlying themes of this book is the development and description of the types of geometric sets on which it is possible to “do calculus.” This leads to the definition of differentiable manifolds. A second, and some- what obvious, theme is how to actually do calculus (e.g., measure rates of change of functions or interdependent variables) on manifolds. A third generalthemeishowto“dogeometry”(e.g.,measuredistances,areas,and angles) on such geometric objects. This theme leads us to the notion of a Riemannian manifold. Applications of differential geometry outside of mathematics first arise in mechanics in the study of the dynamics of a moving particle or system of particles. The study of inertial frames is common to both physics and differential geometry. Most importantly, however, differential geometry is necessarytostudyphysicalsystemsthatinvolvefunctionsoncurvedspaces. Forexample,justtomakesenseofdirectionalderivativesofthesurfacetem- perature at a point on the earth (a sphere) requires analysis on manifolds. The study of mechanics and electromagnetism on a curved surface also requires analysis on a manifold. Finally, arguably the most revolutionary applicationofdifferentialgeometryto physicscame fromEinstein’s theory of generalrelativity. In this theory, Einsteinproposedthat space and time were joined together as a spacetime unit, and he described this spacetime as a 4-manifold that curved more tightly in the presence of mass. Organization of Topics Atypicalcalculussequenceanalyzessingle-variablerealfunctions(R→R), parametriccurves(R→Rn),multivariablefunctions(Rn →R),andvector fields (R2 →R2 or R3 →R3). This does notquite reachthe full generality that one needs for the definition of manifolds. Chapter 1 presents the analysis of functions f :Rn →Rm for any positive integers n and m. Chapter 2 discusses the calculus of moving frames. The concept of moving frames arises as a summary to some results found in Chapters 1,3 and9of[5]butalsointhecontextofinertialframes,animportanttopicin (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Preface ix dynamics. ImplicitinthetreatmentisaviewtowardLiealgebra. However, to retain the chosenlevel of this book, we do not developthis theory here. Chapter 3 defines the category of differentiable manifolds. Manifolds serveastheappropriateandmostcompletegeneralizationtohigherdimen- sions of regular curves and regular surfaces. The chapter also introduces the definition for the tangentspace ona manifold andattempts to provide the underlying intuition behind the abstract definitions. Havingdefinedtheconceptofamanifoldmanifolds,Chapter4develops the analysis on differentiable manifolds, including the differentials of func- tions between manifolds, vector fields, differential forms, and integration. Chapter 5 introduces Riemannian geometry without any pretention of being comprehensive. One can easily take an entire course on Riemannian geometry, the proper context in which one can do both calculus and ge- ometry on a curved space. The chapter introduces the notions of metrics, connections, geodesics, parallel transport, and the curvature tensor. Having developed the technical machinery of manifolds, in Chapter 6 weapplythetheorytoafewareasinphysics. WeconsidertheHamiltonian formulationofdynamics,withaviewtowardsymplecticmanifolds;theten- sorial formulation of electromagnetism; a few geometric concepts involved in string theory, namely the properties of the world sheet that describes a string moving in a Minkowski space; and some fundamental concepts in general relativity. In order to be comprehensive and rigorous and still only require the standard core of most undergraduate math programs, three appendices provide any necessary background from topology, calculus of variations, and multilinear algebra. Using This Book Because of the intended purpose of the book, it can serve well either as a textbook or for self study. The conversationalstyle attempts to introduce new concepts in an intuitive way, explaining why one formulates certain definitions as one does. As a mathematics text, proofs or references to proofs are provided for all claims. On the other hand, this book does not supply all the physical theory and discussion behind the topics we broach. Each section concludes with an ample collection of exercises. The au- thor has marked exercises that require some specific physics background by (Phys) and exercises that require ordinary differential equations with (ODE). Problems marked with (*) indicate difficulty that may be related to technical ability, insight, or length. (cid:2) (cid:2) (cid:2) (cid:2)