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Differential Geometry
of Manifolds
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Differential Geometry
of Manifolds
Stephen Lovett
AKPeters,Ltd.
Natick,Massachusetts
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CRC Press
Taylor & Francis Group
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Version Date: 20110714
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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
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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
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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.
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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
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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.
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