Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists

343 Pages·2013·6.06 MB·English

by  Renteln P.

MANIFOLDS, TENSORS, AND FORMS
Providingasuccinctyetcomprehensivetreatmentoftheessentialsofmoderndif-
ferential geometry and topology, this book’s clear prose and informal style make
itaccessibletoadvancedundergraduateandgraduatestudentsinmathematicsand
thephysicalsciences.
Thetextcoversthebasicsofmultilinearalgebra,differentiationandintegration
on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology,
homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and
degreetheory.Italsofeaturesover250detailedexercises,andavarietyofapplica-
tionsrevealingfundamentalconnectionstoclassicalmechanics,electromagnetism
(including circuit theory), general relativity, and gauge theory. Solutions to the
problemsareavailableforinstructorsatwww.cambridge.org/9781107042193.
PAUL RENTELN is Professor of Physics in the Department of Physics,
California State University San Bernardino, where he has taught a wide range of
courses in physics. He is also Visiting Associate in Mathematics at the California
InstituteofTechnology,whereheconductsresearchintocombinatorics.

MANIFOLDS, TENSORS, AND FORMS
An Introduction for Mathematicians and Physicists
PAUL RENTELN
CaliforniaStateUniversitySanBernardino
and
CaliforniaInstituteofTechnology
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
CambridgeUniversityPressispartoftheUniversityofCambridge.
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107042193
(cid:2)c P.Renteln2014
Thispublicationisincopyright.Subjecttostatutoryexception
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Firstpublished2014
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Renteln,Paul,1959–author.
Manifolds,tensors,andforms:anintroductionformathematiciansandphysicists/PaulRenteln.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-107-04219-3(alk.paper)
1. Geometry,Differential–Textbooks. 2. Manifolds(Mathematics)–Textbooks.
3. Calculusoftensors–Textbooks. 4. Forms(Mathematics)–Textbooks. I. Title.
QA641.R46 2013
516.3(cid:3)6–dc23
2013036056
ISBN978-1-107-04219-3Hardback
Additionalresourcesforthispublicationatwww.cambridge.org/9781107042193
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Contents
Preface pageix
1 Linearalgebra 1
1.1 Vectorspaces 1
1.2 Linearmaps 3
1.3 Exactsequences 4
1.4 Quotientspaces 6
1.5 Matrixrepresentations 7
1.6 Thedualspace 8
1.7 Changeofbasis 9
1.8 Upstairsordownstairs? 11
1.9 Innerproductspaces 14
1.10 TheRieszlemma 19
1.11 Adjointmaps,transposemaps,andduality 20
Additionalexercises 21
2 Multilinearalgebra 30
2.1 Thetensorproduct 30
2.2 Generaltensors 33
2.3 Changeofbasis 34
2.4 Tensorsasmultilinearmaps 34
2.5 Symmetrytypesoftensors 35
(cid:2)
2.6 Alternatingtensorsandthespace pV of p-vectors 38
2.7 Theexterioralgebra 41
(cid:2)
2.8 Theinducedlineartransformation T 42
2.9 TheHodgedual 44
Additionalexercises 49
v
vi Contents
3 Differentiationonmanifolds 54
∗
3.1 Basictopology 54
3.2 Multivariablecalculusfacts 59
3.3 Coordinates 60
3.4 Differentiablemanifolds 62
3.5 Smoothmapsonmanifolds 68
3.6 Immersionsandembeddings 70
3.7 Thetangentspace 73
∗
3.8 ThecotangentspaceT M 79
p
∗
3.9 Thecotangentspaceasjetspace 81
3.10 Tensorfields 83
3.11 Differentialforms 87
3.12 Theexteriorderivative 89
3.13 Theinteriorproduct 93
3.14 Pullback 95
3.15 Pushforward 97
3.16 IntegralcurvesandtheLiederivative 100
Additionalexercises 104
4 HomotopyanddeRhamcohomology 116
4.1 Homotopy 117
4.2 ThePoincarélemma 120
4.3 deRhamcohomology 122
∗
4.4 Diagramchasing 125
∗
4.5 TheMayer–Vietorissequence 128
Additionalexercises 134
5 Elementaryhomologytheory 139
5.1 Simplicialcomplexes 139
5.2 Homology 144
5.3 TheEulercharacteristic 149
Additionalexercises 151
6 Integrationonmanifolds 158
6.1 Smoothsingularhomology 158
6.2 Integrationonchains 159
6.3 Changeofvariables 160
6.4 Stokes’theorem 163
6.5 deRham’stheorem 169
Additionalexercises 174
Contents vii
7 Vectorbundles 176
7.1 Thedefinitions 176
7.2 Connections 181
7.3 Cartan’smovingframesandconnectionforms 183
7.4 CurvatureformsandtheBianchiidentity 184
7.5 Changeofbasis 185
7.6 Thecurvaturematrixandthecurvatureoperator 186
Additionalexercises 188
8 Geometricmanifolds 193
8.1 Indexgymnastics 194
8.2 TheLevi-Civitaconnection 199
8.3 TheRiemanncurvaturetensor 204
8.4 Morecurvaturetensors 206
8.5 Flatmanifolds 208
8.6 Paralleltransportandgeodesics 212
8.7 Jacobifieldsandgeodesicdeviation 215
8.8 Holonomy 216
8.9 Hodgetheory 221
Additionalexercises 225
9 Thedegreeofasmoothmap 249
9.1 ThehairyballtheoremandtheHopffibration 252
9.2 Linkingnumbersandmagnetostatics 255
9.3 The Poincaré–Hopf index theorem and the Gauss–Bonnet
theorem 259
AppendixA Mathematicalbackground 263
AppendixB Thespectraltheorem 271
AppendixC Orientationsandtop-dimensionalforms 274
AppendixD Riemannnormalcoordinates 276
AppendixE Holonomyofaninfinitesimalloop 281
AppendixF Frobenius’theorem 284
AppendixG Thetopologyofelectricalcircuits 296
AppendixH Intrinsicandextrinsiccurvature 308
References 317
Index 321