Fundamentals of Advanced Mathematics 3
Differential Calculus, Tensor Calculus, Differential Geometry, Global Analysis
Henri Bourlès
First published 2019 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Press Ltd
27-37 St George’s Road
Elsevier Ltd
The Boulevard, Langford Lane London SW19 4EU Kidlington, Oxford, OX5 1GB UK UK
www.iste.co.uk
www.elsevier.com
Notices
Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.
For information on all our publications visit our website at http://store.elsevier.com/
© ISTE Press Ltd 2019
The rights of Henri Bourlès to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library Library of Congress Cataloging in Publication Data
A catalog record for this book is available from the Library of Congress ISBN 978-1-78548-250-2
Printed and bound in the UK and US
Chapter1.DifferentialCalculus ........................1
1.1.Introduction.................................1
1.2.Fréchetdifferentialcalculus........................2
1.2.1.Generalconventions...........................2
1.2.2.Fréchetdifferential...........................5
1.2.3.Mappingsofclass C p ..........................9
1.2.4.Taylor’sformulas............................12
1.2.5.Analyticfunctions............................16
1.2.6.Theimplicitfunctiontheoremanditsconsequences.........19
1.3.Otherapproachestodifferentialcalculus.................27
1.3.1.LagrangevariationsandGateauxdifferentials............27
1.3.2.Calculusofvariations:elementaryconcepts.............29
1.3.3.“Convenient”differentials.......................32
1.4.Smoothpartitionsofunity.........................35
1.4.1. C ∞ -paracompactnessofBanachspaces...............35
1.4.2. c∞ -paracompactness..........................36
1.5.Ordinarydifferentialequations.......................37
1.5.1.Existenceanduniquenesstheorems..................37
1.5.2.Lineardifferentialequations......................43
1.5.3.Parameterdependenceofsolutions..................45
Chapter2.DifferentialandAnalyticManifolds ..............49
2.1.Introduction.................................49
2.2.Manifolds:tangentspaceofamanifoldatapoint............50
2.2.1.Notionofamanifold..........................50
2.2.2.Morphismsofmanifolds........................56
2.2.3.Tangentmappings............................58
2.2.4.Tangentvectors.............................58
2.3.Tangentlinearmappings;submanifolds..................65
2.3.1.Tangentlinearmapping;rank.....................65
2.3.2.Differential................................66
2.3.3.Submanifolds..............................67
2.3.4.Immersionsandembeddings......................68
2.3.5.Submersions,subimmersionsand étale mappings..........71
2.3.6.Submanifoldsof Kn ...........................74
2.3.7.Productsofmanifolds..........................75
2.3.8.Transversalmorphismsandmanifolds................76
2.3.9.Fiberproductofmanifolds.......................78
2.3.10.Covectorsandcotangentspaces...................79
2.3.11.Cotangentlinearmapping.......................80
2.4.Liegroupsandtheiractions........................81
2.4.1.Liegroups................................81
2.4.2.Manifoldsoforbitsandhomogeneousmanifolds..........88
Chapter3.FiberBundles ............................93
3.1.Introduction.................................93
3.2.Tangentbundleandcotangentbundle...................94
3.2.1.Tangentbundle..............................94
3.2.2.Cotangentbundle............................96
3.2.3.Tangentbundleandcotangentbundlefunctors............98
3.3.Fibrations...................................98
3.3.1.Notionofafibration...........................99
3.3.2.Fiberproductandpreimageoffibrations...............101
3.3.3.Coverings.................................103
3.3.4.Sections..................................107
3.4.Vectorbundles................................108
3.4.1.Vectorbundles..............................108
3.4.2.Dualofavectorbundle.........................112
3.4.3.Subbundlesandquotientbundles...................113
3.4.4.Whitneysumandtensorproduct....................114
3.4.5.Thecategoryofvectorbundles....................115
3.4.6.Preimageofafiberbundle.......................120
3.5.Principalbundles...............................121
3.5.1.Notionofaprincipalbundle......................121
3.5.2.Verticaltangentvectors.........................123
3.5.3.Morphismsofprincipalbundles....................124
3.5.4.Principalbundlesdefinedbycocycles.................124
3.5.5.Fiberbundleassociatedwithaprincipalbundle...........125
3.5.6.Extension,restriction,quotientizationofthe structuralgroup.................................126
3.5.7.Examplesoftrivialprincipalbundles.................128
Chapter4.TensorCalculusonManifolds
4.1.Introduction.................................131
4.2.Tensorcalculus................................132
4.2.1.Tensors..................................132
4.2.2.Symmetrictensorsandantisymmetrictensors............135
4.2.3.Exterioralgebra.............................138
4.2.4.Dualityintheexterioralgebra.....................139
4.2.5.Interiorproducts.............................141
4.2.6.TensorsonBanachspaces.......................143
4.3.Tensorfields.................................145
4.3.1.Vectorfields...............................145
4.3.2.Covectorfield..............................146
4.3.3.Tensorfieldsandscalarfields.....................146
4.4.Differentialforms..............................148
4.4.1.Differentialformsofdegree p .....................148
4.4.2.Preimageofadifferential p-form...................149
4.4.3.Differentialformstakingvaluesinafiberbundle. Listofformulas.................................151
4.4.4.Orientation................................154
4.4.5.Integralofadifferentialformofmaximaldegree..........157
4.4.6.Differentialformsofoddtype.....................163
4.4.7.Integrationofadifferentialformoverachain............166
4.5.Pseudo-Riemannianmanifolds.......................170
4.5.1.Metric...................................170
4.5.2.Pseudo-Riemannianvolumeelement.................171
Chapter5.DifferentialandIntegralCalculusonManifolds ......173
5.1.Introduction.................................173
5.2.Currentsanddifferentialoperators.....................174
5.2.1.Currentsanddistributions........................174
5.2.2.Differentialoperatorsandpointdistributions............181
5.3.Manifoldsofmappings...........................183
5.3.1.TheBanachframework.........................183
5.3.2.The“convenient”framework......................186
5.4.Liederivatives................................187
5.4.1.Liealgebras...............................187
5.4.2.Liederivativeofafunction.......................190
5.4.3.Liebrackets...............................192
5.4.4.Liederivativeofvector,covectorandtensorfields..........193
5.4.5.Liederivativeofa p-form........................194
5.5.Exteriordifferential.............................195
5.5.1.É.Cartan’stheorem...........................195
5.5.2.Applicationtovectorcalculus.....................198
5.6.Stokes’formulaandapplications......................200
5.6.1.Stokes’formulaonachain.......................200
5.6.2.OstrogradskyandGreenformulas...................203
5.6.3.Hodgedualityandcodifferentials...................206
5.6.4.Gauss’theoremandPoisson’sformula................213
5.6.5.Homology,cohomologyandduality..................215
5.7.Integralcurvesandmanifolds.......................224
5.7.1.First-orderdifferentialequations....................224
5.7.2.Second-orderdifferentialequations..................228
5.7.3.Sprays...................................229
5.7.4.Straighteningofvectorfieldsandframes...............231
5.7.5.Integralmanifolds,foliations......................233
Chapter6.AnalysisonLieGroups
6.1.Introduction.................................245
6.2.Convolution.................................246
6.2.1.Convolutionofdistributions......................246
6.2.2.Haarmeasureandconvolutionoffunctions..............250
6.3.ClassificationofLiealgebras........................256
6.3.1.Additionalnotionsfromalgebra....................256
6.3.2.ClassicalLiealgebras..........................259
6.3.3.GeneralnotionsaboutLiealgebras..................260
6.3.4.NilpotentLiealgebras..........................263
6.3.5.SolvableLiealgebras..........................265
6.3.6.Simpleandsemi-simpleLiealgebras.................267
6.3.7.ReductiveLiealgebras.........................271
6.3.8.RealcompactLiealgebras.......................272
6.4.RelationbetweenLiegroupsandLiealgebras..............273
6.4.1.LiealgebraofaLiegroup.......................273
6.4.2.PassingfromaLiealgebratoaLiegroup...............278
6.4.3.Dictionary................................281
6.5.Harmonicanalysis..............................284
6.5.1.Introduction...............................284
6.5.2.Harmonicanalysison Rn ........................286
6.5.3.FourierseriesandFouriertransformsonthetorus..........296
6.5.4.Fouriertransformonalocallycompactcommutative group.......................................302
6.5.5.Overviewofnon-commutativeharmonicanalysis..........310
Chapter7.Connections .............................315
7.1.Introduction.................................315
7.2.Linearconnections..............................317
7.2.1.Curvilinearcoordinates.........................317
7.2.2.Linearconnectiononavectorbundle.................323
7.2.3.Linearconnectiononamanifold....................325
7.2.4.Paralleltransportandgeodesics....................327
7.2.5.Covariantexteriordifferential.....................330
7.2.6.Curvatureandtorsionofalinearconnection.............331
7.3.Methodofmovingframes.........................333
7.3.1.Movingframeandgaugepotential..................334
7.3.2.Curvature,torsionandcovariantexteriordifferentialofa G-connection..................................337
7.3.3.Quasi-parallelogrammethod......................340
7.3.4.Fundamentalequalities.........................344
7.3.5.Connectionformonthebundleof G-frames............345
7.3.6.Principalconnectionsandparalleltransport.............347
7.3.7.Covariantexteriordifferentiationonaprincipalbundle.......350
7.3.8.Characterizationofa G-connection..................351
7.3.9.Curvatureandtorsionformsofaprincipalconnection.......352
7.3.10.Cartanconnections...........................355
7.4.Riemanniangeometry............................358
7.4.1.Levi-Civitàconnection.........................358
7.4.2.Geodesics.................................360
7.4.3.Flatpseudo-Riemannianmanifolds..................361
7.4.4.RiccitensorandEinsteintensor....................363 References
This page intentionally left blank
Thisthirdvolumeof FundamentalsofAdvancedMathematics (thefirsttwo volumesarereferencedby[P1]and[P2]below)isdedicatedtodifferentialand integralcalculus,examinedfromboththelocalandglobalperspectives.Thisbookis intendedforanyonewhousesmathematics(mathematicians,butalsophysicistsand engineers,andinparticularanyonewhoneedstounderstandthecontrolofnonlinear systems).Somelocalquestionsofintegralcalculuswerealreadypartiallyaddressed in[P2],Chapter4,andthenaturalframeworkofdifferentialcalculus,Banachspaces, ispresentedinChapter1ofthisthirdvolume.Nonetheless,wewillneedtoconsider afewgeneralizationstosketchtheso-called“convenient”contextformorerecent developmentsinglobalanalysis;moreonthislater.Wewillalsopresentthe “Carathédoryconditions”,whicharefinerthantheclassicalCauchy–Lipschitz existenceanduniquenessconditionsforsolutionsoffinite-dimensionalordinary differentialequations.
Globalquestionsdemandanotherperspective.Seenthroughourwindow,theEarth appearsentirelyflat,yeteventheGreeksintheageofPlatoknewbetter,astestified byanexcerptfromhis Phaedo (108,e).KnowledgeoftheEarth’sshapeundoubtedly extendedbacktothePythagoreansinthe6thCenturyBC.Thus,Euclid,whowas anavidstudentofPlato’swork(consider,forexample,hisconstructionofthefive PlatonicsolidsinBookXIIIofthe Elements),certainlyalsoknewthattheEarthis round.Yethisgeometryisquitedifferentfromthegeometryofasphere.Euclidean geometryoffersagoodlocalapproximationofsphericalgeometrybutwasofcourse uselessforthelengthyvoyagesoftheRenaissance.Globalanalysismusttherefore beexpressedintheframeworkof manifolds,aconceptwhichgeneralizescurvesand surfacessinceRiemann.
Eversincetheinventionofvariationalcalculusinthelate17thCentury,ithas becomecommoninmathematicstoargueaboutsetsoffunctions.Ifcertain conditionsaresatisfied,thesesetscanbeendowedwiththestructureofamanifold,
inwhichcasetheyarecalled“functionalmanifolds”andimaginedasdeformed versionsoftheusualfunctionspaces(inthesamewaythattheEarthmightbe imaginedasadeformedversionoftheplane).“Banach”manifoldswerethefirst candidatestobeconsideredasaframeworkforglobalanalysisinthelate1950sand the1960s[EEL66,PAL68](thesemanifoldsaredeformedversionsofaBanach space).However,asbecameclearin[P2],section4.3,manyofthefunctionspaces encounteredinpracticearenotBanachspaces.Forexample,thespace E ofinfinitely differentiablefunctionsonanon-emptyopensubsetof Rn isanuclearFréchetspace. Sincethe1980s,thishasinspiredresearchintomanifoldsthataredeformedversions ofspacesofthistype;thisisthe“convenient”contextforglobalanalysismentioned above(whichmaturedaroundthelate1990s[KRI97]).Althoughwewillnotbeable topresentitexhaustivelyinthisbook,ourdiscussionofmanifoldsofmappingsin section5.3willdemonstratetheconsiderablevalueofthisapproach.
Chapters2to4developtherequiredformalism,withabriefdetourinChapter3to introduceanotionthathasplayedafundamentalroleeversinceÉ.Cartan,theconcept offiberbundles,andinparticularprincipalbundles.Accordingtogeneralrelativity, weliveinapseudo-Riemannianspacethatisthe“base”ofaprincipalbundle;the latterisnamelythemanifoldoforthonormalframe,andits“structuralgroup”(which performschangesofreferenceframe)isa“Liegroup”,namelytheLorentz–Poincaré orthogonalgroupofmatricesleavinginvariantthequadraticform (ds)2 = c2 (dt)2 (dx)2 (dy )2 (dz )2 .Tensorcalculus,astapleofphysicstextbookssincetheearly 20thCentury,ispresentedinChapter4,alongsidethetheoryofdifferential p-forms.
OurformalismfirstbeginstotrulybearfruitinChapter5.Distributions,andthe generalizednotionofcurrents,maynowbedefinedonmanifoldsinsteadofopen subsetsof Rn . Theideaofexteriorderivativesofadifferential p-form(introducedby É.Cartan)allowsustogivehighlycondensedexpressionsfortheclassicalformulas of“vectorcalculus”involvinggradients,divergences,Laplacians,etc.Thefirst fundamentalresultofthischapterisageneralformulationofStokes’theorem encompassingtheOstrogradsky,Gauss,Green–RiemannandGreentheoremswidely usedinphysics,aswellasthe“classical”Stokes’theorem.OnaRiemannian manifold,Stokes’theoremenablesustoformulateHodgeduality,whichsimplifies manyofourcalculationsinvolvingvectors.Fromtheperspectiveofalgebraic topology,Stokes’theoremalsogivesrisetotwoothertypesofduality:Poincaré dualityforhomologiesandDeRhamdualityforcohomologies.On R3 , forinstance, weknowthatthecurlofanyvectorfield −→ E thatderivesfromapotentialiszeroand thedivergenceofanyvectorfield → B thatcanbeexpressedasacurlisalsozero. Stokes’theoremallowsustoprovetheconverseofeachclaim.Thesecond fundamentalresultofChapter5istheFrobeniustheorem,whichgivesnecessaryand sufficientconditionsfortheintegrabilityofa“contactdistribution”.Thisallowsusto
establishtheconceptoffoliation.TheFrobeniustheoremalsoimpliesaresultby RiemanninChapter7thatisessentialforgeneralrelativity,namelythata Riemannianmanifoldisflatifandonlyifitscurvaturetensoriszero(section7.4.3, Theorem7.56).
Liegroupsaremanifoldsbutalsogroups;inChapter6,thegroupstructure enablesustoperformoperationsthatwouldnotmakesenseonanordinarymanifold, specificallyconvolutionoffunctionsordistributions.Furthermore,a“taxonomy”of LiegroupscanbeestablishedfromtheLiealgebrasassociatedwitheachgroup, whicharevectorspacesandthereforeeasiertostudy:asaset,theLiealgebra g = Lie (G) oftheLiegroup G isthetangentspace Te (G) of G atthepoint e, where e istheneutralelementof G.However,thethree“fundamentaltheorems”of S.Lieimplythatthereexistsa“dictionary”thatallowsustocharacterizeLiegroups bythepropertiesoftheirLiealgebras,atleastlocally(andgloballyif G issimply connected).TheclassificationestablishedbyLieiscompleteinthecaseofsimpleor semi-simpleLiegroups(oralgebras).Thisisthemostimportantcase,sincetheseare thegroupsfrequentlyencounteredinparticlephysics,wheretheyplayanessential role(includingtheso-called“exceptional”simpleLiealgebras).Simpleand semi-simpleLiealgebrashavebeenstudiedsinceCartanintermsoftheir“root systems”;theabilitytorepresenttheserootsystemsgraphically(asproposedby CoxeterandDynkin,amongothers)isextremelyuseful,butcannotbepresentedin detailinthisbook1.
OnareductiveLiegroup G,wecanalsofullydevelopthetheoryofharmonic analysis(Fouriertransformsoffunctionsortempereddistributions).Theabeliancase willbepresentedindetail:when G = Rn ,werecovertheusualnotionofFourier transform;when G isthetorus Tn ,werecovertheFourierseriesexpansionof periodicfunctionsordistributions.Thenon-abeliancasewouldfillanotherentire volumeandthuswillonlybebrieflymentioned(eventhoughengineering applicationshaverecentlybeenfound[CHI01]).Readersarewelcometorefertothe bibliographyforthenon-abeliancase[VAR77,VAR89].
Definingageometryonamanifoldisequivalenttoequippingthismanifoldwith aconnection(Chapter7).Liegroupsareimplicitlyequippedwithaconnection. Riemannianorpseudo-Riemannianmanifoldsareoftenimplicitlyequippedwiththe simplestpossibleconnection:theLevi-Civitàconnection.Thisisaspecialcaseofa “G-structure”thatisfrequentlyusedingeneralrelativity.É.Cartanclarifiedthe notionofconnection;hestudiedaffine,projectiveandconformalconnections, summarizinghisideasbyproposingtheconceptof“generalizedspace”[CAR26];
1SeetheWikipediaarticleon Coxeter–Dynkindiagrams
thesespacesareequippedwithconnectionscalled Cartanconnections since Ehresmann(whorephrasedtheseideaswithinthecontextofprincipalconnections). Connectionscanbeequippedwithcurvature(anideathatshouldbefamiliarto relativisticphysicists)andinsomecasestorsion,whichattractedconsiderable interestfromEinstein([EIN54],AppendixII),whohopedtofindawaytounifythe theoriesofgravitationandelectromagnetism.
June2019
HenriB OURLÈS
Volume1(Cont’d)
1)Onp.12,thefifthlineof (V) shouldread R insteadof R.
2)Onp.22,ontheright-handsideof[1.6],itshouldread lim ←− insteadof lim −→
3)Onp.41,line10shouldread“thecardinalof G/H (equaltothecardinalof G\H )”insteadof“thiscardinal”.
4)Onp.45,thefirstlineafter[2.12]shouldread M3 insteadof G
5)Onp.190,inDefinition3.177,itshouldread Δn insteadof Δn .
6)Onp.191,inthefirstlineafter[3.70],add“alsodenotedas dp ”after“operator”.
7)Onp.193,line17shouldread Sp insteadof Sn
8)Onp.220,line7shouldread“π = j ∈J πj wheretheelementarydivisors πj arepairwisenon-associatedandmaximalpowers(amongallelementarydivisors)of irreduciblepolynomials”insteadof“π = n i=1 πi ”.
Volume2
1)Onp.12,line6shouldread“K (α )”insteadof“K (α )”.
2)Onp.17,line19shouldread“=”insteadof“=”;line21shouldread“doesnot exist”insteadof“exists”.
3)Onp.20,line11shouldread“0 ≤ i ≤ r ”insteadof“1 ≤ i ≤ r ”.
4)Onp.24,line28shouldread“x ∈ K”insteadof“x ∈ K”.
5)Onp.27,line10shouldread“K”insteadof“K”.
6)Onp.32,line3shouldread“a isnotoftheform bn ,b ∈ K”insteadof“an / ∈ K”;line5shouldread“ζ ”insteadof“ς ”.
7)Onp.43,inline10,after“yields”,add“with j =0,...,n 2”;thelastsumof line11shouldread“u(j ) i ”insteadof“u(j +1) i ”.
8)Onp.51,replacethesentencebeginningline23by:“Thecoefficients c and d arefreeparameters,theformerin C, thesecondin C× ; indeed,putting M = C t,e t2 /2 wehavethatGalD (N, M)= C, GalD (M, K)= C× , which yieldstheshortexactsequenceofAbeliangroups 0 −→ C −→ G −→ C× −→ 0, in otherwords G isanextensionof C× by C ([P1],section2.2.2(II)).”.
9)Onp.57,line2shouldread“nonemptyset”insteadof“set”.
10)Onp.62,lines27,29;onp.63,lines3,8;onp.64,lines7,9:itshouldread “ xj j ∈J ”insteadof“ xj i∈J ”.
11)Onp.83,line22shouldread“smallest”insteadof“largest”and“=”instead of“=”.
12)Onp.91,line28shouldread“j i0 ”insteadof“j ≥ i0 ”.
13)Onp.92,inline19,addafterthelastsentence:“Thisextensionisthenunique.”.
14)Onp.95,line8shouldread“∀ (x ,x )”insteadof“∀ (x ; x )”.
15)Onp.105,line25shouldread“∀i i0 ”insteadof“∀i i0 ”.
16)Onp.119,line17shouldread“K”insteadof“K”.
17)Onp.128,lines1,2shouldread“(iv)”insteadofthesecond“(iii)”and“(v)” insteadof“(iv)”.
18)Onp.132,line20shouldread“ξx0 → ξ ”insteadof“ξ → ξx0 ”.
19)Onp.133,inline14,delete“in Lcsh”.
20)Onp.135,lines19–21shouldread“reduced ifforall i ∈ I, theprojection pri (E ) , wherepri isthecanonicalprojection i Ei Ei , isdensein Ei ”insteadof “decreasing [ ] Tj )”;inlines27–31,replaceStatement2)ofRemark3.33by:“Every projectivelimitcanbeputintotheformofareducedprojectivelimit:if Fi = pri (E ) and ψ j i istherestrictionof ψ j i to i Fi , then E isthesubspace i j ker ψ j i of i Fi .”.
21)Onp.137,thelastlineshouldread“ p n i=1 xi p ”insteadof “ p n i=1 xi P ”.
22)Onp.141,inlines24,26,delete“decreasing”.
23)Onp.142,inlines10,11,delete“issurjective(ibid.)and”.
24)Onp.144,line11shouldread“boundedin F ”insteadof“boundedin M ”.
25)Onp.168,line13shouldread“x ∈ A”insteadof“x ∈ R”.
26)Onp.150,inline6,delete“decreasing”;line8shouldread“mapping”instead of“surjection”and“→”insteadof“ ”.
27)Onp.172,inline25,itshouldread“exact”insteadof“quasi-exact”;delete “andfrom FSop to Sil”andNote17.
28)Onp.173,inline2,delete“strict”;line5shouldread“an”insteadof“a strict”;inline7,delete“strict”andread“Silva”insteadof“(FS )-”;line8should read“−→”insteadof“ →”and“mapping”insteadof“injection”;inline8,itshould read“mapping”insteadof“surjection”,“−→”insteadof“ →”anddelete“strict”;in line10,itshouldread“(FS )-”insteadof“Silva”;replacelines13–27by:“(1)Since thespaces (Ei )b areFréchet,theyarebornological,andtheirinductivelimitsaswell. ThustheyareMackeyspacesandtheresultfollowsfrom([SCF99],ChapterIV, Section4.4).(2)The Ei arereflexive (DF ) spaces,andtheresultfollowsfrom([SCF 99],Exercise24(f),p.197).”.
29)Onp.174,inline2,delete“increasing”;inline3,itshouldread“mapping” insteadof“injection”and“−→”insteadof“ →”.
30)Onp.175,line14shouldread“L (R; S )”insteadof“L (R,S )”.
31)Onp.177,lines8,9shouldread“ u∈H,x1 ∈A1 u (x1 ,A2 )”insteadof “ u∈H,x1 ∈A1 ”.
32)Onp.185,line5shouldread“such”insteadof“any”;lines15,23shouldread “≥”insteadof“≤”.
33)Onp.186,line11shouldread“| ei |x |2 ”insteadof“ ei |x 2 ”.
34)Onp.188,inline27,add“salient”after“closed”.
35)Onp.193,inline12,add“,v = r i=1 xi ⊗ yi ”attheendoftheline.
36)Onp.200,inline4,change δa (a) to δa (A.)
37)Onp.204,inline5,suppressthesign .
38)Onp.204,line21shouldread“ p |f |p .dμ”insteadof“ p |f | .dμ”.
39)Onp.218,thethirdlineoftheproofshouldread “ 1 h (f (x + h) f (x)) g (c) ”insteadof“ 1 h |f (x + h) f (x)|”.
40)Onp.230,inline16,add“nonzero”after“many”.
41)Onp.253,inlines1,3,delete“strict”;inline26,add“denotedby Cr (0)”after “plane”.
42)Onp.299,line18shouldread“isan A-module”insteadof“isthe Amodule W ”.
43)Onp.204,line13shouldread“equation”insteadof“solution”.
44)Onp.247,changelines1,2to“givenbythefunctionswithabsolutevalue upperboundedbytheabsolutevalueofapolynomial,andtheirpartialderivativesof allordersaswell”.
ListofNotations
Chapter1:DifferentialCalculus
∞,ω, NK , N× K ,p.2
O (f ) ,o (f ):Landaunotation,p.2
E∨ :dualof E,p.2
|.|γ , . γ ,p.2
Ln,s (E ; F ):spaceofsymmetricelementsof Ln (E; F),p.2
u.hn ,p.2
D f (a), df (a), f (a):(Fréchet)differentialof f atthepoint a,p.6
˙ f (a):derivativeof f atthepoint a,p.6
Da (A; F ),p.6
rka (f ),p.6
grada (f ) , ∇a f :gradientof f atthepoint a,p.6
Di f (a) ,∂i f (a) , ∂f ∂xi (a):partialdifferential,partialderivative,p.8
∂ (f 1 ,...,f n )
∂ (x1 ,...,xn ) (a):Jacobian,p.8
C p (A; F ),p.10
Hf (a):Hessianmatrix,p.11
N :Nemitskyoperator,p.11
ev:evaluationoperator,p.11,186
N = N ∪{+∞},p.26
FB ,p.12
b a f (t) dt,p.13
D α :partialdifferentialoforder α,p.16
S (E; F):spaceofformalseries,p.16
S (E; F):spaceofconvergentseries,p.17
ρ (S):radiusofconvergence,p.17
C ω (A; F ),p.17
fa ,p.17
δ f (a) ,δ 2 f (a):first,secondLagrangevariation,p.28
D G f (a):Gateauxdifferential,p.28
c∞ E,p.32
c∞ , cω ,p.33
ϕ (.; t0 ,x0 ),p.41
Φ(t2 ,t1 ):resolvent,p.44
ϕλ = ϕ (.,t0 (λ) ,x0 (λ)),p.46
Chapter2:DifferentialandAnalyticManifolds
(U,ϕ, E) , (U,ϕ,m):chart,p.51,52
dim(M ) , dimx (M ):dimension,p.52
Ta (M ):tangentspace,p.59
θ
c : Ta (M ) → E,ϑc = θ 1 c ,p.59
S r a (M ):spaceofgermsofstationaryfunctionsatthepoint a,p.61
∂
∂ξ j a ,p.61
LXa :Liederivative,p.62
γi∗0 ,∂i (a) ,∂i |a ,p.64
Xa .f ,p.64
Ta (f ) ,f∗a ,f∗ (a):tangentlinearmapping,p.65
rka (f ):rankofamorphism,p.66
da f :differential,p.66
T i (a1 ,a2 ) (f ),p.75
M1 ×Z M2 , f1 ×Z f2 :fiberproduct,p.79
t fa :cotangentlinearmapping,p.80
LieGrp:categoryofLiegroups,p.81
G◦ :neutralcomponentof G,p.83
Tn : n-dimensionaltorus,p.82
H K = K H = H ×τ K:semi-directproductofsubgroups(H normal)
V ∞ ,p.84
λ (s) , ρ (s):left,righttranslation,p.88
GL (E):automorphismgroupof E,p.82
Un (C) , On (R) , SLn (K) , SOn (R), SUn (C):unitary,orthogonal,speciallinear, specialorthogonal,specialunitarygroup,p.86
Z (G):centerof G,p.87
PGLn (K) , PSLn (R) , POn (K) , PSOn (K) , PUn (C):projectivegenerallinear, projectivespeciallinear,projectiveorthogonal,projectivespecialorthogonal, projectiveunitarygroup,p.87
Sp2n (K) , USpn , An , En , SEn :symplectic,unitarysymplectic,generalaffine, affineorthogonal,specialaffineorthogonalgroup,p.87
Dn (K) , Tn (K) , STn (K) , Nn (K),p.87
Ad:adjointmapping,p.88
M/G, G\M :manifoldoforbits,p.89
g:tangentspace Te (G),p.90
Affn (K) , Eucn :affine,affineEuclideanspaces,p.91
Chapter3:FiberBundles
T (B ):tangentfiberbundle,p.94
T ∨ (B ):cotangentfiberbundle,p.96
λ =(M,B,π ):fibration,p.99
Sn : n-dimensionalsphereofradius1,p.98
λ ×B λ ,M ×B M :fiberproductoffibrations,p.101
λ × λ :productoffibrations,p.102
f 0∗ (λ):preimageofafibration,p.102
˜
G:universalcoveringoftheLiegroup G,p.106
Spinn (K):spinorgroup,p.106
Γ(k ) (U,M ) , Γ(U,M ):setofsectionsofclass C k of U in M ,ofmorphismsof class C k from U into M ,p.108
(M,B,π ):vectorbundle M withbase B andprojection π ,p.108
rkb (M ):rankofthevectorbundle M ,p.108
EB :trivialbundle,p.109
M ∨ ,M ∗ :dualbundleof M ,p.112
s∗i = dξ i :dualoftheframe (si )= ∂ ∂ξ i ,p.112
M/M :quotientbundle,p.113
M ⊗ M ,M ⊕ M :tensorproduct,Whitneysumoftwofiberbundles,p.114
M(C) :complexificationoftherealfiberbundle M ,p.115
VB:categoryofvectorbundles,p.117
ker(u) , im (u) , coker (u):kernel,image,cokernelofthelocallydirectmorphism u,p.118
f 0∗ (M ):preimageofthevectorbundle M under f 0 ,p.120
(P,B, G,π ):principalbundle P ofbase B ,structuralgroup G andprojection π , p.121
(B × G,B,pr1 ):trivialprincipalbundle,p.122
Vq (P ):spaceofverticaltangentvectors,p.123
P ×G F, G\ (P × F ) , P ×G F,B,πF ,p.126
Chapter4:TensorCalculusonManifolds
Tp q (E):spaceof p-timescontravariantand q -timescovarianttensors(alsocalled tensorsoftype (p,q )),p.133
T (E):tensoralgebraof E,p.133
ci j : Tp q (E) → Tp 1 q 1 (E):indexcontractionmapping,p.134
( ),p.134
σ.t:imageofthetensor t underthepermutation σ ,p.135
s.t, a.t:symmetrization,antisymmetrizationofthetensor t,p.135
alt,p.136
TSn (E):spaceofsymmetriccontravarianttensorsoforder n,p.135
An (E):spaceofantisymmetriccontravarianttensorsoforder n,p.135
zp ∧ zq :exterior(orwedge)productof zp ∈ Ap (E) and zq ∈ Aq (E),p.138
n E = An (E): n-thexteriorpowerof E,p.138
det(E)= m E,p.138
E = 0≤p≤m p E:exterioralgebraof E,p.139
An (E; F):vectorspaceofalternating n-linearmappingsfrom E n into F,p.140
Sh (p,q ),p.140 ,iv ,i (s):interiorproduct,p.141,153
Altq (E ; E):spaceofcontinuousantisymmetric q -linearmappingsfrom E q into E,p.143,151
λ:vectorfunctor,p.144
∧Φ ,p.145,152
T 1 0 (U ):spaceofvectorfieldsofclass C r ,p.145
T 0 1 (U ) , Ω1 (U ):spaceofcovectorfieldsofclass C r (Pfaffforms),p.146
T p q (U ):tensorfieldoftype (p,q ),p.147
Ωp (U ; N ) , Ωp (U ; F):spaceof p-forms,p.148,151
f ∗ (ω ):preimage,p.149,154,164
Ω(U ; A):deRhamalgebra,p.149,153
OrT (B ) = B :orientationcovering,p.155
B :orientedmanifold,p.157
B ω ,p.160
[ω ]:volumeform,p.161
f :orientationofthemorphism f ,p.162
˜
O :canonicalorientationof ˜ B ,p.163
˜
R:fiberbundleofscalarsofoddtype,p.163
ω, ω,ω ,p.163,165
∂ps τ,∂τ :pseudoboundary,(regular)boundaryofthechain τ ,p.170
g ,gij :metricofapseudo-Riemannianmanifold,p.171
Chapter5:DifferentialandIntegralCalculusonManifolds
Ωp c (B ):spaceofcompactlysupported p-formsofclass C ∞ ,p.174
Ωp , Ωp c :spaceofodd p-forms,p.175
Ωp∨ c , Ωp c ∨ :spaceof(even) p-currents,p.175
T ,ϕ ,p.175
δzb :Dirac p-current,p.176
T ∧ β :exteriorproductofaneven p-currentandanodd q -form,p.176
D (B ) , E (B ):spaceofdistributions,ofcompactlysupporteddistributionson B , p.178
S ⊗ T :tensorproductofcurrentsordistributions,p.178
u (T ) ,D α ξ T , π ∗ (T ),p.178
Diff (B ; M,N ) , Diff (B ): E (B )-moduleofdifferentialoperators,p.181
T ∞ b (B ):spaceofpointdistributionsat b,p.182
T ∞ (B ):spaceoffinitelysupporteddistributionson B ,p.182
C k (B ; Y ) , C p,q (X × Y ; Z ) , c∞ (X ; Y ):manifoldsofmappings,p.183,186, 186
[X,Y ]:Liebracket,p.187,192
LieAl:categoryofLiealgebras,p.188
gl (E),p.188
a1 ⊕ a2 , h ⊕σ k:directsum,semi-directsumofLiealgebras,p.190
