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
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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
Der (A):algebraofderivationsofthealgebra A,p.189
ad:adjointlinearrepresentation,p.190
z (g):centerof g,p.190
LX :Liederivative,p.191,193,194
d:exteriordifferential,p.195
grad, −→ ∇ :gradient,p.198,209
div:divergence,p.199,211
∇2 , Δ:BeltramiLaplacian,deRhamLaplacian,p.199,212
curl,p.199
∂n ϕ:normalderivativeof ϕ,p.204
∗:Hodgeoperator,p.208
, :musicaloperators,p.208
| :bilinearsymmetricforminducedbythemetric,p.208
δ :codifferential,p.210
.|. 2 :bilinearsymmetricform,p.210
L2 B ; Altk (B ) :Hilbertspaceof k -forms,p.210
Zp c (B ; R) , Bp c (B ; R):spaceof p-cocycles,ofcompactlysupported p-coboundaries,p.215
Hp c (B ; R): p-thcompactlysupporteddeRhamcohomologyspace,p.215
Zp (B ) , Bp (B ):spaceof p-cycles,of p-boundaries(forcurrents),p.216
Hp (B ): p-thhomologyspace(forcurrents),p.216
Hp (B ; R)= H∞ p (B ; R): p-thhomologyspace(forchainsofsmoothsimplexes), p.216
bp (B ): p-thBettinumber,p.221
φt :flow,p.226
φ∗ t f = f ◦ φt ,p.227
expZ :exponentialmappingdefinedbythefield Z ,p.229
O (Δ):graduatedidealof Ω(M ) generatedby Γ(r 1) M, Δ0 ,p.238
Chapter6:AnalysisonLieGroups
f g,T S,f S :convolutionproduct,p.246,247,253,254
ϕΔ ,K Δ ,p.247
K(G) :algebraofthegroup G over K,p.250
ΔG :modulusofthegroup G,p.251
ˇ ϕ, ˇ μ:imageof ϕ,of μ underthediffeomorphism s → s 1 ,p.251
μl ,μr :left,rightHaarmeasure,p.251
μ/μ :Haarmeasureonaquotientgroup,p.253
m = mG :leftHaarmeasure,p.253
C0 (G; C):spaceofcontinuousfunctionsconvergingto 0 atinfinity,p.254
Tr (u):traceoftheendomorphism u,p.256
gln (K) , sln (K) , sp (2n, K) , on (K) , dn (K) , tn (K) , stn (K) , nn (K): Liealgebras,p.259
un (C) , sun (C) , an , sen , cn :Liealgebras,p.259
Bg :Killingform,p.260
g = D g:derivedidealof g,p.261
C p g,D p g, p.261
U (g):envelopingalgebraof g,p.261
G:infinitesimalalgebra,p.274
Lie (G)= g:Liealgebraof G,p.274
ZX : q → q.X :fieldoffundamental(orKilling)vectorsassociatedwith X ∈ Lie (G),p.275
expG :exponentialmapping,p.280
Lie:functor LieGrp → LieAl,p.281
F , F :Fouriertransform,cotransform,p.287,300,302
C k b (Rn ) , C k 0 (Rn ),p.289
χa : t → exp(2πia.t),p.287
W k,p (Rn ) ,W k,p 0 (Rn ):Sobolevspaces,p.289
ˆ qs,m ,p.290
M α ,p.289
Fn :normalizedFouriertransform,p.292
OM (Rn ) , Oc (Rn ):spaceofmultipliers,ofconvolutors,p.293
v.p.:Cauchyprincipalvalue,p.295
I =[ 1/2, 1/2]n :unithypercube,p.296
EI (Rn ) , DI (Rn ):spaceof C ∞ functions,ofperiodicdistributions,p.298
U (respectively f ):distribution(respectivelyfunction)onthetorusassociatedwith theperiodicdistribution(respectivelyfunction) U ,p.299
S (Zn ) (respectively S (Zn )):spaceofrapidlydecreasing(respectivelyslowly increasing)sequences,p.299
qm ,p.299
Z Uz .dπ (z ):continuoussumofunitaryrepresentations,p.303
χ:unitarycharacter,p.304
U:groupofcomplexnumberswithunitmodulus,p.304
G:dualgrouporspace,p.304,313
x,x :dualitybracket,p.305
x → x∗ ,f → ˜ f :involution,p.306
A (G),p.306
S (G):spaceoftempereddistributionsonthegroup G,p.309
H⊥ :orthogonalcomplementofthesubgroup H of G,p.310
π :Plancherelmeasure,p.313
F G ,p.313
Θ x :character,p.314
Chapter7:Connections
Γk ij = k ij ,γ k ij :coefficientsoftheconnection,Christoffelsymbolofthesecond kind,p.319,324
ω k j ,ω ,gaugepotential,p.319,336
∇j , ∇Z :covariantderivative,p.320,324
gij ,p.320
Γpq,j =[pq,j ]:Christoffelsymbolofthefirstkind,p.321
g ij ,p.321
∇, d:covariantdifferentialofthelinearconnection C,p.323,326,330
∇0 :covariantdifferentialofthetrivialconnection,p.324
ak ,j ,ak :j ,ak ;j ,p.322
r:curvature,p.331
Rl i,jk :Riemann–Christoffeltensor,p.332
t,T k ij :torsion,p.332
σ ,p.335
ΩU , Ωl i :curvature2-form,p.339
ΘU , Θk :torsion2-form,p.339
R (B ) ,RG (B ):bundleofframes,of G-frames,p.334,345,351
ρg : q −→ q.g :p.346
Hq :spaceofhorizontalvectors,p.347
ω :connection1-formofaprincipalconnection P,p.347
cq :horizontalliftingofthecurve c,p.349
[ω,ω ],p.350
D:covariantdifferentialofaprincipalconnection P,p.351
Ω:curvature2-form,p.352
σ :soldering1-form,p.353
Θ:torsion2-form,p.353
b (ξ ):basicvectorfieldcorrespondingto ξ ,p.354
Rik :componentsoftheRiccitensor Ric,p.363
R:Riccicurvature,p.363
K :Gaussiancurvature,p.363
Sik :componentsoftheEinsteintensor S,p.363
