Table Of ContentFundamentals of Advanced Mathematics 3
New Mathematical Methods, Systems and Applications Set
coordinated by
Henri Bourlès
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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Preface
This third volume of Fundamentals of Advanced Mathematics (the first two
volumes are referenced by [P1] and [P2] below) is dedicated to differential and
integralcalculus,examinedfromboththelocalandglobalperspectives.Thisbookis
intended for anyone who uses mathematics (mathematicians, but also physicists and
engineers,andinparticularanyonewhoneedstounderstandthecontrolofnonlinear
systems). Some local questions of integral calculus were already partially addressed
in[P2],Chapter4,andthenaturalframeworkofdifferentialcalculus,Banachspaces,
ispresentedinChapter1ofthisthirdvolume.Nonetheless,wewillneedtoconsider
a few generalizations to sketch the so-called “convenient” context for more recent
developments in global analysis; more on this later. We will also present the
“Carathédory conditions”, which are finer than the classical Cauchy–Lipschitz
existence and uniqueness conditions for solutions of finite-dimensional ordinary
differentialequations.
Globalquestionsdemandanotherperspective.Seenthroughourwindow,theEarth
appearsentirelyflat, yeteventheGreeksintheageofPlatoknewbetter, astestified
byanexcerptfromhisPhaedo(108,e).KnowledgeoftheEarth’sshapeundoubtedly
extended back to the Pythagoreans in the 6th Century BC. Thus, Euclid, who was
an avid student of Plato’s work (consider, for example, his construction of the five
Platonic solids in Book XIII of the Elements), certainly also knew that the Earth is
round. Yet his geometry is quite different from the geometry of a sphere. Euclidean
geometryoffersagoodlocalapproximationofsphericalgeometrybutwasofcourse
useless for the lengthy voyages of the Renaissance. Global analysis must therefore
beexpressedintheframeworkofmanifolds,aconceptwhichgeneralizescurvesand
surfacessinceRiemann.
Ever since the invention of variational calculus in the late 17th Century, it has
become common in mathematics to argue about sets of functions. If certain
conditions are satisfied, these sets can be endowed with the structure of a manifold,
xii FundamentalsofAdvancedMathematics3
in which case they are called “functional manifolds” and imagined as deformed
versions of the usual function spaces (in the same way that the Earth might be
imagined as a deformed version of the plane). “Banach” manifolds were the first
candidatestobeconsideredasaframeworkforglobalanalysisinthelate1950sand
the 1960s [EEL66, PAL68] (these manifolds are deformed versions of a Banach
space). However, as became clear in [P2], section 4.3, many of the function spaces
encounteredinpracticearenotBanachspaces.Forexample,thespaceE ofinfinitely
differentiablefunctionsonanon-emptyopensubsetofRnisanuclearFréchetspace.
Sincethe1980s,thishasinspiredresearchintomanifoldsthataredeformedversions
ofspacesofthistype; thisisthe“convenient”contextforglobalanalysismentioned
above(whichmaturedaroundthelate1990s[KRI97]).Althoughwewillnotbeable
to present it exhaustively in this book, our discussion of manifolds of mappings in
section5.3willdemonstratetheconsiderablevalueofthisapproach.
Chapters2to4developtherequiredformalism,withabriefdetourinChapter3to
introduceanotionthathasplayedafundamentalroleeversinceÉ.Cartan,theconcept
of fiber bundles, and in particular principal bundles. According to general relativity,
we live in a pseudo-Riemannian space that is the “base” of a principal bundle; 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,alongsidethetheoryofdifferentialp-forms.
Our formalism first begins to truly bear fruit in Chapter 5. Distributions, and the
generalized notion of currents, may now be defined on manifolds instead of open
subsetsofRn.Theideaofexteriorderivativesofadifferentialp-form(introducedby
É.Cartan)allowsustogivehighlycondensedexpressionsfortheclassicalformulas
of “vector calculus” involving gradients, divergences, Laplacians, etc. The first
fundamental result of this chapter is a general formulation of Stokes’ theorem
encompassingtheOstrogradsky,Gauss,Green–RiemannandGreentheoremswidely
used in physics, as well as the “classical” Stokes’ theorem. On a Riemannian
manifold, Stokes’ theorem enables us to formulate Hodge duality, which simplifies
many of our calculations involving vectors. From the perspective of algebraic
topology, Stokes’ theorem also gives rise to two other types of duality: Poincaré
dualityforhomologiesandDeRhamdualityforcohomologies.OnR3,forinstance,
−→
weknowthatthecurlofanyvectorfield E thatderivesfromapotentialiszeroand
(cid:2)→
the divergence of any vector field B that can be expressed as a curl is also zero.
Stokes’ theorem allows us to prove the converse of each claim. The second
fundamentalresultofChapter5istheFrobeniustheorem,whichgivesnecessaryand
sufficientconditionsfortheintegrabilityofa“contactdistribution”.Thisallowsusto
Preface xiii
establish the concept of foliation. The Frobenius theorem also implies a result by
Riemann in Chapter 7 that is essential for general relativity, namely that a
Riemannian manifold is flat if and only if its curvature tensor is zero (section 7.4.3,
Theorem7.56).
Lie groups are manifolds but also groups; in Chapter 6, the group structure
enablesustoperformoperationsthatwouldnotmakesenseonanordinarymanifold,
specifically convolution of functions or distributions. Furthermore, a “taxonomy” of
Lie groups can be established from the Lie algebras associated with each group,
which are vector spaces and therefore easier to study: as a set, the Lie algebra
g = Lie(G) of the Lie group G is the tangent space Te(G) of G at the point e,
where e is the neutral element of G. However, the three “fundamental theorems” of
S.Lieimplythatthereexistsa“dictionary”thatallowsustocharacterizeLiegroups
by the properties of their Lie algebras, at least locally (and globally if G is simply
connected).TheclassificationestablishedbyLieiscompleteinthecaseofsimpleor
semi-simpleLiegroups(oralgebras).Thisisthemostimportantcase,sincetheseare
the groups frequently encountered in particle physics, where they play an essential
role (including the so-called “exceptional” simple Lie algebras). Simple and
semi-simple Lie algebras have been studied since Cartan in terms of their “root
systems”; the ability to represent these root systems graphically (as proposed by
Coxeter and Dynkin, among others) is extremely useful, but cannot be presented in
detailinthisbook1.
On a reductive Lie group G, we can also fully develop the theory of harmonic
analysis(Fouriertransformsoffunctionsortempereddistributions).Theabeliancase
will be presented in detail: when G = Rn, we recover the usual notion of Fourier
transform; when G is the torus Tn, we recover the Fourier series expansion of
periodic functions or distributions. The non-abelian case would fill another entire
volume and thus will only be briefly mentioned (even though engineering
applicationshaverecentlybeenfound[CHI01]).Readersarewelcometorefertothe
bibliographyforthenon-abeliancase[VAR77,VAR89].
Definingageometryonamanifoldisequivalenttoequippingthismanifoldwith
a connection (Chapter 7). Lie groups are implicitly equipped with a connection.
Riemannianorpseudo-Riemannianmanifoldsareoftenimplicitlyequippedwiththe
simplest possible connection: the Levi-Cività connection. This is a special case of a
“G-structure” that is frequently used in general relativity. É. Cartan clarified the
notion of connection; he studied affine, projective and conformal connections,
summarizing his ideas by proposing the concept of “generalized space” [CAR26];
1SeetheWikipediaarticleonCoxeter–Dynkindiagrams.
xiv FundamentalsofAdvancedMathematics3
these spaces are equipped with connections called Cartan connections since
Ehresmann (who rephrased these ideas within the context of principal connections).
Connections can be equipped with curvature (an idea that should be familiar to
relativistic physicists) and in some cases torsion, which attracted considerable
interestfromEinstein([EIN54],AppendixII),whohopedtofindawaytounifythe
theoriesofgravitationandelectromagnetism.
HenriBOURLÈS
June2019
Errata for Volume 1 and Volume 2
Volume1(Cont’d)
1)Onp.12,thefifthlineof(V)shouldreadRinsteadofR.
2)Onp.22,ontheright-handsideof[1.6],itshouldreadl←im−insteadofl−i→m.
3)On p. 41, line 10 should read “the cardinal of G/H (equal to the cardinal of
G\H)”insteadof“thiscardinal”.
4)Onp.45,thefirstlineafter[2.12]shouldreadM3insteadofG.
5)Onp.190,inDefinition3.177,itshouldreadΔninsteadofΔn.
6)Onp.191,inthefirstlineafter[3.70],add“alsodenotedasdp”after“operator”.
7)Onp.193,line17shouldreadSpinsteadofSn.
(cid:2)
8)Onp.220,line7shouldread“π = j∈Jπj wheretheelementarydivisorsπj
arepairwisenon-associatedandmaximal(cid:2)powers(amongallelementarydivisors)of
irreduciblepolynomials”insteadof“π = ni=1πi”.
Volume2
1)Onp.12,line6shouldread“K(cid:3)(α(cid:3))”insteadof“K(α(cid:3))”.
2)Onp.17,line19shouldread“(cid:4)=”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“aisnotoftheformbn,b ∈ K”insteadof“an ∈/
K”;line5shouldread“ζ”insteadof“ς”.
xvi FundamentalsofAdvancedMathematics3
7)Onp.43,inline10,after“yields”,add“withj =0,...,n−2”;thelastsumof
(j) (j+1)
line11shouldread“u ”insteadof“u ”.
i i
8)On p. 51, replace the sentence beginning line 23 by: “The coefficients c
and d ar(cid:3)e free par(cid:4)ameters, the former in C, the second in C×; indeed, putting
M = C t,e−t2/2 we have that GalD(N,M) = C, GalD(M,K) = C×, which
yieldstheshortexactsequenceofAbeliangroups0−→C−→G−→C× −→0,in
otherwordsGisanextensionofC×byC([P1],section2.2.2(II)).”.
9)Onp.57,line2shouldread“nonemptyset”insteadof“set”.
(cid:5)10(cid:6))Onp.62, lines2(cid:5)7, 2(cid:6)9; onp.63, lines3, 8; onp.64, lines7, 9: itshouldread
“ x(cid:3) ”insteadof“ x(cid:3) ”.
j j∈J j i∈J
11)On p. 83, line 22 should read “smallest” instead of “largest” and “=” instead
of“(cid:4)=”.
12)Onp.91,line28shouldread“j (cid:7)i0”insteadof“j ≥i0”.
13)Onp.92,inline19,addafterthelastsentence:“Thisextensionisthenunique.”.
14)Onp.95,line8shouldread“∀(x(cid:3),x(cid:3)(cid:3))”insteadof“∀(x(cid:3);x(cid:3)(cid:3))”.
15)Onp.105,line25shouldread“∀i(cid:7)i(cid:3)0(cid:3)”insteadof“∀i(cid:7)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 (cid:10)→ξ”insteadof“ξ (cid:10)→ξx0”.
19)Onp.133,inline14,delete“inLcsh”.
20)On p. 135, lines 19–21 should read “re(cid:2)duced if for all i ∈ I, the projection
pri(E),wherepriisthecanonicalprojection iEi (cid:2)Ei,isdenseinEi”insteadof
“decreasing[...]Tj)”;inlines27–31,replaceStatement2)ofRemark3.33by:“Every
projectivelimitcanbeputintothef(cid:2)ormofareducedprojectivelim(cid:7)it: ifFi =(cid:3)pri(cid:4)(E)
a(cid:2)nd ψ˜ij is the restriction of ψij to iFi, then E is the subspace i(cid:4)jker ψ˜ij of
iFi.”.
(cid:8)(cid:9)
(cid:10)21)On p. 137, the last line should read “p ni=1(cid:11)xi(cid:11)p” instead of
(cid:9)
“p ni=1(cid:11)xi(cid:11)P”.
22)Onp.141,inlines24,26,delete“decreasing”.
23)Onp.142,inlines10,11,delete“issurjective(ibid.)and”.
24)Onp.144,line11shouldread“boundedinF”insteadof“boundedinM”.
