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Jacques Lafontaine
An Introduction to Differential Manifolds
Grenoble Sciences
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Grenoble Sciences is a department of the Joseph Fourier University
supported by the ministère de l’Enseignement supérieur
et de la Recherche and the région Rhône-Alpes.
An Introduction to Differential Manifolds is a translation of
the original book Introduction aux variétés différentielles (2nd ed.)
by Jacques Lafontaine, EDP Sciences, Grenoble Sciences Series, 2010,
ISBN 978 2 7598 0572 3.
ThereadingcommitteeoftheFrenchversionincludedthefollowingmembers:
Pierre Averbuch, Emeritus CNRS Senior Researcher, Grenoble
Pierre Bérard, Professor, Joseph Fourier University, Grenoble I
Gaël Meigniez, Professor, Bretagne-Sud University
Jean-Yves Mérindol, Professor, President of Sorbonne Paris Cité
Translation from original French version performed by Éric Bahuaud;
excerpt from le Livre de mon ami by Anatole France translated
by Jonathan Upjohn; typesetting: ARCHITEX; figures: Sylvie Bordage;
cover illustration: Alice Giraud, after Klein bottle by Thomas Banchoff
and Jeff Beall (Brown University), The “figure 8” immersion of the
Klein bottle byFropuffandInductiveload(KleinBottle-Figure8-01.png,2007,
Wikimedia Commons) and elements provided by the author.
Jacques Lafontaine
An Introduction
to Differential Manifolds
JacquesLafontaine
Département de Mathématiques
Université Montpellier 2
Montpellier, France
Based on a translation from the French language edition:
‘Introduction aux variétés différentielles’ (2ème édition) by Jacques Lafontaine
Copyright © EDP Sciences, 2010 All Rights Reserved
ISBN 978-3-319-20734-6 ISBN978 -3-319-20735-3 (eB ook)
DOI 10.1007/978-3-319-20735-3
Library of Congress Control Number: 2015946989
Springer Cham Heidelberg New York Dordrecht London
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Preface
Itwasatthatmoment,thatFontanetcameupwithathirdbrain-
wave. “Andwhat”,heexclaimed,“ifwedidahistoryofFrance,in
50volumes,puttingineverysingledetail?” Ifoundthesuggestion
fabulous,clappingmyhandsandshoutingforjoyinapprobation.
We were sent off to bed. But, this sublime idea of a 50-volume
history of France, containing every single detail, kindled such
excitement that, for a full quarter of an hour, I lay there, unable
to sleep.
Andsowelaunchedoutintothishistory. Totellthetruth,Iamno
longerquitesurejustwhywebeganwithKingTeutobochus. But,
we had to; it was what the project demanded. Our first chapter,
then, broughtus face to face with King Teutobochus who, as the
measurementofhisbones(which,incidentally,werediscoveredby
accident) can testify, was 30 feet tall.1 To be confronted by such
a giant, right from the start! Even Fontanet was taken aback.
“We’llhavetoskipTeutobochus”,hesaidtome. But,Ijustdidn’t
have the courage.
And so it was that the 50-volume history of France came to an
end at Teutobochus.
Anatole France, My Friend’s Book
This charming little lesson of methodology applies admirably to the subject
of this book. It is for the reader to judge what I have made of it. The first
steps inthetheoryofmanifoldscan,ifonefollowsFontanet’sfootsteps,have
dire consequences; there is the danger of demotivation, of being discouraged
by the subject before realizing that the real difficulties lie elsewhere.
1. Anatole France (1844–1924, Nobel Prize 1921) is a French writer who has, unfortu-
nately,somewhatfallenintooblivion. Hewasapacifistandadefenderofhumanrights.
Inthisstoryheisreferringtotheperiodbeforethedevelopmentofpalaeontology,when
the bones of prehistoric animals were taken to be the remains of monsters or giants.
The interested reader is referred to the articles “Anatole France” and “Teutobochus”
inWikipedia.
V
VI An Introduction to Differential Manifolds
Smooth manifolds arethe naturalgeneralizationof curvesandsurfaces. The
idea of a manifold appeared for the first time (and without discussion!)
in 1851, in Riemann’s inaugural lecture, and allowed him to construct a
satisfactory solution of the problem of analytic continuation of holomorphic
functions.
It took some 50 years for a precise definition to emerge. It is a question of
conceptualising,notthepartsofaspaceRnwithlargen,definedbyacertain
number of equations, but, in a more abstract way, objects which, “a priori”,
are not within the “ordinary” space of dimension n, for which the notion of
smooth function still makes sense.
Therearenumerousreasonstobeinterestedin“higher” dimensions. Perhaps
oneofthemoreevidentcomesfromclassicalmechanics. Describingthespace
ofconfigurationsofamechanicalsystemrapidlydependsonmorethanthree
parameters: one already needs six for a solid.
The fact that it is not always desirable to consider objects as subsets of Rn
is more subtle. For example, the set of directions in three-dimensional space
dependsontworealparameters,andnaturallyformsamanifoldofdimension
two,calledtheprojectiveplane. Thismanifoldadmits numerousrealizations
as a subspace of Euclidean space, but these realizationsare not immediately
obvious and it is not clear how to select a “natural” one amongst them.
These“abstract” manifoldsfurnishthenaturalmathematicalsettingforclas-
sical mechanics (both configuration and phase space), but also for general
relativity and particle physics.
I wanted to write a text which introduced manifolds in the most direct way
possibleandprincipallyexplorestheirtopologicalproperties,whileremaining
elementary. Inthis wayaspherestretchedanddentedremainsa sphere,and
in the same setting as curves and surfaces. We will mostly be interested in
topological and differential properties over metric properties (length, curva-
ture, etc.).
The reader is expected to have a good knowledge of the basics of differential
calculus and a little point-set topology. Certain remarks, always enclosed
with (cid:2)(cid:2) will require a more elaborate foundation. The first chapter is dedi-
cated to classical differential calculus discussed in a way that will extend
easily to the manifold setting.
Ourproperstudyofmanifoldsstartsinthesecondandthirdchapters. Itried
to give significant examples and results as rapidly as possible.
One class ofexamples – Lie groupstheir homogeneousspaces – struckme as
deserving its own chapter. Chapters 5, 6 and 7 are devoted to differential
forms and their relationship to the topology of manifolds. Each chapter
depends on its predecessors with one exception: if Lie groups (Chapter 4)
Preface VII
arise in subsequent chapters, it’s only through examples and occasionally
in exercises. Finally the last chapter treats the Gauss-Bonnet theorem for
surfaces. One attractive feature of this result is the variety of techniques it
brings into play. Above allit illuminates a phenomena thathas neverceased
to fascinate me through the years: the appearance of integers (perhaps we
could say a quantization?) in geometry.
Each chapter starts with a relatively detailed introduction in which I give
motivations and an informal description of the contents appealing to the
reader’s geometric intuition. A section entitled “Comments” gives possible
extensions on the subjects introduced.
As I explained above, I decided mostly to limit myself to discussing differ-
ential structures. Except in the last chapter, metric structure is discussed
infrequently, and symplectic structure is omitted. I make up a little for this
in the “Comments” section and the annotated bibliography.
The numerous exercises (more than 150) are for the most part easy. Those
labelled with a star are a little more delicate for beginners. Those labelled
with two stars are not necessarily technical but of the “sit and think” style.
Many exercises can be thought of as complementary material to the book.
For this reason I have included the solutions to many of them.
Throughout the years that I taught the course in differential geometry at
Montpellier,Ibenefitedfromanagreeable,attentiveanddemandingaudience
that would leave no question behind. Their attitude deeply encouraged me
as I was preparing the notes which became the first version of this book.
After this book was submitted to Grenoble Sciences, I benefited from
numerous remarks and stimulating suggestions from the reading committee.
I benefited greatly from the broad scientific perspective and temperaments
of these colleagues,anditwas they who encouragedme to write the detailed
introductions I described above.
IwishalsotothankThomasBanchoffandJeffBeallforallowingthepublisher
toreproducetheirbeautifulrealizationoftheKleinbottleonthefrontcover.
Last, but not least, I have been profoundly influenced by my mentor Marcel
Berger.
ThetranslationintoEnglishwasperformedbyEricBahuaud. Iwishtothank
him for an excellent coordination. Moreover, he pointed out and helped me
to correct several bugs in the French version. I am of course responsible for
the remaining ones!
OurjobwassupervisedbyStéphanieTrinewithefficiencyandbonnehumeur.
VIII An Introduction to Differential Manifolds
How to use this book
The first chapter and a good part of Chapters 5 and 6 give a relatively
complete discussion of classical differential calculus, from the beginning to
Stokes’s theorem.
The ambitionofChapter2 isto explainwhatsmoothmanifolds areandhow
tousethemtothosethatmightfindthisnotiontooabstractortootechnical.
Chapter3ismoretechnical,preciselybecauseitexplainstechniquesthattoo
often pass without mention.
Thefinaltwochapterscanbe directlyreadassoonasonemastersalittle bit
of the notions of manifolds and differential forms.
We also note that a reader who, starting with the word “holomorphic” in
the index, completes all of the exercises referred to, will get a sense of the
different world of complex manifolds.
The book is self-contained as far as differential calculus goes. However it
would be in vain to discuss manifolds without a little topology. We very
briefly discuss simple connectedness and covering maps, and I have given
precise and usable statements of results, but at the expense of the book
being self-contained (as far as topology is concerned).
The French version is provided with electronic complements:
https://grenoble-sciences.ujf-grenoble.fr/pap-ebooks/lafontaine/home.
You will find in particular:
some perquisites: connexity, proper maps;
more exercises, more solutions;
Poincaré-Hopf theorem in any dimension;
sporadic isomorphisms between small-dimensional Lie groups.
Jacques Lafontaine
October 2014
Contents
Chapter 1 – Differential Calculus 1
1.1. Introduction................................................................. 1
1.1.1.What Is Differential Calculus?.................................... 1
1.1.2.In This Chapter ..................................................... 3
1.2. Differentials.................................................................. 4
1.2.1.Definition and Basic Properties................................... 4
1.2.2.Three Fundamental Examples..................................... 7
1.2.3.Functions of Class Cp............................................... 10
1.3. The Chain Rule............................................................. 11
1.4. Local Invertibility .......................................................... 14
1.4.1.Diffeomorphisms..................................................... 14
1.4.2.Local Diffeomorphisms............................................. 16
1.4.3.Immersions, Submersions .......................................... 18
1.5. Submanifolds................................................................ 21
1.5.1.Basic Properties..................................................... 21
1.5.2.Examples: Spheres, Tori, and the OrthogonalGroup......... 23
1.5.3.Parametrizations .................................................... 25
1.5.4.Tangent Vectors, Tangent Space.................................. 26
1.6. One-Parameter Subgroups of the Linear Group....................... 29
1.7. Critical Points............................................................... 33
1.8. Critical Values .............................................................. 36
1.9. Differential Calculus in Infinite Dimensions............................ 38
1.10.Comments ................................................................... 40
1.11.Exercises..................................................................... 42
IX
