Analysis and algebra on differentiable manifolds : a workbook for students and teachers

635 Pages·2013·3.969 MB·English

by  P M Gadea, J Muñoz Masqué, I V Mikiti︠u︡k

Problem Books in Mathematics
SeriesEditor
PeterWinkler
DepartmentofMathematics
DartmouthCollege
Hanover,NH03755
USA
Forfurthervolumes:
www.springer.com/series/714
Pedro M. Gadea (cid:2) Jaime Muñoz Masqué (cid:2)
Ihor V. Mykytyuk
Analysis and Algebra
on Differentiable
Manifolds
A Workbook for Students and Teachers
Second Edition
PedroM.Gadea IhorV.Mykytyuk
InstitutodeFísicaFundamental InstituteofMathematics
CSIC PedagogicalUniversityofCracow
Madrid,Spain Cracow,Poland
and
JaimeMuñozMasqué InstituteofAppliedProblemsofMechanics
InstitutodeSeguridaddelaInformación andMathematics
CSIC NASU
Madrid,Spain L’viv,Ukraine
ISSN0941-3502 ProblemBooksinMathematics
ISBN978-94-007-5951-0 ISBN978-94-007-5952-7(eBook)
DOI10.1007/978-94-007-5952-7
SpringerLondonHeidelbergNewYorkDordrecht
LibraryofCongressControlNumber:2012955855
Mathematics Subject Classification: 53-01, 53A04, 53A05, 53A07, 53A45, 53C05, 53C07, 53C22,
53C30,53C35,53C43,53C50,53C55,53C80,53D05,53D22,55R10,57R20,58-01,58A05,58A10,
58A12,58A14,58A15,58A17,58A20,58A25,58A30
Firstedition:©KluwerAcademicPublishers2001
Correctededitionofthefirstedition:©SpringerScience+BusinessMediaB.V.2009
©Springer-VerlagLondon2013
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To Mary
Foreword
GeometryhascomealongwayfromthetimeofEuclid,PythagorasandArchimedes.
Manydevelopmentsinthesubjecthaveaccompaniedrevolutionsinwaysofthink-
ing about and describing our world. The creation of the calculus by Newton and
Leibniz was married to analytic descriptions of geometric forms, gravitational
forces and motions of planets. The realisation of non-Euclidean geometries by
Lobachevsky,BolyaiandGaussleadtoabstractdescriptionsofgeometriesofsur-
faces and Riemann’s introduction of the concept of manifold. The consequent di-
minishingofthecentralroleofcoordinatesaccompaniedEinstein’srelativitytheo-
ries. Similar ideas from Hamilton’sreformulation of mechanics lead to Poincaré’s
fundamentalideasindynamics.
Formalisationandextensionofsuchconceptsresultinthefascinatinginterplay
between tensorial geometry and symmetries. This provides foundational building
blocksfortheoreticalmodelsinphysics,buthasalsobecomeanessentialpartofthe
modern treatment of statistical models, engineering design and a significant arena
foranalysis.
Thisisamathematicalsubjectthatisundercontinualdevelopmentandspectac-
ular advances based on these ideas are still being made. Of most popular recent
noteisPerelman’sresolutionofthePoincaréconjectureandtheresultingproofof
Thurston’sgeometrisationconjectureforthree-manifolds.
For those wishing to make good use of these ideas and concepts, there are a
number of excellent texts. However, an exposition of theory is often not enough
andthereisalwayslimitedspacefordemonstratinghands-oncomputations.Incon-
trast to many others, this book is centred around providing a useful set of worked
examples, carefully designed to help develop the reader’s skills and intuition in a
systematicway.
Thisneweditionaddsfreshexamplesandextendsthereferencematerial.Itstays
withinthegeneralscopeofthefirstedition,butalsoprovideswelcomematerialon
newtopics,detailedinthePreface,mostnotablyintheareaofsymplecticgeometry
andHamiltoniandynamics.Thestudentorteacherofacourseinmoderndifferential
geometrywillfindthisavaluableresource.
AarhusUniversity AndrewSwann
vii
Foreword to the First Edition
AfamousSwissprofessorgaveinBaselastudent’scourseon“Riemannsurfaces.”
Afteracoupleoflecturesastudentaskedhim,“Professor,untilnowyoudidnotgive
an exact definition of a Riemann surface.” The professor answered, “Concerning
Riemannsurfaces,themainthingistoUNDERSTANDthem,nottodefinethem.”
This episode really happened. The student’s objection was necessary and rea-
sonable. From a formal viewpoint, it is, of course, necessary to start as soon as
possiblewithstrictdefinitions.Buttheansweroftheprofessoralsohasasubstan-
tial background. Namely, the pure definition of a Riemann surface—as a complex
1-dimensionalcomplexanalyticmanifold—doesabsolutelynothingcontributetoa
realunderstanding.IttakesreallyalongtimetounderstandwhataRiemannsurface
is.
Thisexampleistypicalfortheobjectsofglobalanalysis—manifoldswithstruc-
tures.Ontheonehand,therearecomplexconcretedefinitions,butontheotherhand,
thesedonotautomaticallyexhibitwhattheyreallymean,whatwecandowiththem,
whichoperationstheyreallyadmit,howrigidthisallis.Hencetherearisesthenat-
uralquestion:Howtosubmitadeeperunderstanding,whatisthebest—oratleasta
good—waytodothis?
One well-known way for this is to underpin the definitions, theorems and con-
structionsbyhierarchiesofexamples,counterexamplesandexercises.Theirchoice,
construction and logical order is for any teacher in global analysis an interesting,
importantandfuncreatingtask.
This workbook is a really succeeded attempt to submit to the reader by very
clevercomposedseriesofexercisesandexamplescoveringthewholeareaofmani-
folds,Liegroups,fibrebundlesandRiemanniangeometryadeepunderstandingand
feeling.
Thechoiceandorderoftheexamplesandexerciseswillbeextraordinarilyhelp-
fulandusefulforanystudentorteacherofmanifoldsanddifferentialgeometry.
GreifswaldUniversity JürgenEichhorn
ix
Preface
As stated in the Preface to the first edition, this book intends to provide material
for the practical side of standard courses on analysis and algebra on differentiable
manifoldsatamiddlelevel,correspondingtoadvancedundergraduateandgraduate
years.TheexercisesfocusonLiegroups,fibrebundles,andRiemanniangeometry.
Aims, approach and structure of the book remain largely the same as in the first
edition.Inthepresentedition,thenumberoffiguresis68.
Theprerequisitesarelinearandmultilinearalgebra,calculusofseveralvariables,
various concepts of point–set topology, and some familiarity with linear algebraic
groups,thetopologyoffibrebundles,andmanifoldtheory.
Wewouldliketoexpressourappreciationtotheauthorsofsomeexcellentbooks
as those which appear in the references in chapters. These books have served us
asasourceofideas,inspiration,statementsandsometimesofresults.Westrongly
recommendthesebookstothereader.
Weintroducenowabriefoverviewofthecontents.
Chapters1to6contain412solvedproblems,sortedaccordingtotheaforemen-
tioned topics and in almost the same vein, notations, etc., as in the first edition,
but 39 problems of the first edition have been deleted and 76 new problems have
beenaddedinthepresentedition.Thefirstsectionofeachchaptergivesaselection
of those definitions and theorems whose terminology, with ample use throughout
thebook,couldbemisleadingduetothelackofuniversalacceptance.However,we
shouldliketoinsistonthefactthatwedonotclaimthatthisisanykindorpartofa
bookonthetheoryofdifferentiablemanifolds.
Wenowunderlinesomeofthechangesinthisedition.
Unlikethefirsteditionofthebook,inthepresenteditiontheEinsteinsummation
conventionisnotused.
WeconsiderinSect.1.3(andonlythere)differentiablestructuresdefinedonsets,
analysing what happens when one of the properties of being Hausdorff or second
countablefailstohold.Wethustrytoelicitinthereaderabetterunderstandingof
themeaningandimportanceofthesetwoproperties.
In Chap. 1 of the present edition, we have added, as an instructive example, a
problem where we prove in detail that the manifold of affine straight lines of the
xi
xii Preface
plane, the 2-dimensional real projective space RP2 minus a point, and the infinite
Möbiusstriparediffeomorphic.
InChap.4,twonewproblemshavebeenaddedinthesectionconcerningtheex-
ponentialmap,wherethesimplyconnectedLiegroupcorrespondingtoagivenLie
algebraisobtained.Thesectiondevotedtotheadjointrepresentation,containssix
new problems concerning topics such as Weyl group, Cartan matrix, Dynkin dia-
grams,etc.Similarly,thesectiondevotedtoLiegroupsoftransformationshasbeen
increasedintennewapplicationproblemsinsymplecticgeometry,Hamiltonianme-
chanics,andotherrelatedtopics.Finally,wehaveaddedinthesectionconcerning
homogeneous spaces two problems on homogeneous spaces related to the excep-
tionalLiegroupG .
2
The section on characteristic classes in Chap. 5 includes two new problems on
theGodbillon–Veyclassinthepresentedition.Moreover,thelastsection,devotedto
almostsymplecticmanifolds,Hamilton’sequations,andtherelationwithprincipal
U(1)-bundles, contains five new problems, including topics as Hamiltonianvector
fields.
In the present edition, the section of Chap. 6 concerning Riemannian connec-
tionshasbeenenlarged,includingsixnewproblemsonalmostcomplexstructures.
The section on Riemannian geodesics also includes four new problems on spe-
cial metrics. Moreover, a completely new section is devoted to a generalisation of
Gauss’ Lemma. The section on homogeneous Riemannian and Riemannian sym-
metricspacescontainstwonewproblemsaboutgeneralpropertiesofhomogeneous
Riemannian manifolds and two new problems on specific three-dimensional Rie-
mannian spaces. Furthermore, a short novel section deals with some properties of
theenergyofHopfvectorfields.Thesectiononleft-invariantmetricsonLiegroups
containsinparticulartwonewproblems:Onegivesinadetailedwaythestructureof
theKodaira–Thurstonmanifold;andtheotherfurnishesthedeRhamcohomology
ofaspecificnilmanifold.
Chapter 7 offers an expanded 56-page long collection of formulae and tables
concerning frequent spaces and groups in differential geometry. Many of them do
notactuallyappearintheproblems,buthavingthemcollectedtogethermayprove
usefulasanaide-mémoire,eventoteachersandresearchers.
Attheendofthereferencestoeachchapter,severalbooks(orpapers)appearthat
havenotbeenexplicitlycited,butsuchthattheyhaveinspiredseveralideasofthe
chapterand/orareveryusefulreferences.
All in all, we hope that this new edition of the book will again render a good
servicetopractitionersofdifferentialgeometryandrelatedtopics.
We acknowledge the anonymous referees for their thorough, enlightening and
suggestivereports;theirinvaluablesuggestionsandcorrectionshavecontributedto
improveseveralaspectsofcontentsaswellaspresentationofthebook.
Our hearty thanks to Professor José A. Oubiña for his generous help and
wise advices. We are also indebted to Mrs. Dava Sobel and Professors William
M.Boothby,JoséC.González-Dávila,SigurdurHelgason,A.MontesinosAmilibia,
Kent E. Morrison, John O’Connor, Peter Petersen, Edmund F. Robertson, Waldyr
Preface xiii
A. Rodrigues, Jr., Chris M. Wood, and John C. Wood who kindly granted us per-
missiontoreproduceheresomeoftheirniceandinterestingtexts,resultsandcon-
structions.
WeareindebtedtoDonatasAkmanavicˇiusandhisteamfortheircarefulediting.
We are also indebted to José Ignacio Sánchez García for his excellent work on
thegraphics.
OurspecialthankstoAndrewSwann,whokindlyacceptedourinvitationtowrite
theForeword.
Madrid,Spain PedroM.Gadea
JaimeMuñozMasqué
L’viv,Ukraine IhorV.Mykytyuk