First steps in differential geometry: Riemannian, Contact, Symplectic PDF
Preview First steps in differential geometry: Riemannian, Contact, Symplectic
Undergraduate Texts in Mathematics Andrew McInerney First Steps in Diff erential Geometry Riemannian, Contact, Symplectic Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Forfurthervolumes: http://www.springer.com/series/666 Andrew McInerney First Steps in Differential Geometry Riemannian, Contact, Symplectic 123 AndrewMcInerney DepartmentofMathematics andComputerScience BronxCommunityCollege CityUniversityofNewYork Bronx,NY,USA ISSN0172-6056 ISBN978-1-4614-7731-0 ISBN978-1-4614-7732-7(eBook) DOI10.1007/978-1-4614-7732-7 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013941656 MathematicsSubjectClassification:53-01,53B20,53D05,53D10,53A35,53A45,58A10,58-01,37J05, 37C05,37C10,34A26,78A05 ©SpringerScience+BusinessMediaNewYork2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ToArlen Preface This text grew out of notes designed to prepare second-year undergraduate stu- dents, primarily mathematics majors, to work on introductory research projects in mathematics. Most of these projects were under the auspices of the Louis Stokes Alliance for Minority Participation at Bronx Community College, City University ofNewYork. Assuch,thetexthastwodistinctparts.Thefirstthreechaptersareintroductory, andshouldbeseenas“everythingonewouldneedtoknow”tounderstandamodern approachtodifferentialgeometry.Thesecondpartofthebookisthecoreofthetext, and showcases three geometric structures that are all prominent areas of current mathematicalresearch.Itisahallmarkofthistexttopresentthethreetogetherinan introductoryway. Thereareseveralconceivableapproachestothistext,dependingonthelevelof the class. To cover the entire text in detail, presupposing only minimal exposure to matrix algebra and multivariable calculus, could take two semesters. For that reason,inaone-semesterdifferentialgeometryclass,Irecommendaquickreview of Chap.2 (focused on Sects.2.8–2.10) and Chap.3, followed by a more careful treatmentofChap.4,especiallySect.4.7.Thisleavestimeforapproximatelythree weeks for each of Chaps.5–7. The student who completes this regimen should haveagoodsenseofdifferentialgeometryasthestudyofsmoothlyvaryingtensor structuresonthetangentbundle.Alternatively,aninstructormightchoosetospend moretimeonChap.3,supplementingthetextwithamorerigoroustreatmentofthe inverseandimplicitfunctiontheorems,andthenfocusingonjustoneortwoofthe geometricstructureslaterinthetext. The price of the early introduction to geometric structures presented here is that it is purely “local.” As mentioned throughout the text, the text stops short of introducingmanifolds,andhasdownplayedtheroleoftopologysignificantly.Ihope thatthetreatmenthereprovidesfirmpreparationtotakethestepsinthatdirection. Ithankthecolleagueswhohavehelpedmeinproofreading,withtheusualcaveat thatallremainingerrorsareminealone:QuanleiFang,MichaelHarrison,Alexander Kheyfits, Mohamed Messaoudene, Cormac O’Sullivan, Philippe Rukimbira, and Anthony Weaver. Prof. Fang also provided helpful guidance in creating figures vii viii Preface in LATEX. Prof. Augustin Banyaga and Prof. Dusa McDuff provided welcome encouragement(nottomentioninspiration!)atvariousstagesoftheproject. IthanktheeditorialteamatSpringer,especiallyKaitlinLeachandthereviewers, fortheirpatienceandhelpfulguidanceinseeingthistextthrough. Finally, I would like to give special thanks to the many students who have in onewayoranotherencouragedmetowritethistext:ChristianCastillo,AliouDiop, Linus Mensah, Rosario Tate, Stephany Soria, Raysa Martinez, Aida Wade, Feraz Mohamed, Frances Villar, Mandie Solo, Jean Yao, and Keysi Peralta, to mention only a few. My MTH 35 class in Fall 2012 helped me reshape parts of the first threechapters;DorianWhytewasespeciallyhelpfulinproofreading.Withoutthese students,thistextwouldnothavecometobe. Bronx,NY AndrewMcInerney Contents 1 BasicObjectsandNotation................................................. 1 1.1 Sets..................................................................... 1 1.2 Functions............................................................... 5 2 LinearAlgebraEssentials................................................... 9 2.1 VectorSpaces.......................................................... 10 2.2 Subspaces .............................................................. 12 2.3 ConstructingSubspacesI:SpanningSets ............................ 13 2.4 LinearIndependence,Basis,andDimension......................... 16 2.5 LinearTransformations................................................ 21 2.6 ConstructingLinearTransformations................................. 23 2.7 ConstructingSubspacesII:SubspacesandLinear Transformations........................................................ 27 2.8 TheDualofaVectorSpace,Forms,andPullbacks.................. 30 2.9 GeometricStructuresI:InnerProducts............................... 37 2.10 GeometricStructuresII:LinearSymplecticForms.................. 44 2.11 ForFurtherReading................................................... 61 2.12 Exercises............................................................... 61 3 AdvancedCalculus........................................................... 67 3.1 TheDerivativeandLinearApproximation........................... 68 3.2 TheTangentSpaceI:AGeometricDefinition ....................... 74 3.3 GeometricSetsandSubspacesofT (Rn)........................... 79 p 3.4 TheTangentSpaceII:AnAnalyticDefinition....................... 91 3.5 TheDerivativeasaLinearMapBetweenTangentSpaces .......... 99 3.6 Diffeomorphisms ...................................................... 103 3.7 VectorFields:FromLocaltoGlobal.................................. 110 3.8 IntegralCurves......................................................... 116 3.9 DiffeomorphismsGeneratedbyVectorFields ....................... 121 3.10 ForFurtherReading................................................... 126 3.11 Exercises............................................................... 127 ix
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