Table Of ContentUndergraduate 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
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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
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