Geometry from a Differentiable Viewpoint

374 Pages·2012·5.253 MB·English

by  John McCleary

Geometry from a Differentiable Viewpoint
The development of geometry from Euclid to Euler to Lobachevski˘ı, Bolyai,
Gauss, and Riemann is a story that is often broken into parts – axiomatic geome-
try, non-Euclideangeometry, and differential geometry. This poses a problem for
undergraduates: Which part is geometry? What is the big picture to which these
partsbelong?
Inthisintroductiontodifferentialgeometry,thepartsareunitedwithalloftheir
interrelations, motivated by the history of the parallel postulate. Beginning with
the ancient sources, the author first explores synthetic methods in Euclidean and
non-Euclideangeometry and then introduces differential geometry in its classical
formulation, leading to the modern formulation on manifolds such as space-time.
The presentation is enlivened by historical diversions such as Hugyens’s clock
andthe mathematics of cartography. The intertwined approacheswill help under-
graduatesunderstandtheroleofelementaryideasinthemoregeneral,differential
setting.
This thoroughly revised second edition includes numerous new exercises,
togetherwithnewlypreparedsolutionstoselectedexercises.
JOHN MCCLEARY isprofessorofmathematicsatVassarCollegeontheElizabeth
StillmanWilliamsChair.Hisresearchinterestslieattheboundarybetweengeom-
etry and topology, especially where algebraic topology plays a role. His research
papers have appeared in journals such as Inventiones Mathematicae and the
American Journal of Mathematics, and he has also written expository papers in
theAmericanMathematicalMonthly. Heis interestedin the historyofmathemat-
ics, particularlythe historyofgeometryin thenineteenthcenturyandoftopology
inthetwentiethcentury.HeistheauthorofAUser’sGuidetoSpectralSequences
andAFirstCourseinTopology:ContinuityandDimension,andhehaseditedpro-
ceedingsin topologyandin history, aswellasa volumeofthe collectedworksof
JohnMilnor.

Geometry
from a Differentiable Viewpoint
Second Edition
JOHN MCCLEARY
VassarCollege
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521133111
(cid:1)c CambridgeUniversityPress1995,2013
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished1995
Secondeditionpublished2013
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
McCleary,John,1952–
Geometryfromadifferentiableviewpoint/JohnMcCleary.–2nded.
p. cm.
Includesbibliographicalreferencesandindexes.
ISBN978-0-521-11607-7(hardback)–ISBN978-0-521-13311-1(pbk.)
1. Geometry,Differential. I. Title.
QA641.M38 2012
(cid:2)
516.36–dc23 2012017159
ISBN978-0-521-11607-7Hardback
ISBN978-0-521-13311-1Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLs
forexternalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnot
guaranteethatanycontentonsuchWebsitesis,orwillremain,accurateorappropriate.
To my sisters
MaryAnn,Denise,Rose

•
Contents
Prefacetothesecondedition pageix
Introduction xi
PARTA
Preludeandthemes:Syntheticmethodsandresults
1 Sphericalgeometry 3
2 Euclid 12
Euclid’stheoryofparallels 19
Appendix:TheElements:BookI 24
3 Thetheoryofparallels 27
Uniquenessofparallels 28
Equidistanceandboundednessofparallels 29
Ontheanglesumofatriangle 31
Similarityoftriangles 34
TheworkofSaccheri 37
4 Non-Euclideangeometry 43
TheworkofGauss 43
Thehyperbolicplane 47
Digression:Neutralspace 55
Hyperbolicspace 62
Appendix:TheElements:SelectionsfromBookXI 72
PARTB
Development:Differentialgeometry
5 Curvesintheplane 77
Earlyworkonplanecurves(Huygens,Leibniz,Newton,andEuler) 81
Thetractrix 84
Orientedcurvature 86
Involutesandevolutes 89
6 Curvesinspace 99
Appendix:OnEuclideanrigidmotions 110
7 Surfaces 116
Thetangentplane 124
Thefirstfundamentalform 128
Lengths,angles,andareas 130
7bis Mapprojections 138
Stereographicprojection 143
vii
viii Contents
Central(gnomonic)projection 147
Cylindricalprojections 148
Sinusoidalprojection 152
Azimuthalprojection 153
8 Curvatureforsurfaces 156
Euler’sworkonsurfaces 156
TheGaussmap 159
9 Metricequivalenceofsurfaces 171
Specialcoordinates 179
10 Geodesics 185
EuclidrevisitedI:TheHopf–RinowTheorem 195
11 TheGauss–BonnetTheorem 201
EuclidrevisitedII:Uniquenessoflines 205
Compactsurfaces 207
Adigressiononcurves 211
12 Constant-curvaturesurfaces 218
EuclidrevisitedIII:Congruences 223
TheworkofMinding 224
Hilbert’sTheorem 231
PARTC
Recapitulationandcoda
13 Abstractsurfaces 237
14 Modelingthenon-Euclideanplane 251
TheBeltramidisk 255
ThePoincarédisk 262
ThePoincaréhalf-plane 265
15 Epilogue:Wherefromhere? 282
Manifolds(differentialtopology) 283
Vectorandtensorfields 287
Metricalrelations(Riemannianmanifolds) 289
Curvature 294
Covariantdifferentiation 303
Riemann’sHabilitationsvortrag:Onthehypotheseswhichlieatthe
foundationsofgeometry 313
Solutionstoselectedexercises 325
Bibliography 341
Symbolindex 351
Nameindex 352
Subjectindex 354
•
Preface to the second edition
Givinganauthorachanceto rewrite a textis amixedblessing. Thetemptation to
rewrite every sentence is strong, as is the temptation to throw everything out and
start over. When I asked colleagues who had taught from the book what I might
change, I was surprised in one case to hear that I should change nothing. That
presentedachallengetopreparingasecondedition.
SeveralfolkswerekindenoughtopointouterrorsinthefirsteditionthatIhave
fixed.Manythankstoyou.Amongtheseerrorswasamishandlingofcongruences
that has led to additional material in this edition. The notion of congruenceleads
totheimportanttheoryoftransformationgroupsandtoKlein’sErlangenProgram.
I have taken some of the opportunities to apply arguments using transformations,
exposinganotherofthepillarsofgeometry(Stillwell2005)tothereader.
Succumbing to the second temptation, I have reordered the material signifi-
cantly, resulting in one fewer chapter and a better story line. Chapter 4 is a more
or less self-contained exposition of non-Euclideangeometry, and it now parallels
Chapter14better.IhaveaddedmaterialincludingEuclid’sgeometryofspace,fur-
therresultsoncycloids,anothermapprojection,Clairaut’srelation,andreflections
intheBeltramidisk.Ihavealsoaddednewexercises.Duetoalossoffilesforthe
first edition, all of the pictures have been redrawn and improved. In many places
I have made small changes that I hope improve the clarity of my telling of this
amazingandrichstory.
ix