Geometric Trilogy III. An Differential Approach to Geometry PDF
Preview Geometric Trilogy III. An Differential Approach to Geometry
Francis Borceux A Diff erential Approach to Geometry Geometric Trilogy III A Differential Approach to Geometry Francis Borceux A Differential Approach to Geometry Geometric Trilogy III FrancisBorceux UniversitécatholiquedeLouvain Louvain-la-Neuve,Belgium ISBN978-3-319-01735-8 ISBN978-3-319-01736-5(eBook) DOI10.1007/978-3-319-01736-5 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013954709 MathematicsSubjectClassification(2010): 53A04,53A05,53A45,53A55,53B20,53C22,53C45 ©SpringerInternationalPublishingSwitzerland2014 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. 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Coverimage:CarlFriedrichGaussbyChristianAlbrechtJensen Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Océane,Anaïs,Magali,Lucas, Cyprien, Théophile, Constance,Léonard, Georges, ... and thosestilltocome Preface Thereaderisinvitedtoimmersehimselfina“lovestory”whichhasbeenunfolding for35centuries:thelovestorybetweenmathematiciansandgeometry.Inaddition toaccompanyingthereaderuptothepresentstateoftheart,thepurposeofthisTril- ogyispreciselytotellthisstory.TheGeometricTrilogywillintroducethereaderto themultiplecomplementaryaspectsofgeometry,firstpayingtributetothehistori- calworkonwhichitisbasedandthenswitchingtoamorecontemporarytreatment, making full use of modern logic, algebra and analysis. In this Trilogy, Geometry isdefinitelyviewedasanautonomousdiscipline,neverasasub-productofalgebra or analysis. The three volumes of the Trilogy have been written as three indepen- dent but complementary books, focusing respectively on the axiomatic, algebraic and differential approaches to geometry. They contain all the useful material for a wide range of possibly very different undergraduate geometry courses, depending onthechoicesmadebytheprofessor.Theyalsoprovidethenecessarygeometrical background for researchers in other disciplines who need to master the geometric techniques. Inthe1630sFermatandDescarteswerealreadycomputingthetangentstosome curves using arguments which today we would describe in terms of derivatives (see[4],TrilogyII).However,theseargumentsconcernedalgebraiccurves,thatis, curveswhoseequationisexpressedbyapolynomial,andthederivativeofapolyno- mialissomethingthatonecandescribealgebraicallyintermsofitscoefficientsand exponents,withouthavingtohandlelimits.Somedecadeslater,thedevelopmentof differentialcalculusbyNewtonandLeibnizallowedtheseargumentstobeformal- izedintermsofactualderivatives,forratherarbitrarycurves.Inthepresentbook, wefocusonthisgeneralsettingofcurvesandsurfacesdescribedbyfunctionswhich arenolongerdefinedbypolynomials,butarearbitraryfunctionshavingsufficiently wellbehavedpropertieswithrespecttodifferentiation. We have deliberately chosen to restrict our attention to curves in the 2- and 3- dimensionalrealspacesandsurfacesinthe3-dimensionalrealspace.Althoughwe occasionallygiveahintonhowtogeneralizeseveralofourresultstohigherdimen- sions,ourfocusonlowerdimensionsprovidesthebestpossibleintuitionofthebasic notionsandtechniquesusedtodayinadvancedstudiesofdifferentialgeometry. vii viii Preface Animportantnotionistheconsiderationofparametricequations,followingan ideaofEuler(see[4],TrilogyII).Acloserlookatsuchequationssuggeststhatwe shouldviewa curvenotas theset of points whosecoordinatessatisfy some equa- tion(s), but as a continuous deformation of the real line in R2 or in R3, according to the case. When a parameter t varies on the real line, the parametric equations describesuccessivelyallthepointsofthecurve.Analogously,asurfacecanbeseen asacontinuousdeformationoftherealplaneinthespaceR3.Thisisthenotionof aparametricrepresentation,whichisthebasictoolthatweshalluseinourstudy. Ourfirstchapterisessentiallyhistorical:itspurposeistoexplainwheretheideas ofdifferentialgeometrycamefromandwhywechoosethisorthatprecisedefinition andnotanotherpossibleone. TheformalizedstudyofcurvesthenbeginswithChap.2,wherewerestrictour attention to the simplest case: the plane curves. We pay special attention to basic notionsliketangency,lengthandcurvature,butwealsoproveverydeeptheorems, such as the Hopf theorem for simple closed curves. Working in the plane makes certainly things easier to grasp in a first approach. However, it is also a matter of factthatthestudyofplanecurvesoffersmanyinterestingaspects,suchasenvelopes, evolutes,involutes,whichhavebeautifulapplications.Manyoftheseaspectsdonot generalizeelegantlytohigherdimensions. Our Chap. 3 is a kind of parenthesis in our theory of differential geometry: we present a museum of some specimens of curves which have played an important historicalroleinthedevelopmentofthetheory. Chapter4isthendevotedtothestudyofcurvesinthreedimensionalspace:the so-called skew curves. We focus our attention on the main aspects of the theory, namely, the study of the curvature and the torsion of skew curves and the famous Frenettrihedron. Next we switch to surfaces in R3. In Chap. 5 we concentrate our attention on the local properties of surfaces, that is, properties “in a neighborhood of a given pointofthesurface”,suchasthetangentplaneatthatpointorthevariousnotionsof curvature at that point: normal curvature, Gaussian curvature, and the information that we can get from these on the shape of the surface in a neighborhood of the point. Chapter 6 then begins by repeating many of the arguments of Chap. 5, but us- ing a different notation: the notation of Riemannian geometry. Our objective is to provideinthiswayagoodintuitiveapproachtonotionssuchasthemetrictensor, the Christoffel symbols, the Riemann tensor, and so on. We provide evidence that theseapparentlyverytechnicalnotionsreduce,inthecaseofsurfacesinR3,tovery familiarnotionsstudiedinChap.5.Wealsodevotespecialattentiontothecaseof geodesicsandestablishthemainproperties(includingtheexistence)ofthesystems ofgeodesiccoordinates. The last chapter of this book is devoted to some global properties of surfaces: properties for which one has to consider the full surface, not just what happens in a neighborhood of one of its points. We start with a basic study of the surfaces of revolution, the ruled and the developable surfaces and the surfaces with constant Preface ix curvature. Next we switch to results and notions such as the Gauss–Bonnet theo- rem and the Euler characteristic, which represent some first bridges between the elementarytheoryofsurfacesandmoreadvancedtopics. Each chapter ends with a section of “problems” and another section of “exer- cises”.Problemsaregenerallystatementsnottreatedinthisbook,butoftheoretical interest,whileexercisesaremoreintendedtoallowthereadertopracticethetech- niquesandnotionsstudiedinthebook. Ofcoursereadingthisbookassumessomefamiliaritywiththebasicnotionsof linear algebra and differential calculus, but these can be found in all undergradu- ate courses on these topics. An appendix on general topology introduces the few ingredientsofthattheorywhichareneededtoproperlyfollowourapproachtoRie- manniangeometryandtheglobaltheoryofsurfaces.Asecondappendixstateswith fullprecision(butwithoutproofsthistime)sometheoremsontheexistenceofsolu- tionsofdifferentialequationsandpartialdifferentialequations,whicharerequired insomeadvancedgeometricalresults. Aselectivebibliographyforthetopicsdiscussedinthisbookisprovided.Certain items,nototherwisementionedinthebook,havebeenincludedforfurtherreading. Theauthorthanksthenumerouscollaboratorswhohelpedhim,throughtheyears, toimprovethequalityofhisgeometrycoursesandthusofthisbook.Amongthem aspecialthankstoPascalDupont,whoalsogaveusefulhintsfordrawingsomeof theillustrations,realizedwithMathematicaandTikz. The Geometric Trilogy I.AnAxiomaticApproachtoGeometry 1. Pre-HellenicAntiquity 2. SomePioneersofGreekGeometry 3. Euclid’sElements 4. SomeMastersofGreekGeometry 5. Post-HellenicEuclideanGeometry 6. ProjectiveGeometry 7. Non-EuclideanGeometry 8. Hilbert’sAxiomatizationofthePlane Appendices A. Constructibility B. TheThreeClassicalProblems C. RegularPolygons II.AnAlgebraicApproachtoGeometry 1. ThebirthofAnalyticGeometry 2. AffineGeometry 3. MoreonRealAffineSpaces 4. EuclideanGeometry 5. HermitianSpaces 6. ProjectiveGeometry 7. AlgebraicCurves Appendices A. PolynomialsoveraField B. PolynomialsinSeveralVariables C. HomogeneousPolynomials D. Resultants E. SymmetricPolynomials F. ComplexNumbers xi xii TheGeometricTrilogy G. QuadraticForms H. DualSpaces III.ADifferentialApproachtoGeometry 1. TheGenesisofDifferentialMethods 2. PlaneCurves 3. AMuseumofCurves 4. SkewCurves 5. TheLocalTheoryofSurfaces 6. TowardsRiemannianGeometry 7. ElementsoftheGlobalTheoryofSurfaces Appendices A. Topology B. DifferentialEquations
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