Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry

848 Pages·2010·15.26 MB·English

by  Marcel Berger (auth.)

GEOMETRY REVEALED
SelectedworksofMarcelBerger
RESEARCH
Lespectred’unevarie´te´ riemanniene(withPaulGauduchon andEdmondMazet),
Springer,1971
(Collaboratively under the pseudonym Arthur L. Besse) Manifolds all of whose
geodesicsareclosed,Springer,1978
(CollaborativelyunderthepseudonymArthurL.Besse)EinsteinManifolds,Springer,
1978
PEDAGOGYANDPOPULARIZATION
GeometryI,II.Springer,1987,2009
Differential Geometry: Manifolds, Curves and Surfaces (with Bernard Gostiaux),
Springer,1987
“Peut-on de´finir la ge´ome´trie aujourd’hui?”, in Results in Mathematics, vol. 40,
pp.37–87,2001
APanoramicViewofRiemannianGeometry,Springer,2003
Cinqsie`clesdemathe´matiquesenFrance,E´d.ADPF(Associationpourladiffusion
delapense´efranc¸aise),2005
Convexite´ dansleplanetdansl’espace:delapuissanceetdelacomplexite´ d’une
notionsimple(withPierreDamphousse),Ellipses,Paris,2006
“Geometry in the 20th century”, 2002, in History of mathematics, edited by
V.L.HansenandJ.J.Gray,EncyclopediaofLifeSupportSystems(EOLSS),Devel-
opedundertheauspicesofUNESCO,EolssPublishers,Oxford,UK
MARCEL BERGER
Geometry Revealed
A Jacob’s Ladder to Modern Higher
Geometry
123
Author Translator
MarcelBerger LesterJ.Senechal
InsitutdesHautesE´tudesScientifiques(IHES) ProfessorEmeritus
Bures-sur-Yvette DepartmentofMathematics
France MountHolyokeCollege
SouthHadley,MA10475
USA
[email protected]
Springer-Verlagthankstheoriginalpublishersofthefiguresforpermissiontoreprinttheminthisbook.
Wehavemadeeveryefforttoidentifythecopyrightownersofallillustrationsincludedinthisbookin
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ISBN978-3-540-70996-1 e-ISBN978-3-540-70997-8
DOI10.1007/978-3-540-70997-8
SpringerHeidelbergDordrechtLondonNewYork
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About the Author
MarcelBergerhasplayedauniqueroleinthedevelopmentofgeometryinFrance.
Theimportantexceptionalholonomyisduetohim,andpracticallyallthegeometry
thatisdeartoheartsofphysicistsisrelatedtoit.Theeleganceofhistheoremonthe
1=4pinchingcontinuestoattractyoungmathematicianstoRiemanniangeometry.
A veritable school of geometry formed about him in the 1970s. His students,
andhisstudents’students(nowabout90innumber),formthenucleusofgeometry
inFrance.Hemaintainscontactwiththeassociation“ArthurBesse”whereheand
hisstudentshavewrittenseveralbooks:theoneonthespectraofRiemannianman-
ifoldswasforawhilethebibleofthesubject.Hisbookonclosedgeodesicsturned
outtobeascientificadventurestorybecauseimportantproblemsweresolveddur-
ingthewritingprocess.TheoneonEinsteinmanifoldsbecameabest-sellerbecause
itappearedatanopportunemomentandpopularizedalittle-knownsubjectwhose
methodswereprimitive.Itshowedgreatflair,fortodaythesemanifoldsconstitute
acommonareaofresearchfortheoreticalphysicistsandmathematiciansofallspe-
cialties.
Lastbutnotleast,MarcelBergerrecognizedMikhaelGromov’stalentandcon-
vinced him to remain in Paris, which has been a determining factor for the devel-
opmentofgeometryinFrance.MarcelBergerhimselfhadlikewisebeenthebenefi-
ciaryofthesupportofAndréLichnerowicz,aleadingfigureindifferentialgeometry
V
VI ABOUTTHEAUTHOR
andrelativisticmechanics,whogaveBergerathesistopicpreciselyontheholonomy
groupsthatsubsequentlyplayedanimportantroleinhiswork.
Asateacher,MarcelBergerhasbeenabletosharehispassionforgeometry(cid:2)
fromtheoutsetthatofelementarygeometryinallitsforms(cid:2)withgenerationsof
students. His book Geometry remains unequaled as a survey, disclosing and mod-
ernizingallthevariouspointsofviewthatcompriseelementarygeometry,whichof
courseisneithersimplenoreasy.
Marcel Berger was Professor at the Universities of Strasbourg (1953–1964),
Nice(1964–1966),Paris(1966–1974),DirectorofResearchatCNRS(1974–1985
and 1994–1996), Director of IHES (1985–1994). He was President of the French
MathematicalSociety1979–1981.
Introduction
Jacobhadadream:andbehold,aladderwasplaceduponthe
earth,anditreachedtoheaven;andbehold,theangelsofGodas-
cendedanddescendedonit;andtheLordstoodaboveitandspoke
(Genesis28,12–13)
Jacobawokefromhissleepandsaid:“Truly,theLordisinthis
placeandIdidnotknowit”... SoJacobaroseearlyinthemorning
andtookthestonethathadservedashispillowandsetitupasa
pillarandpouredoiluponit(Genesis28:16,18)
Numerousproblemsofgeometrythatarequitevisualandcanbepresentedina
verysimplemannerhaveoneormoreofthefollowingpropertiesincommon:
(cid:3) theyremainunsolved,orhavebeensolvedonlyrecentlyfollowinggreatefforts;
(cid:3) for being well understood (cid:2) and eventually completely or partly solved (cid:2) they
requirethecreationofconceptsandtoolsthatvaryintheirdegreeofabstraction,
whichisinanycasegreaterthanwhatisrequiredforstatingtheproblem;
(cid:3) themathematicaltoolsusedinsolvingthemwereconceivedforquiteotherpur-
poses.
In this work we present a whole series of such problems while showing the
necessity of abstract concepts and how they enter progressively into the solution.
These are conceptual notions, each built “above” the preceding and permitting an
increaseinabstraction,representedmetaphoricallybyJacob’sladderwithitsrungs.
Thisparableappearsasaleitmotivthroughoutthisbook.
Wedon’tneglectmentioningproblemsthatstillremainopen,an“openness”that
mayseemaprioriastonishing,butlesssoonceweunderstandthetotalityoftheef-
fortsandconceptualprogressneededforsolvingsimilarproblems.Theseclassical
problems are ever the object of vigorous research, all the while mathematical re-
searchisconstantlysuggestingnewones.Andthusso-calledelementarygeometry
isindeedverymuchaliveandattheveryheartoftheworkofnumerouscontempo-
rarymathematicians.
This book pursues another goal: to show of course the unceasingly renewed
vigor of the spirit of geometry, but also to offer readers the elements of a modern
geometric culture. For mathematical instruction in our time presents a disquieting
paradox. On the one hand, geometry is increasingly present in daily life; we live
inacivilizationofimages.Virtualreality,roboticvision,aerialnavigationandthe
conquest of space require more and more specialists: engineers, aerial controllers,
navigatorsinspace,etc.Butatthesametimegeometry(cid:2)inanycasespatialgeom-
etry(cid:2)isalmostcompletelymissingfromtheinstructionalprogramsofschoolsand
VII
VIII INTRODUCTION
universities. The small amount of three-dimensional geometry that nominally still
belongstothoseprogramsisinfactpracticallynevertreated.
It should seem obvious that geometry is of relatively large importance in the
whole of mathematics, but the reality is quite different: the language of geometry,
geometricalmetaphors,havetakenholdthroughoutasanexpedientinmodernmath-
ematics.Themostbanalofthesemetaphorsconsistsofcallingasetofobjectsofthe
same type a “space” and the elements “points”. This has spread to such an extent
(andit’snoparadox)thatmathematiciansthrowoffpicturesofthespaceinorderto
warmtotheobjectscreatedbythem.Justthinkforexampleofthefunctionspaces,
whoseintroductionhasthrownlightonnumerousproblemsinanalysis.
“It may seem surprising that a simple change in language has brought such
progress. The impact that it produces seems to come from what might be called
atransferofintuition,”writesJeanDieudonnéinanarticleentitledDominationuni-
verselledelagéometrie.Headds:“inbreakingoutofitstraditionalboundaries,[ge-
ometry]has revealed itshidden powers, itsflexibilityanditsextraordinary adapta-
tionabilities,thusbecomingoneofthemostusedanduniversaltoolsinallbranches
ofmathematics.”
It also happens that a theory breathes new life by renewing contact with ge-
ometry in arather unexpected way, so as Antaeus revived his strengths by contact
withmotherEarth:it’sbeenthecase,inrecenthistory,withprobabilitytheory,with
topology under the geometrization of Thurston and Gromov, and with that part of
functional analysis which, under Alain Connes’ influence has become noncommu-
tativegeometry.
For Alain Connes, “a geometer is a person with sufficient vision to be able to
createsufficientmentalimagesthatpermittreatingvariedmathematicalproblems.”
For“whatisdifficultandessentialinmathematicsisthecreationofenoughmental
imagestoallowthebraintofunction.”
AnattemptatexplanationwasgivenbyMichaelAtiyahinhis2000FieldsLec-
tureMathematicsinthetwentiethcentury:“Vision...usesupsomethinglike80or
90 percent of the cortex of the brain. There are about 17 different centers in the
brain,...somepartsareconcerned withvertical,somepartswithhorizontal,some
parts with color, perspective, finally some parts are concerned with meaning and
interpretation.Understanding,andmakingsenseof,theworldthatweseeisavery
important part of our evolution. Therefore spatial intuition or spatial perception is
anenormouslypowerfultool,andthatiswhygeometryisactuallysuchapowerful
partofmathematics(cid:2)notonlyforthingsthatareobviouslygeometrical,buteven
forthingsthatarenot.”
There remains in the cortex a place for algebra, which Atiyah associates with
time, with the succession of events, of operations. For the geometer, this one-
dimensionality evokes the necessity of complying with the logical rules for proof,
of metaphorically projecting our intuition onto a single axis. Geometers are often
tempted to reject these rules when it is perceived that they bully the intuition, but
knowfromexperiencethatit’spreciselythesethatleadtopushingourimagination
INTRODUCTION IX
beyonditslimitssoastocreatecompletelynewmathematicalobjects,asweattempt
toshowinthiswork.
Asforthebook’sstructure,wehaveontheonehandgroupedtheseproblemsby
theirnature,affinityandsimilarity.Forthemostpartthechaptercanbereadinde-
pendently.Nevertheless,togetherwithourbookGeometry[B]fordetailsbeyondthe
conceptual,thisworkcanserveforacourseingeometryasseenfromthecultural
aspect.Wecaninfactperceivethevariouschaptersasextensionsof[B],illuminat-
ing it with some very recent results. We have therefore used [B] as a systematic
referencefor“elementary”geometry.Althoughbiased,thischoiceisjustifiedsince
only[B]treatsallthenotionsusedhere;conversely,foreachparticularsubjectthere
aresomanybooksthatitishopelesstogivesystematicreferences.
Butmoreimportantisthefactthat,incontrastto[B],theresultsstudiedarenot,
withrareexceptions,provedindetail.Tolightenthereading,somedefinitionshave
beenplacedattheendsofchaptersundertherubricXYZ.Onlythecrucialideasand
abovealltheabstractconceptsintroducedforattainingtheseresultsareelucidated.
Inthisrespectwefollowinthestepsinauguratedbytheabsolutelyremarkable
book Geometry and the Imagination by Hilbert & Cohn-Vossen (original German,
1932, English translation, 1952), which filled a need for modern and easily acces-
sible cultural geometry. It is this book that we hope to emulate, for it seems to us
thatamodernversionisnowmuchneeded.Thiscanonlybeatthepriceofahuge
increaseinsize,giventheexponentialgrowthofresultssincetheappearanceinGer-
manin1932ofHilbert’scourse.Inourprefacetothe1996republication(Hilbert&
Cohn-Vossen, 1996) we emphasized that the work is not intended to be read from
the first to the last page, but that we rather hoped that the reader would open it at
randomandpagethroughitandplungeintothisorthatchapterwithsomepleasure
depending on intuition and inclination. Will we likewise here be able to transmit
ourconvictionthatgeometryisespeciallyaliveandthattherearestillinnumerable
waystobeexploredandconceptstobecreated?
It is important to state that we have by no means covered all the directions of
contemporary geometry. Thus we have made but little room for geometric prob-
ability, very little for combinatorics and none at all for some recent extensions
of the notions of space and point. For this Cartier (1998) and Chap. 3 of Gro-
mov (1999) may be consulted. A good reference for combinatorial geometry is
Pach and Argawal (1995); we also find much that is well presented in Aigner
and Ziegler (1998). A good idea of several new directions in mathematics and
contemporary geometry can be obtained from the recent Carbone, Gromov and
Prusinkiewisz(2000).
We should add that mathematics today is advancing extremely fast. We must
thereforealertthereader(cid:2)andespeciallytheresearcher(cid:2)thatwearecertainlynot
completelyup-to-dateinallsubjectstreated.
ThisbookbeganasacoursegivenattheUniversityofPennsylvaniainthefallof
1994,atSUNYStonyBrookinthewinterof1995andatETHZürichin1995–1996.
We extend our gratitude to those mathematics departments for making it possible
forustooffercourseswhichdivergedverynoticeablyfromtraditionalinstruction.