An Alternative Approach to Lie Groups and Geometric Structures

226 Pages·2018·7.57 MB·English

by  Ercüment H. Ortaçgil

an alternative approach to lie
groups and geometric structures

An Alternative Approach to Lie Groups
and Geometric Structures
ERCÜMENT H. ORTAÇG˙IL
1
3
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FOREWORD
WhenIreceivedthefirstfewchaptersofthisbookfromtheauthor,Iwasbothsurprisedand
intrigued,askingmyselfwhyanyoneshouldwriteawholebookonsuchaseeminglynaive
andtrivialsubjectaslocalLiegroups.Thenextday,however,Ichangedmymindentirely.
Sincethelate1930s,wehavesufferedfromarathernastyaffliction,the“Syndromeof
Globality,”undertheleadershipofGénéralNicolasBourbaki(asplayedmainlybyAndré
Weil,whoneverraisedhisswordinSevastopolattheendoftheCrimeanWar)—although
IhavealwaysthoughtthatthenameofEudoxusofCniduswouldhavebeenamuchbetter
choiceforthemainactorinthissaga.Ithasseemedasiftheonlyacceptableapproachtoany
mathematicalproblemorsettingisaglobalone.Ofcourse,thishasitscharmandbeauty,
a global Lie group, for instance, being a much more expressive structure than any local
counterpart,evenpossessingatopologyandadifferentiablestructureandnolongerbeing
justsomefinitesetofnumericalparametersdifficulttolookforandpinpoint—Humphrey
BogartversusCharlieChaplin.ThenIsuddenlyandunexpectedlynoticedinthesechapters
anaspectofamostcontradictorynature,namely,thedefinitionofaglobalobjectcalleda
localLiegroup.Withthehelpofparallelismandacovariantderivative,theauthorofthis
bookmanagestoproduceaveryusefulglobalobjectthattranscendslocality.Itnotonly
helpsbyprovidingabetterunderstandingandgreaterprecisiontomuchofLieandCartan’s
work,butalsofitslikeaglovewithmanypracticalapplicationsinphysicsandtechnology.It
cannotbedeniedthatweliveandthriveinacompletelylocalspatialcontextandhencemust
adaptourthoughtstothis.Formanyyears,thepresentauthorhashadagreatadmirationfor
globalobjectsandfortheirharmoniousinterlacingandcoexistence,themostspectacular
examplebeingLeonardodaVinci’sLastSupper,whererenaissanceartcametocoexistforthe
firsttimewithprojectivegeometry(perspective).Ishouldalsomentionthatotherauthors
haverecentlyfocusedtheirattentiononlocalitywithinaglobalcontext,anexamplebeing
RonaldBrown’sapproachinhiswritingsonthephilosophyofmathematics.
Asforthebookitself,itisaveryagreeableread,containingexcellentideas,withthorough
argumentationconcerningitspurposes,presentedmostlythroughcompleteproofs.Asa
finalsuggestion,Ibelievethatsimilarconsiderationshouldbegiventotheinfinitegroupsof
transformationsofLieandCartanthatnowadaysshowupinthedisguiseofLiegroupoids
of higher-order Ehresmannian jets or Lie pseudogroups of local transformations in the
terminologyofMatsushimaandKuranishi.
AntonioKumpera

ACKNOWLEDGMENTS
IamgratefultoAntonioKumpera,whobecamemydistantteacherwithhisbookyearsago
andmyclosefriendlater,forkindlyacceptingtowritetheforeworduponmyrequestand
forhismanysuggestionstoimprovemyexposition.
I am also gratefulto Peter Olver for his encouragingtaps on my back over the years.
Withouthissupport,thisbookwouldnotbeinprint.
IexpressmyheartythankstoAnthonyBlaomforreadingthewholebookandmaking
valuablesuggestions.Overtheyears,hegenerouslysharedhisideasandpreprintswithme,
fromwhichIbenefitedgreatly.
WhileIwasstumblingwiththefirstideasforthisbookmanyyearsago,somepeople
gave me invaluable support and protection, and I feel a particular debt of gratitude to
them.TheyaremyvenerableteacherCahitArf,ErgünTog˘rol(RectorofBU,1982–92),
Yalçın Koç (Dean of Arts and Sciences, 1990–92), and my friends Alp Eden (Head of
theMathematicsDepartment,1999–2001)andTeomanTurgut(HeadoftheFezaGürsey
Institute,2005–2011).
Gregor Weingart pointed out to me the mistake mentioned in Chapter 16 and Jan
Draismahelpedmewithitscorrection.MyfriendsEnderAbadog˘lu,YılmazAkyıldız,Yücel
andAydanEg˘eciog˘lu,AlihanNe¸seliler,˙IlhanÖzemekandGeorgeSimpsonmadevarious
suggestionsandcorrections.Ithankthemallverymuch.
IalsothankJean-FrançoisPommaretforourintensecorrespondencein1994,fromwhich
Ilearnedagreatdealaboutjets.
Last,butnotleast,IexpressmyheartythankstoDanielTaber,myeditoratOUP,for
hisfriendlyandprofessionalassistanceduringtheprocessofevaluationandpreparation
ofthisbook.
ErcümentOrtaçgil
[e-mail]
Bodrum(Halicarnassus)2017