Table Of ContentUniversitext
Editorial Board
(North America):
S. Axler
K.A. Ribet
Thomas E. Cecil
Lie Sphere Geometry
With Applications to Submanifolds
Second Edition
Thomas E. Cecil
Department of Mathematics and Computer Science
College of the Holy Cross
1 College Street
Worcester, MA 01610
[email protected]
Editorial Board
(North America):
S. Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
ISBN: 978-0-387-74655-5 e-ISBN: 978-0-387-74656-2
Library of Congress Control Number: 2007936690
Mathematics Subject Classifi cation (2000): 53-02, 53A07, 53A40, 53B25, 53C40, 53C42
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Tomysons,
Tom,Mark,andMichael
Preface to the First Edition
The purpose of this monograph is to provide an introduction to Lie’s geometry of
orientedspheresanditsrecentapplicationstothestudyofsubmanifoldsofEuclidean
space. Lie [104] introduced his sphere geometry in his dissertation, published as
a paper in 1872, and used it in his study of contact transformations. The subject
was actively pursued through the early part of the twentienth century, culminating
with the publication in 1929 of the third volume of Blaschke’s [10] Vorlesungen
über Differentialgeometrie, which is devoted entirely to Lie sphere geometry and
its subgeometries. After this, the subject fell out of favor until 1981, when Pinkall
[146]useditastheprincipaltoolinhisclassificationofDupinhypersurfacesinR4.
Since that time, it has been employed by several geometers in the study of Dupin,
isoparametricandtautsubmanifolds.
This book is not intended to replace Blaschke’s work, which contains a wealth
of material, particularly in dimensions two and three. Rather, it is meant to be a
relativelybriefintroductiontothesubject,whichleadsthereadertothefrontiersof
currentresearchinthispartofsubmanifoldtheory. Chapters2and3(chapternumbers
fromthesecondedition)areaccessibletoabeginninggraduatestudentwhohastaken
courses in linear and abstract algebra and projective geometry. Chapters 4 and 5
containtheapplicationstosubmanifoldtheory. Thesechaptersrequireafirstgraduate
courseindifferentialgeometryasanecessarybackground. Adetaileddescriptionof
thecontentsoftheindividualchaptersisgivenintheintroduction,whichalsoserves
asasurveyofthefieldtothispointintime.
Iwishtoacknowledgecertainworkswhichhavebeenespeciallyusefultomein
writingthisbook. MuchofChapters2and3isbasedonBlaschke’sbook. Theproof
of the Cartan and Dieudonné theorem in Section 3.2 is taken from E.Artin’s book
[4],GeometricAlgebra. TwosourcesareparticularlyinfluentialinChapters4and5.
ThefirstisPinkall’sdissertation[146]andhissubsequentpaper[150], whichhave
proven to be remarkably fruitful. Secondly, the approach to the study of Legendre
submanifoldsusingthemethodofmovingframesisduetoShiing-ShenChern,and
waspresentedintwopapersbyChernandmyself[37]–[38]. Thesetwopapersand
indeedthismonographgrewoutofmyworkwithProfessorChernduringmy1985–
viii PrefacetotheFirstEdition
1986sabbaticalatBerkeley. IamverygratefultoProfessorChernformanyhelpful
discussionsandinsights.
I also want to thank several other mathematicians for their personal contribu-
tions. Katsumi Nomizu introduced me to Pinkall’s work and Lie sphere geometry
in1982, andhisseminaratBrownUniversityhasbeenthesiteofmanyenlighten-
ing discussions on the subject since that time. Thomas Banchoff introduced me to
thecyclidesofDupinintheearlyseventies,whenIwasagraduatestudent,andhe
has provided me with several key insights over the years, particularly through his
films. PatrickRyanhascontributedsignificantlytomyunderstandingofthissubject
throughmanylecturesanddiscussions. Ialsowanttoacknowledgehelpfulconver-
sations and correspondence on various aspects of the subject with Steven Buyske,
SheilaCarter,LeslieCoghlan,JosefDorfmeister,ThomasHawkins,Wu-YiHsiang,
Nicolaas Kuiper, Martin Magid, Reiko Miyaoka, Ross Niebergall, Tetsuya Ozawa,
Richard Palais, Ulrich Pinkall, Helmut Reckziegel, Chuu-Lian Terng, Gudlaugur
Thorbergsson,andAlanWest.
ThisbookgrewoutoflecturesgivenintheBrownUniversityDifferentialGeom-
etrySeminarin1982–83andsubsequentlecturesgiventotheClaviusGroupduring
thesummersof1985–1989attheUniversityofNotreDame,theUniversityofCali-
forniaatBerkeley,FairfieldUniversityandtheInstituteforAdvancedStudy. Iwant
tothankmyfellowmembersoftheClaviusGroupfortheirsupportoftheselectures
andmanyenlighteningremarks. Ialsoacknowledgewithgratitudethehospitalityof
theinstitutionsmentionedabove.
IwishtothankmycolleaguesintheDepartmentofMathematicsattheCollege
oftheHolyCross, severalofwhomaremyformerteachers, formanyinsightsand
much encouragement over the years. I especially wish to mention my first teacher
inlinearalgebraandrealanalysis,LeonardSulski,whorecentlypassedawayaftera
courageousbattleagainstleukemia. ProfessorSulskiwasasuperb,dedicatedteacher,
andagoodandgenerousman. Hewillbemissedbyallwhoknewhim.
While writing this book, I was supported by grants from the National Science
Foundation(DMS-8907366andDMS-9101961)andbyaFacultyFellowshipfrom
theCollegeoftheHolyCross. Thissupportisgratefullyacknowledged.
I want to thank my three undergraduate research assistants from Holy Cross,
MicheleIntermont,ChristopherButlerandKarenPurtell,whowerealsosupported
by the NSF. They worked through various versions of the manuscript and made
manyhelpfulcomments. Ialsowishtothankthemathematicseditorialdepartment
of Springer-Verlag for their timely professional help in preparing this manuscript
for publication, and Kenneth Scott of Holy Cross for his assistance with the word-
processingprogram.
Finally, I am most grateful to my wife, Patsy, and my sons, Tom, Mark, and
Michael, for their patience, understanding and encouragement during this lengthy
project.
August,1991
Preface to the Second Edition
Themostsignificantchangesinthesecondeditionarethefollowing. Firstofall,this
editionofthemanuscriptwaspreparedusingtheLATEXdocumentpreparationsystem,
and thus all of the numbers of the equations, theorems, etc., are different from the
firstedition.
IhaveaddedanewsectionSection4.7whichdescribestheconstructiondueto
Ferus,Karcher,andMünzner[73]ofisoparametrichypersurfaceswithfourprincipal
curvatures,basedonrepresentationsofCliffordalgebras. Ourtreatmentfollowsthe
originalpaperofFerus,Karcher,andMünznerquiteclosely. Ihavealsosubstantially
revisedthepresentationoftheinvarianceoftautnessunderLiespheretransformations
in Section 4.6, giving a different proof, due to J. C. Álvarez Paiva [2], who used
functions whose level sets form a parabolic pencil of spheres rather than the usual
distancefunctionstoformulatetautness. ThisleadstoaverynaturalproofoftheLie
invarianceoftautness.
Sections 5.2–5.4 regarding reducible Dupin hypersurfaces and the cyclides of
Dupinhavealsobeensignificantlyrevised,and11newfiguresillustratingthecyclides
ofDupinhavebeenaddedtoSection5.4. Theintroductionandseveralotherplaces,
forexampleSection4.5,inthetextwheresurveysofknownresultsaregivenhave
allbeenupdatedtoreflectthecurrentstateofresearch.
All 14 of the figures from the first edition were redone, and the second edition
contains14additionalfigures.Allofthesefigureswereconstructedbymycolleague
atHolyCross,AndrewD.Hwang,usinghisePiXprogramforconstructingfiguresin
theLATEXpictureenvironment. IammostgratefultoProfessorHwangfortheexcellent
qualityofthefigures,andforhistimeandeffortinconstructingthem. Theprojectpage
fortheePiXprogramis: http://math.holycross.edu/∼ahwang/software/ePiX.html
In addition to the many mathematicians that I acknowledged in the preface to
thefirstedition,IwishtothankQuo–ShinChiandGaryJensen,withwhomIhave
collaboratedonjointresearchoverthepastdecade. Thiscollaborationhasbeenmost
stimulatingandhasledmetoadeeperunderstandingofmanyaspectsofthissubject,
inparticularthemethodofmovingframesanditsapplicationstoisoparametricand
Dupinhypersurfaces.
x PrefacetotheSecondEdition
I would also again like to thank the members of the Clavius Group for their
encouragementandfortheirsupportofmylecturesonLiespheregeometrygivenat
theUniversityofNotreDameinJuly,2005andtheCollegeoftheHolyCrossinJuly,
2006. IalsowanttothankthoseinstitutionsfortheirhospitalityduringtheClavius
Groupmeetings.
Whilecompletingthissecondedition,IwassupportedbyagrantfromtheNational
ScienceFoundation(DMS–0405529)andbyasabbaticalleavefromtheCollegeof
the Holy Cross. This support is gratefully acknowledged. I also wish to thank my
three recent undergraduate research assistants from Holy Cross, Ellen Gasparovic,
HeatherJohnsonandReneeLaverdiere,whowerealsosupportedbymyNSFgrant,
andwhohelpedmeinrevisingthisbookinvariousways.
IamalsogratefultoAnnKostantandherstaffatSpringerfortheirsupportand
encouragementtocompleteasecondeditionofthebook.
Finally, as with the first edition, I wish to thank my wife, Patsy, and my sons,
Tom, Mark, and Michael, for their warm encouragement and support of all of my
scholarlyefforts,inparticular,thesecondeditionofthisbook.
CollegeoftheHolyCross ThomasE.Cecil
Worcester,Massachusetts
August,2007
Contents
PrefacetotheFirstEdition ........................................ vii
PrefacetotheSecondEdition ...................................... ix
1 Introduction................................................. 1
2 LieSphereGeometry ......................................... 9
2.1 Preliminaries .............................................. 9
2.2 MöbiusGeometryofUnorientedSpheres ...................... 11
2.3 LieGeometryofOrientedSpheres ............................ 14
2.4 GeometryofHyperspheresinSnandHn....................... 16
2.5 OrientedContactandParabolicPencilsofSpheres............... 19
3 LieSphereTransformations ................................... 25
3.1 TheFundamentalTheorem................................... 25
3.2 GenerationoftheLieSphereGroupbyInversions ............... 30
3.3 GeometricDescriptionofInversions........................... 34
3.4 LaguerreGeometry......................................... 37
3.5 SubgeometriesofLieSphereGeometry........................ 46
4 LegendreSubmanifolds ....................................... 51
4.1 ContactStructureon(cid:1)2n−1 .................................. 51
4.2 DefinitionofLegendreSubmanifolds.......................... 56
4.3 TheLegendreMap ......................................... 60
4.4 CurvatureSpheresandParallelSubmanifolds ................... 64
4.5 LieCurvaturesandIsoparametricHypersurfaces ................ 72
4.6 LieInvarianceofTautness ................................... 82
4.7 IsoparametricHypersurfacesofFKM-type ..................... 95
4.8 CompactProperDupinSubmanifolds.......................... 112
xii Contents
5 DupinSubmanifolds.......................................... 125
5.1 LocalConstructions ........................................ 125
5.2 ReducibleDupinSubmanifolds............................... 127
5.3 LieSphereGeometricCriterionforReducibility................. 141
5.4 CyclidesofDupin .......................................... 148
5.5 LieFrames ................................................ 159
5.6 CovariantDifferentiation .................................... 165
5.7 DupinHypersurfacesin4-Space .............................. 168
References ...................................................... 191
Index........................................................... 201
