Complex, contact and symmetric manifolds: In honor of L. Vanhecke

277 Pages·2004·1.269 MB·English

by  Oldrich Kowalski, Emilio E. Musso, Domenico Perrone

Progress in Mathematics
Volume234
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
Complex, Contact and
Symmetric Manifolds
In Honor of L. Vanhecke
Old˘rich Kowalski
Emilio Musso
Domenico Perrone
Editors
Birkha¨user
Boston • Basel • Berlin
EmilioMusso
Old˘richKowalski
Universita`diL’Aquila
CharlesUniversity
DipartimentodiMatematicaPura
FacultyofMathematicsandPhysics
edApplicata
18675Praha
67100L’Aquila
CzechRepublic
Italy
DomenicoPerrone
Universita`degliStudidiLecce
DipartimentodiMatematica“E.DeGiorgi”
73100Lecce
Italy
AMSSubjectClassifications:Primary:53Cxx,53Bxx,53Dxx,57Sxx,58Kxx,22Exx;Secondary:
53C15,53C20,53C21,53C22,53C25,53C26,53C30,53C35,53C40,53C43,53C50,53C55,53C65,
53B05,53B20,53B25,53B30,53B35,53B40,53D10,53D15,55S30,55P62,57S17,57S25,58K05,
22E15,22E67
ISBN0-8176-3850-4 Printedonacid-freepaper.
(cid:1)c2005Birkha¨userBoston
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tenpermissionofthepublisher(SpringerScience+BusinessMediaInc.,RightsandPermissions,233
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Contents
Preface .......................................................... vii
CurvatureofContactMetricManifolds
DavidE.Blair..................................................... 1
ACaseforCurvature:theUnitTangentBundle
H.EricBoeckx .................................................... 15
ConvexHypersurfacesinHadamardManifolds
A.A.Borisenko .................................................... 27
ContactMetricGeometryoftheUnitTangentSphereBundle
G.Calvaruso...................................................... 41
Topological–antitopologicalFusionEquations,PluriharmonicMapsand
SpecialKa¨hlerManifolds
VicenteCorte´s,LarsScha¨fer ......................................... 59
Z andZ-DeformationTheoryforHolomorphicandSymplecticManifolds
2
PaolodeBartolomeis ............................................... 75
Commutative Condition on the Second Fundamental Form of
CR-submanifoldsofMaximalCR-dimensionofaKa¨hlerManifold
MirjanaDjoric´ .................................................... 105
TheGeographyofNon-FormalManifolds
MarisaFerna´ndezandVicenteMun˜oz.................................. 121
vi Contents
Total Scalar Curvatures of Geodesic Spheres and of Boundaries of
GeodesicDisks
J.C.D´ıaz-Ramos,E.Garc´ıa-R´ıo,andL.Hervella ........................ 131
CurvatureHomogeneousPseudo-RiemannianManifoldswhicharenot
LocallyHomogeneous
CoreyDunnandPeterB.Gilkey....................................... 145
OnHermitianGeometryofComplexSurfaces
A.FujikiandM.Pontecorvo.......................................... 153
UnitVectorFieldsthatareCriticalPointsoftheVolumeandoftheEnergy:
CharacterizationandExamples
OlgaGil-Medrano ................................................. 165
On3D-RiemannianManifoldswithPrescribedRicciEigenvalues
OldrˇichKowalskiandZdeneˇkVla´sˇek................................... 187
TwoProblemsinRealandComplexIntegralGeometry
A.M.Naveira ..................................................... 209
NotesontheGoldbergConjectureinDimensionFour
TakashiOguroandKoueiSekigawa.................................... 221
CurvedFlats,ExteriorDifferentialSystems,andConservationLaws
Chuu-LianTerngandErxiaoWang .................................... 235
SymmetricSubmanifoldsofRiemannianSymmetricSpacesandSymmetric
R-spaces
KazumiTsukada ................................................... 255
ComplexformsofQuaternionicSymmetricSpaces
JosephA.Wolf..................................................... 265
Preface
Thisvolumecontainstheextendedversionsofalmostalllecturesdeliveredduringthe
InternationalConference“CurvatureinGeometry”heldinLecce(Italy),11–14June
2003,inhonourofProfessorLievenVanhecke.
Prof.LievenVanheckebeganhisprofessionalcareerattheCatholicUniversityof
Leuven (Belgium) where he obtained his PhD in 1966. He has been teaching at that
University since the academic year 1965–1966 and was appointed full professor in
1972.Since1972,hehasbeentheheadtheSectionofGeometryoftheMathematics
DepartmentoftheCatholicUniversityofLeuven.From1972until1985healsotaught
attheUniversityofAntwerpasapart-timeprofessorandbecameanHonoraryProfessor
therein1985.
Prof.LievenVanheckehasdoneresearchmainlyinthefieldofdifferentialgeometry
and,moreparticularly,inRiemannianandpseudo-Riemanniangeometry.Throughout
his scientific work, the study of curvature and of its properties has always played a
centralrole.Hestartedwithclassicaltopicsonlinecongruencesandminimalvarieties.
Later,heinvestigatedLorentzian,HermitianandKaehlerianmanifolds,almostcomplex
andalmostcontactmanifolds,volumesofgeodesicspheresandtubes,homogeneous
structuresonRiemannianmanifolds,harmonicspaces,generalizedHeisenberggroups
and Damek-Ricci spaces, geodesic symmetries and reflections on Riemannian mani-
folds,Sasakianmanifolds,variousgeneralizationsofsymmetricspaces(e.g.,naturally
reductive,weaklysymmetricandD’Atrispaces),curvaturehomogeneousspaces,fo-
liations,thegeometryofthetangentbundleandoftheunittangentbundle,geodesic
transformations,specialvectorfieldsonRiemannianmanifolds(minimal,harmonic),
etc.
He has given more than one hundred lectures in almost as many universities and
researchcentersaroundtheworld,andvisitedmanyoftheseuniversitiesasaresearcher.
The almost 80 mathematicians from many different countries with whom Prof.
LievenVanheckehascollaboratedtestifybothtothewiderangeofinterestingproblems
coveredbyhisresearchand,aboveall,tohisuncommonpersonalqualities.Thishas
madehimoneoftheworld’sleadingresearchersinthefieldofRiemanniangeometry.
Mostofthepaperspublishedinthisvolumearewrittenbymathematicianswhohave
beenatsomepointeitherhisstudentsorcollaborators.
viii Preface
We dedicate this volume to Professor Lieven Vanhecke with great affection and
deeprespect.
Acknowledgements Wewouldliketothank:
DipartimentodiMatematica“E.DeGiorgi”dell’Universita`diLecce,Universita`degli
StudidiLecce,Indam(GNSAGA),MIURproject“Proprieta`geometrichedellevarieta`
reali e complesse” ( Unita` di ricerca dell’Universita` di L’Aquila e dell’Universita` di
Roma “La Sapienza”). The Conference would have not been possible without their
financialsupport.
The Scientific Committee: D. Alekseevsky (Hull, England), V. Ancona (Firenze,
Italy),J.-P.Bourguignon,(Paris,France),M.Cahen(Brussels,Belgium),L.A.Cordero
(SantiagodeCompostela,Spain),M.Fernandez(Bilbao,Spain),O.Kowalski(Prague,
Czech Republic), L. Lemaire (Brussels, Belgium), S. Marchiafava (Roma, Italy), E.
Musso(L’Aquila,Italy),D.Perrone(Lecce,Italy),S.Salamon(Torino,Italy),I.Vais-
man(Haifa,Israel).TheiradviceensuredtheinternationalinterestintheConference.
Therefereesfortheircarefulwork.
The Organizing Committee: R. A. Marinosci (coordinator) (Lecce, Italy), G. De
Cecco (Lecce, Italy), E. Boeckx (Leuven, Belgium), G. Calvaruso (Lecce, Italy), L.
Nicolodi(Parma,Italy),E.Musso(L’Aquila,Italy),D.Perrone(Lecce,Italy).Wewant
to express special thanks to Prof. R. A. Marinosci whose hard work contributed so
muchtothesuccessoftheConference.
WearealsogratefultoMrs.FaustaGuzzoni(Parma,Italy)forhervaluablesupport
inthetechnicalpreparationoftheseProceedings.
Finally,ourthanksgototheparticipants,thespeakers,andtoallwhocontributed
inmanywaystotherealizationoftheConference.
Jerusalem YaakovFriedman
October,2004
Curvature of Contact Metric Manifolds(cid:1)
DavidE.Blair
DepartmentofMathematics,
MichiganStateUniversity,EastLansing,MI48824
[email protected]
DedicatedtoProfessorLievenVanhecke
Summary. Thisessaysurveysanumberofresultsandopenquestionsconcerningthecurvature
ofRiemannianmetricsassociatedtoacontactform.
In1975,whentheauthorwasonsabbaticalinStrasbourg,itwasanopenquestion
whetherornotthe5-toruscarriedacontactstructure.Theauthor,beinginterestedinthe
Riemanniangeometryofcontactmanifolds,provedatthattime([4])thatonacontact
manifold of dimension ≥ 5, there are no flat associated metrics. Shortly thereafter,
R.Lutz[31]provedthatthe5-torusdoesindeedadmitacontactstructureandhencethe
naturalflatmetriconthe5-torusisnotanassociatedmetric.Thenon-flatnessresultof
1975wasgeneralizedbyZ.Olszak[35],whoprovedin1978thatacontactmetricman-
ifoldofconstantcurvaturecanddimension≥5isSasakianandofconstantcurvature
+1.Indimension3,theonlyconstantcurvaturecasesareofcurvature0and1aswewill
notebelow.Sometimesonehasanintuitivesensethattheexistenceofacontactform
tendstotightenupthemanifold.Thenon-existenceofflatassociatedmetricsdoesraise
thequestionastowhether,asidefromtheflat3-dimensionalcase,anycontactmetric
manifoldmusthavesomepositivesectionalcurvature.Ifthemanifoldiscompact,itis
known([7]p.99)thatthereisnoassociatedmetricofstrictlynegativecurvature.This
followsfromadeepresultofA.Zeghib[48]ongeodesicplanefields.Recallthataplane
fieldonaRiemannianmanifoldissaidtobegeodesicifanygeodesictangenttotheplane
fieldatsomepointiseverywheretangenttoit.Zeghibprovesthatacompactnegatively
curvedRiemannianmanifoldhasnoC1geodesicplanefield(ofnon-trivialdimension).
Sinceforanyassociatedmetrictheintegralcurvesofthecharacteristicvectorfield,or
Reebvectorfield,aregeodesics,thecharacteristicvectorfielddeterminesageodesic
linefieldtowhichwecanapplythetheoremofZeghibtoobtainthefollowingresult.
Theorem. On a compact contact manifold, there is no associated metric of strictly
negativecurvature.
(cid:1)Thisessayisanexpandedversionoftheauthor’slecturegivenattheconference“Curvature
inGeometry”inhonorofProfessorLievenVanheckeinLecce,Italy,11–14June2003.