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. 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(TXQ/HP) 987654321 SPIN10944936 www.birkhauser.com 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.