Real Hypersurfaces in Hermitian Symmetric Spaces PDF
Preview Real Hypersurfaces in Hermitian Symmetric Spaces
JürgenBerndt,YoungJinSuh RealHypersurfacesinHermitianSymmetricSpaces Advances in Analysis and Geometry | Editor in Chief Jie Xiao, Memorial University, Canada EditorialBoard Der-ChenChang,GeorgetownUniversity,USA GoongChen,TexasA&MUniversity,USA AndreaColesanti,UniversityofFlorence,Italy RobertMcCann,UniversityofToronto,Canada De-QiZhang,NationalUniversityofSingapore,Singapore KeheZhu,UniversityatAlbany,USA Volume 5 Jürgen Berndt, Young Jin Suh Real Hypersurfaces in Hermitian Symmetric Spaces | MathematicsSubjectClassification2020 53-02,53C15,53C35,53C40,53C55,53D15 Authors Prof.JürgenBerndt Prof.YoungJinSuh King’sCollegeLondon KyungpookNationalUniversity StrandCampus ResearchInstituteof LondonWC2R2LS Real&ComplexManifolds UnitedKingdom Daegu41566 [email protected] RepublicofKorea [email protected] ISBN978-3-11-068978-5 e-ISBN(PDF)978-3-11-068983-9 e-ISBN(EPUB)978-3-11-068991-4 ISSN2511-0438 LibraryofCongressControlNumber:2022930118 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2022WalterdeGruyterGmbH,Berlin/Boston Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck www.degruyter.com Preface Thismonographoriginatesfromresearchcollaborationsofthetwoauthorsoverape- riodofmorethan20yearsandcoversaspectsofthegeometryofrealhypersurfaces in Hermitian symmetric spaces. The research has been motivated by an attempt to generalizeclassicaltopicsaboutthegeometryofrealhypersurfacesincomplexpro- jectivespacestootherKählermanifolds.Weinitiallyfocusedoncomplexquadricsand complex2-planeGrassmannians,andtheirnon-compactdualmanifolds,thecomplex hyperbolicquadricsandthecomplexhyperbolic2-planeGrassmannians.Morerecent researchdealswithamoregeneralapproachinHermitiansymmetricspacesofcom- pacttype. Thefocusinthismonographisontheresearchresultsobtainedbythetwoauthors andtheircollaborators,butrelatedworkisalsoincluded.Theexhaustivebibliography attheendofthemonographenablesthereadertofindrelevantliteraturewithmore details. ThesecondauthorwassupportedbyGrantProj.No.NRF-2018-R1D1A1B-05040381 fromtheNationalResearchFoundationofKorea. WewouldliketothankDr.HyunjinLeeforprovidingtechnicalsupport.Dr.Lee wassupportedbyGrantProj.No.NRF-2019-R1I1A1A-01050300fromtheNationalRe- searchFoundationofKorea. JürgenBerndt YoungJinSuh https://doi.org/10.1515/9783110689839-201 Introduction Realhypersurfacesincomplexprojectivespacesandincomplexhyperbolicspaces have been of interest to geometers since many years. A thorough introduction and overview to this topic can be found in the excellent monograph [33] by Thomas E. CecilandPatrickJ.Ryan. InthismonographweextendsomeofthisresearchtootherHermitiansymmetric spaces, with particular focus on the complex quadric Qn = SOn+2/SOnSO2 (n ≥ 3), the complex hyperbolic quadric Qn∗ = SOn,2/SOnSO2 (n ≥ 3), the complex 2-plane GrassmannianG2(ℂk+2)=SUk+2/S(UkU2)(k ≥3)andthecomplexhyperbolic2-plane GrassmannianG2∗(ℂk+2)=SUk,2/S(UkU2)(k ≥3). ThereaderisexpectedtohavepriorknowledgeofbasicconceptsofRiemannian geometryandsubmanifolds.InChapter1wereviewconceptsfromRiemanniange- ometrythataremostrelevanttothecontentsofthismonograph.Thisincludesabrief introductiontoKählergeometry,toRiemanniansymmetricspacesandtoHermitian symmetricspaces.Wewillnotprovideproofsbutreferthereadertosuitablelitera- ture.Wediscussindetailtheconstructionofcomplexquadrics,complexhyperbolic quadrics, complex 2-plane Grassmannians and complex hyperbolic 2-plane Grass- mannians. The complex quadrics and complex hyperbolic quadrics are equipped withtwogeometricstructures,namelyaKählerstructureandacirclebundleofreal structures. The complex 2-plane Grassmannians and complex hyperbolic 2-plane Grassmannians are also equipped with two geometric structures, namely a Kähler structureandaquaternionicKählerstructure.Wewillexplainhowthesegeometric structuresareconstructed,astheyareimportantfortheinvestigationsofrealhyper- surfacesinthesespaces. InChapter2wepresentrelevantbasicconceptsfromsubmanifoldgeometry.This includesthefundamentalequationsofsubmanifoldgeometry.Importantclassesof submanifoldswillbediscussed,suchastotallygeodesicsubmanifoldsandcurvature- adaptedsubmanifolds.Wewillthendescribeafundamentaltechnique,basedonJa- cobifields,forinvestigatingthegeometryoffocalsetsofhypersurfacesandoftubes aroundsubmanifolds.Inthefinalpartofthischapterwewillencounterhomogeneous hypersurfaces. InChapter3wediscussthegeometryofrealhypersurfacesinKählermanifolds. TheKählerstructureofaKählermanifoldinducesaso-calledalmostcontactmetric structureonarealhypersurfaceinthemanifold.Wediscussbasicpropertiesofal- mostcontactmetricstructuresandwhattheytellusaboutthegeometryofrealhyper- surfacesinKählermanifolds.Oneimportantingredientofanalmostcontactmetric structureisaunitvectorfieldontherealhypersurfacethatisknownastheReebvec- torfield.WhentheReebvectorfieldisaprincipalcurvaturevectoroftherealhyper- surfaceeverywhere,thentherealhypersurfaceiscalledaHopfhypersurface.Hopf hypersurfacesareimportantsubmanifoldsinKählergeometry.Wewillderivesome https://doi.org/10.1515/9783110689839-202 VIII | Introduction fundamentalequationsforHopfhypersurfacesinvolvingtheshapeoperatorandcur- vaturequantities.Wewillthenrefinetheseequationsforcontacthypersurfacesand forrealhypersurfaceswithisometricReebflow.Wewillalsoreviewsomeimportant classificationresultsforrealhypersurfacesincomplexprojectivespacesandincom- plexhyperbolicspaces,whichtosomeextentmotivatedtheresearchinthismono- graph. Inthenextfourchaptersweinvestigatethegeometryofrealhypersurfacesincom- plex2-planeGrassmannians(Chapter4),incomplexhyperbolic2-planeGrassmanni- ans(Chapter5),incomplexquadrics(Chapter6)andincomplexhyperbolicquadrics (Chapter7).Ineachofthesechapterswewillfirstderivesomebasicequationsforreal hypersurfacesinvolvingthealmostcontactmetricstructure,theshapeoperatorand curvaturequantities.Theseequationswillbefundamentalforinvestigatingthegeom- etryofrealhypersurfacesinthesespaces.Wewillthendiscusstheirtotallygeodesic submanifoldsandtheirhomogeneousrealhypersurfaces.Forsomeofthehomoge- neousrealhypersurfaceswewillinvestigatetheirgeometryinmoredetailandderive someinterestinggeometricproperties.Wewilltheninvestigateinhowfartheseho- mogeneousrealhypersurfacescanbecharacterizedbysomeoftheirgeometricprop- erties. In the final Chapter 8 we will generalize some of this to Hermitian symmetric spacesofcompacttype.Forthiswewillfirstexplainthegeneralstructuretheoryof Hermitiansymmetricspacesofcompacttype,whichinvolvesstructuretheoryofcer- tainsemisimplerealLiealgebras.Wethenapplythegeneralstructuretheorytoinves- tigaterealhypersurfaceswithisometricReebflowandcontacthypersurfaces.
The list of books you might like
