Table Of ContentJürgenBerndt,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
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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.
