CP2 CH2 Parallel *-Ricci tensor of real hypersurfaces in and GEORGE KAIMAKAMIS AND KONSTANTINA PANAGIOTIDOU ABSTRACT:Inthispapertheideaofstudyingrealhypersurfacesinnon-flatcomplexspaceforms,whose*-Ricci 4 tensor satisfies geometric conditions is presented. More precisely, the non-existence of three dimensional real 1 0 hypersurfacesinnon-flatcomplexspaceformswithparallel*-Riccitensorisproved.Attheendofthepaperideas 2 forfurtherresearchon∗-Riccitensoraregiven. n a Keywords: Realhypersurface,Parallel,*-Riccitensor,Complexprojectiveplane,Complexhyperbolicplane. J 7 2 MathematicsSubjectClassification(2010):Primary53B20;Secondary53C15,53C25. ] G 1 Introduction D . h Acomplexspaceformisann-dimensionalKaehlermanifoldofconstantholomorphicsectionalcurvature t a c. Acompleteandsimplyconnected complexspaceformiscomplexanalytically isometricto m [ • acomplexprojective spaceCPn ifc > 0, 1 v • acomplexEuclidean spaceCn ifc= 0, 4 9 • oracomplexhyperbolic spaceCHn ifc < 0. 7 6 . The symbol Mn(c) is used to denote the complex projective space CPn and complex hyperbolic space 1 0 Cn, when it is not necessary to distinguish them. Furthermore, since c 6= 0 in previous two cases the 4 notion ofnon-flatcomplexspaceformreferstoboththem. 1 : Let M be a real hypersurface in a non-flat complex space form. An almost contact metric structure v i (ϕ,ξ,η,g) is defined on M induced from the Kaehler metric G and the complex structure J on Mn(c). X The structure vector field ξ is called principal if Aξ = αξ, where A is the shape operator of M and r a α = η(Aξ)isasmoothfunction. ArealhypersurfaceiscalledHopfhypersurface, ifξ isprincipalandα iscalledHopfprincipal curvature. TheRiccitensor S ofaRiemannianmanifoldisatensorfieldoftype(1,1)andisgivenby g(SX,Y)= trace{Z 7→ R(Z,X)Y}. IftheRiccitensorofaRiemannianmanifoldsatisfiestherelation S = λg, whereλisaconstant, thenitiscalledEinstein. Realhypersurfaces innon-flatcomplex space formshavebeenstudied intermsoftheirRiccitensor S,whenitsatisfiescertaingeometricconditions extensively. Differenttypesofparallelism orinvariance oftheRiccitensorareissuesofgreatimportance inthestudyofrealhypersurfaces. 1 2 G.KaimakamisandK.Panagiotidou In[4]itwasprovedthenon-existence ofrealhypersurfaces innon-flatcomplexspaceformsM (c), n n ≥ 3 with parallel Ricci tensor, i.e. (∇ S)Y = 0, for any X, Y ∈ TM. In [5] Kim extended X the result of non-existence of real hypersurfaces with parallel Ricci tensor in case of three dimensional real hypersurfaces. Another type of parallelism which was studied is that of ξ-parallel Ricci tensor, i.e. (∇ S)Y = 0 for any Y ∈ TM. More precisely in [6] Hopf hypersurfaces in non-flat complex space ξ formswithconstantmeancurvatureandξ-parallelRiccitensorwereclassified. Moredetailsonthestudy ofRiccitensorofrealhypersurfaces areincluded inSection6of[7]. Motivated by Tachibana, who in [9] introduced the notion of *-Ricci tensor on almost Hermitian manifolds, in [2] Hamada defined the *-Ricci tensor of real hypersurfaces in non-flat complex space formsby 1 ∗ g(S X,Y) = (trace{ϕ◦R(X,ϕY)}), forX,Y ∈TM. 2 ∗ ∗ The -Ricci tensor S is a tensor field of type (1,1) defined on real hypersurfaces. Taking into account the work that so far has been done in the area of studying real hypersurfaces in non-flat complex space formsintermsoftheirtensorfields,thefollowingissueraisesnaturally: ∗ ∗ Thestudyofrealhypersurfacesintermsoftheir -RiccitensorS ,whenitsatisfiescertaingeometric conditions. In this paper three dimensional real hypersurfaces in CP2 and CH2 equipped with parallel ∗-Ricci tensorarestudied. Therefore, thefollowingcondition issatisfied ∗ (∇ S )Y = 0, X,Y ∈TM. (1.1) X Moreprecisely thefollowingTheoremisproved. MainTheorem: Theredonotexistrealhypersurfaces inCP2 andinCH2,whose*-Riccitensor is parallel. Thepaperisorganizedasfollows: InSection2preliminaries relationsforrealhypersurfaces innon- flatcomplexspaceformsarepresented. InSection3theproofofMainTheorem isprovided. Finally,in Section4ideasforfurtherresearchontheaboveissueareincluded. 2 Preliminaries ∞ Throughoutthispaperallmanifolds,vectorfieldsetcareassumedtobeofclassC andallmanifoldsare assumedtobeconnected. Furthermore, therealhypersurfaces M aresupposed tobewithoutboundary. LetM bearealhypersurface immersedinanon-flatcomplexspaceform(M (c),G)withcomplex n structureJ ofconstantholomorphic sectionalcurvature c. LetN bealocallydefinedunitnormalvector fieldonM andξ = −JN thestructure vectorfieldofM. ForavectorfieldX tangenttoM thefollowingrelation holds JX = ϕX +η(X)N, where ϕX and η(X)N are the tangential and the normal component of JX respectively. The Rieman- nianconnections ∇inM (c)and∇inM arerelatedforanyvectorfieldsX,Y onM by n ∇ Y = ∇ Y +g(AX,Y)N, X X whereg istheRiemannianmetricinduced fromthemetricG. Parallel*-Riccitensor 3 Theshapeoperator Aoftherealhypersurface M inM (c)withrespecttoN isgivenby n ∇ N = −AX. X The real hypersurface M has an almost contact metric structure (ϕ,ξ,η,g) induced from the complex structureJ onM (c),whereϕisthestructuretensoranditisatensorfieldoftype(1,1). Moreover,ηis n an1-formonM suchthat g(ϕX,Y)= G(JX,Y), η(X) = g(X,ξ) = G(JX,N). Furthermore, thefollowingrelations hold 2 ϕ X = −X +η(X)ξ, η◦ϕ= 0, ϕξ = 0, η(ξ) = 1, g(ϕX,ϕY) = g(X,Y)−η(X)η(Y), g(X,ϕY) = −g(ϕX,Y). SinceJ iscomplexstructure implies∇J = 0. Thelastrelationleadsto ∇ ξ = ϕAX, (∇ ϕ)Y = η(Y)AX −g(AX,Y)ξ. (2.1) X X Theambient space M (c)isofconstant holomorphic sectional curvature cand thisresults intheGauss n andCodazziequations tobegivenrespectively by c R(X,Y)Z = [g(Y,Z)X −g(X,Z)Y +g(ϕY,Z)ϕX (2.2) 4 −g(ϕX,Z)ϕY −2g(ϕX,Y)ϕZ]+g(AY,Z)AX −g(AX,Z)AY, c (∇ A)Y −(∇ A)X = [η(X)ϕY −η(Y)ϕX −2g(ϕX,Y)ξ] X Y 4 whereRdenotestheRiemanniancurvature tensoronM andX,Y,Z areanyvectorfieldsonM. ThetangentspaceT M ateverypointP ∈M canbedecomposed as P T M = span{ξ}⊕D, P whereD = kerη = {X ∈ T M : η(X) = 0}andiscalledholomorphicdistribution. Duetotheabove P decomposition thevectorfieldAξ canbewritten Aξ = αξ +βU, (2.3) 1 whereβ = |ϕ∇ ξ|andU = − ϕ∇ ξ ∈ ker(η)providedthatβ 6= 0. ξ β ξ Since the ambient space M (c) is of constant holomorphic sectional curvature c following similar n calculations to those in Theorem 2 in [3] and taking into account relation (2.2), it is proved that the ∗ *-RiccitensorS ofM isgivenby cn ∗ 2 2 S = −[ ϕ +(ϕA) ]. (2.4) 2 4 G.KaimakamisandK.Panagiotidou 3 ProofofMainTheorem Let M be a non-Hopf hypersurface in CP2 or CH2, i.e. M2(c). Then the following relations hold on everynon-Hopfthree-dimensional realhypersurface inM2(c). Lemma3.1 LetMbearealhypersurface inM2(c). Thenthefollowingrelations holdonM AU = γU +δϕU +βξ, AϕU = δU +µϕU, (3.1) ∇ ξ = −δU +γϕU, ∇ ξ = −µU +δϕU, ∇ ξ = βϕU, (3.2) U ϕU ξ ∇UU = κ1ϕU +δξ, ∇ϕUU = κ2ϕU +µξ, ∇ξU = κ3ϕU, (3.3) ∇UϕU = −κ1U −γξ, ∇ϕUϕU = −κ2U −δξ, ∇ξϕU = −κ3U −βξ, (3.4) whereγ,δ,µ,κ1,κ2,κ3 aresmoothfunctions onMand{U,ϕU,ξ}isanorthonormal basisofM. FortheproofoftheaboveLemmasee[8] Let M be a real hypersurface in M2(c), i.e. CP2 or CH2, whose ∗-Ricci tensor satisfies relation (1.1),whichismoreanalytically written ∗ ∗ ∇ (S Y) = S (∇ Y), X,Y ∈TM. (3.5) X X Weconsider theopensubsetNofM suchthat N = {P ∈ M : β 6= 0, inaneighborhood ofP}. InwhatfollowsweworkontheopensubsetN. OnNrelation(2.3)andrelations(3.1)-(3.4)ofLemma3.1hold. Sorelation(2.4)forX ∈{U,ϕU,ξ} takingintoaccountn = 2andrelations (2.3)and(3.1)yields ∗ ∗ 2 ∗ 2 S ξ = βµU −βδϕU, S U = (c+γµ−δ )U and S ϕU = (c+γµ−δ )ϕU. (3.6) Theinner product ofrelation (3.5)forX = Y = ξ withξ duetothe firstandthe third of(3.6),the first of(2.1)forX = ξ andthethirdofrelations (3.3)and(3.4)implies δ = 0. (3.7) Moreover,theinnerproductofrelation(3.5)forX = ϕU andY = ξ withξ becauseof(3.7),thefirstof (2.1)forX = ϕU,thefirstandthesecondof(3.6)andthesecondof(3.3)resultsin µ = 0. Finally,theinnerproductofrelation(3.5)forX = ξandY = ϕU withξ takingintoaccountµ = δ = 0, thefirstandthethirdof(3.6)andthelastrelation of(3.4)leadsto c= 0, whichisacontradiction. SotheopensubsetNisemptyandweleadtothefollowingProposition. ∗ Proposition3.2 Everyrealhypersurface inM2(c)whose -Riccitensorisparallel, isaHopfhypersur- face. Parallel*-Riccitensor 5 SinceM isaHopfhypersurface, the structure vector fieldξ is aneigenvector oftheshape operator, i.e. Aξ = αξ. Due to Theorem 2.1 in [7] α is constant. We consider a point P ∈ M and choose a unit principalvectorfieldW ∈DatP,suchthatAW = λW andAϕW = νϕW. Then{W,ϕW,ξ}isalocal orthonormal basisandthefollowingrelationholds(Corollary 2.3[7]) α c λν = (λ+ν)+ . (3.8) 2 4 Thefirstof relation (2.1)and relation (2.4)for X ∈{W,ϕW,ξ} because of Aξ = αξ, AW = λW andAϕW = νϕW impliesrespectively ∇ ξ = λϕW and ∇ ξ = −νW (3.9) W ϕW ∗ ∗ ∗ S ξ = 0, S W = (c+λν)W and S ϕW = (c+λν)ϕW. (3.10) Relation(3.5)forX = W andY = ξ becauseofthefirstof(3.9)andthefirstandthirdrelationof(3.10) yields λ(c+λν)= 0. Suppose that (c+ λν) 6= 0 then the above relation results in λ = 0. Moreover, relation (3.5) for X = ϕW andY = ξ becauseofthesecond of(3.9)andthefirstandsecondrelation of(3.10)yields ν = 0. Substitution ofλ = ν = 0in(3.8)resultsinc= 0,whichisacontradiction. Sorelationc = −λν holds. Thelastoneimpliesλν 6= 0sincec6= 0. Supposethatλ = −c. Substitution ofthelastonein(3.8)leadsto ν 2 2αν +5cν −2αc = 0. (3.11) In case of CP2 we have that c = 4 and from equation (3.11) there is always a solution for ν. So ν is constant and λ will be also constant. Therefore, the real hypersurface has three distinct constant eigenvalues. The latter results in M being a real hypersurface of type (B), i.e. a tube of radius r over complexquadric. Substitution oftheeigenvalues oftype(B)inλν = −cleadstoacontradiction. Sono realhypersurface inCP2 hasparallel ∗-Riccitensor(eigenvalues canbefoundin[7]). In case of CH2 we have that c = −4 and from equation (3.11) there is a solution for ν if 0 ≤ 2 25 α ≤ . If α = 0 equation (3.11) implies cν = 0, which is impossible. So there is a solution for 4 2 25 ν if 0 < α ≤ and ν will be constant. The latter results in that λ is also constant and so the real 4 hypersurface is of type (B), i.e. a tube of radius r around totally geodesic RHn. Substitution of the eigenvalues of type (B) in λν = −c leads to a contradiction and this completes the proof of our Main Theorem(eigenvalues canbefoundin[1]). 4 Discussion-OpenProblems ∗ Inthispaperthreedimensional realhypersurfaces innon-flat complexspaceformswithparallel -Ricci tensorarestudiedandthenon-existenceofthemisproved. Therefore,aquestionwhichraisesinanatural wayis Are there real hypersurfaces in non-flat complex space forms of dimension greater than three with 6 G.KaimakamisandK.Panagiotidou ∗ parallel -Riccitensor? Generally, the next step in the study of real hypersurfaces in non-flat complex space forms is to study them when a tensor field P type (1,1) of them satisfies other types of parallelism such as the D- parallelismorξ-parallelism. ThefirstoneimpliesthatP isparallelinthedirectionofanyvectorfieldX orthogonal toξ, i.e. (∇ P)Y = 0, forany X ∈D,and thesecond one implies that P isparallel inthe X directionofthestructurevectorξ,i.e. (∇ P)Y = 0. Sothequestionswhichshouldbeansweredarethe ξ following ∗ Arethererealhypersurfaces innon-flat complex space formswhose -Riccitensor satisfies thecon- ditionofD-parallelism orξ-parallelism? Finally, other types of parallelism play important role in the study of real hypersurfaces is that of semi-parallelism and pseudo-parallelism. A tensor field P of type (1, s) is said to be semi-parallel if it satsfiesR·P = 0,whereRistheRiemanniancurvaturetensorandactsasaderivationonP. Moreover, P is said to be pseudo-parallel if there exists a function L such that R(X,Y)·P = L{(X ∧Y)·P}, where(X ∧Y)Z = g(Y,Z)X −g(Z,X)Y. Sothequestions are: Are there real hypersurfaces in non-flat complex space forms with semi-parallel or pseudo-parallel ∗ -Riccitensor? Theimportance of answering the above question lays in the fact that the class ofreal hypersurfaces ∗ withparallel∗-Riccitensorisincludedintheclassofrealhypersurfaceswithsemi-parallel -Riccitensor. ∗ Furthermore,thelastoneisincludedintheclassofrealhypersurfaceswithpseudo-parallel -Riccitensor. References [1] J.BERNDT:Realhypersurfaceswithconstantprincipalcurvaturesincomplexhyperbolicspace,J.ReineAngew.Math.,395(1989), 132-141. [2] T.Hamada,”RealhypersurfacesofcomplexspaceformsintermsofRicci*-tensor”,TokyoJ.Math.25(2002),473-483. [3] T.IveyandP.J.Ryan,”The*-RiccitensorforhypersurfacesinCPnandCHn”,TokyoJ.Math.34(2011),445-471. [4] U-H.Ki,”RealhypersurfaceswithparallelRiccitensorofacomplexspaceform”,TsukubaJ.Math,13(1989),73-81. [5] U.K.Kim,”NonexistenceofRicci-parallelrealhypersurfacesinCP2andCH2”,Bull.KoreanMath.Soc.41(2004),699-708. [6] M.KimuraandS.Maeda,”OnrealhypersurfacesofacomplexprojectivespaceII”,TsukubaJ.Math,15(1991),547-561. [7] R.NiebergallandP.J.Ryan,”Realhypersurfacesincomplexspaceforms”,Math.Sci.Res.Inst.Publ.32(1997),233-305. [8] K.PanagiotidouandPh.J.Xenos,RealhypersurfacesinCP2andCH2whosestructureJacobioperatorisLieD-parallel,NoteMat. 32(2012).89-99. [9] S.Tachibana”Onalmost-analyticvectorsinalmostKa¨hlerianmanifolds”,TohokuMathJ.11(1959),247-265. G.KAIMAKAMIS,FACULTYOFMATHEMATICSANDENGINEERINGSCIENCES,HELLENICMILITARYACADEMY,VARI,ATTIKI, GREECE E-MAIL:[email protected] K.PANAGIOTIDOU,FACULTYOFENGINEERING,ARISTOTLEUNIVERSITYOFTHESSALONIKI,THESSALONIKI54124,GREECE E-MAIL:[email protected]