Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

Canad. Math. Bull. Vol. 61(3),2018pp. 543–552 http://dx.doi.org/10.4153/CMB-2017-049-5 ©Canadian MathematicalSociety2018 Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians Imsoon Jeong,Juande DiosPérez,Young Jin Suh, and ChanghwaWoo Abstract.On areal hypersurface Min a complextwo-planeGrassmannianG2(Cm+2)wehavethe Lie derivation Land a differential operatoroforderone associated withthegeneralized Tanaka– Websterconnection  L(k).Wegiveaclassification ofreal hypersurfacesMon G2(Cm+2)satisfying  L(k) ξS=LξS,whereξis theReeb vectorfieldon MandStheRiccitensorofM. 1 Introduction Itisoneofthemost classicalandinteresting parts in differential geometrytofind geometricpropertiesofsubmanifoldson asymmetricspace equipped withaKähler structureJ,i.e.,aHermitiansymmetricspace.Among Hermitiansymmetricspaces asahigher rank space of complexprojectivespace Pn(C),the authors haveinvestigated the complextwo-planeGrassmannianG2(Cm+2),whichconsists ofthesetof all complextwo-dimensional linear subspacesin Cm+2.Thespace G2(Cm+2)isdiffeomorphictothehomogeneous space SUm+2/S(U2⋅Um),thespecialunitarygroup SUm+2acts transitivelyon Cm+2, andS(U2⋅Um)means theisotropicsubgroup of SUm+2.Cartandecomposition oftheLie algebraofS(U2⋅Um)isexpressed byk= su2⊕sum⊕u1.Wehave a Kähler structureJfrom u1,theone-dimensionalcenter ofk.Remarkably,we also have a quaternionicKähler structureJfrom su2satisfying JJν=JνJ(ν=1,2,3),where{Jν}ν=1,2,3isan orthonormalbasisofJ.Whenm=1, G2(C3)isisometrictothetwo-dimensionalcomplexprojectivespace CP2withconstantholomorphicsectionalcurvature eight.Whenm=2,wenotethat theisomorphismSpin(6)≃SU(4)yieldsan isometrybetweenG2(C4)andthereal Grassmann manifoldG+ 2(R6)oforiented two-dimensional linear subspacesin R6. In thispaper we assumem≥3. Toclassifyreal hypersurfaceswithcertain geometric conditions,let us give a explanation ofthegeometryofreal hypersurfaceson G2(Cm+2). Let us considerareal hypersurface Min G2(Cm+2)andletNdenote a localunitnormalvectorfieldon M Received bythe editors March 18, 2017; revised July31,2017. Published electronicallyJune 8, 2018. Thisworkwas supported byGrantProj. No. NRF-2015-R1A2A1A-01002459andProj. No. NRF2018-R1D1A1B-05040381. AuthorI. J. is supported byNRFGrantfunded bytheKorean Government GrantProj. No. 2017R1A2B4005317.AuthorJ. D. P. ispartiallysupported byMINECO-FEDERGrant MTM2013-47828-C2-1-P. AuthorC.W. is supported byNRFGrantfunded bytheKorean Government GrantProj. No. 2017R1C1B1010265. AMS subjectclassification:53C40, 53C15. Keywords:real hypersurface, complextwo-planeGrassmannian,Hopfhypersurface, shapeoperator,Riccitensor,Lie derivation. 544 I. Jeong,J. D. Pérez,Y. J. Suh, and C.Woo in G2(Cm+2).TheReeb vectorfieldξ=−JN ∈TpMatp∈Misinduced from the Kähler structureJ. LetCbe the distribution givenbytheorthogonalcomplementof [ξ]in TpMatp∈M. Ifξisinvariant under theshapeoperatorA,itis saidtobe Hopf. The1-dimensionalfoliation ofMbytheintegral manifoldsoftheReeb vectorfieldξ is saidtobe a Hopf foliation ofM.WesaythatMisaHopfhypersurface in G2(Cm+2) if andonlyiftheHopf foliation ofMis totallygeodesic. Itis the complexmaximal subbundleofTpM=C⊕C.Thereal hypersurface Mis saidtobe HopfifAC⊂C,or equivalently,theReeb vectorfieldξisprincipal,whereAis theshapeoperatorofthe real hypersurface M. IfXisatangentvectoron M,we can put JX=ϕX+η(X)NandJνX=ϕνX+ην(X)N whereϕX(resp. ϕνX) is thetangential part ofJX(resp. JνX)andη(X)=g(X,ξ) (resp. ην(X)=g(X,ξν)) is the coefficientofnormal part ofJX(resp. JνX). In this case, we call ϕthestructuretensorfieldofM.Using theGauss andWeingartenformulasin [6,Section 1 and2], theKählercondition ¯ ∇J=0gives∇Xξ=ϕAXforany tangentvectorfieldXon M,where∇(resp. ¯ ∇)denotes the covariantderivativeon M (resp. G2(Cm+2)). From this,itcanbe easilychecked thatMisHopfif andonlyifthe Reeb vectorfieldξisHopf. In thiscase, theprincipalcurvatureα=g(Aξ,ξ)is saidtobe a Reeb curvature ofM. From thequaternionicKähler structureJofG2(Cm+2),therenaturallyexist almost contact 3-structurevectorfields{ξ1,ξ2,ξ3} defined byξν=−JνN,ν=1,2,3. Nowlet us denote byQ=Span{ξ1,ξ2,ξ3}a3-dimensionaldistribution in thetangent space TpMatp∈M. In addition,Qstandsfor theorthogonalcomplementof Qin TpM.Then itbecomesaquaternionicmaximalsubbundleofTpM.Thus,the tangent space ofMconsists ofthe direct sum ofQandQasfollows:TpM=Q⊕Q. For twodistributionsCandQdefined above, we canconsider two natural invariantgeometricproperties under theshapeoperatorAofM,thatis,AC⊂Cand AQ⊂Q.The following theorem isfrom apaperduetoSuh[13,Theorem 1.1]. TheoremALetMbe a real hypersurface in G2(Cm+2),m≥3.Thenboth[ξ]and Qareinvariant under theshapeoperatorofM if andonlyif (A)M isan open part of a tube around a totallygeodesicG2(Cm+1)in G2(Cm+2),or (B)m iseven,saym=2n, andM isan open part of a tube around a totallygeodesic HPnin G2(Cm+2). In the caseof(A),wewant tosayMisofType(A).Similarly,in the caseof(B), wesayMisofType(B). Until now,manygeometers haveinvestigated some characterizationsofHopfhypersurfacesin G2(Cm+2)that satisfycommuting conditionsinvolving geometric quantitieslikeshapeoperator,structure(ornormal) Jacobi operator,Riccitensor, and so on. ForatangentvectorX,ϕXis thetangential part ofJX; thenϕis saidtobe the structuretensorfield.Commuting Riccitensormeans that theRiccitensorSandthe structuretensorfieldϕcommutewitheach other,thatis,Sϕ =ϕS.From suchapoint Lie DerivativesandRicciTensor545 ofview,Suh[12]hasgivena characterization ofreal hypersurfacesofType(A)with commuting Riccitensor. On theotherhand, a Jacobifield along geodesicsof a givenRiemannian manifold(M,g)isan important tool in thestudyof differential geometry. It satisfies awell-knowndifferentialequation thatinspiresJacobi operators. Itisdefined by (RX(Y))(p)=(R(Y,X)X)(p),whereRdenotes the curvaturetensorofMand X,Ydenote any vectorfieldson M. Itisknowntobe a self-adjointendomorphism on thetangent space TpM,p∈M.Clearly, eachtangentvectorfieldXtoMprovides aJacobi operatorwithrespect toX.Thus,theJacobi operatoron areal hypersurface MofG2(Cm+2)withrespect toξ(resp. N) is saidtobe a structureJacobi operator (resp. normal Jacobi operator)andwill be denoted byRξ(resp. RN). Among manygeometric conditions,in thispaperwe focus on commuting conditions thathave a strong relationship with hypersurfacesoftube typewhenthe Reeb vectorfieldξbelongs toQ,thatis tosay,the commuting conditionsbetween (1,1)typetensorfieldson real hypersurfacesin complextwo-planeGrassmannians G2(Cm+2)areused to givesameresults to isometricReeb flow. Fora commuting problemconcerned withstructureJacobi operatorRξandstructuretensorϕofMin G2(Cm+2),thatis,Rξϕ=ϕRξ,SuhandYang [16]gave a characterization of a real hypersurface ofType(A)in G2(Cm+2).Also, concerned with a commuting problemfor thenormal Jacobi operator¯ RN,Pérez,Jeong, andSuh[9] gaveacharacterization of a real hypersurface ofType(A)in G2(Cm+2). Related totheLevi–Civita connection ∇,Tanno [18]introduced thegeneralized Tanaka–Websterconnection (GTWconnection) forcontactmetricmanifoldsasa generalization oftheTanaka–Websterconnection. Itisdefined asa canonicalaffine connection on anon-degenerate, pseudo-HermitianCR-manifold(see [17,19]). Then theGTWconnection coincideswithTanaka–Websterconnection ifthe associated CR-structureisintegrable.Cho defined theGTWconnection forareal hypersurface in aKählermanifoldin suchawaythat  ∇(k) XY=∇XY+ F(k) XY, wherek(∈R∖{0})denotesanon-zeroconstantand F(k) XYisdefined by  F(k) XY=g(ϕAX,Y)ξ−η(Y)ϕAX−kη(X)ϕY. Theskew-symmetric(1,1) typetensor F(k) Xis saidtobe a Tanaka–Webster(or k-th-Cho)operatorwithrespect toX. In particular,ifthereal hypersurface satisfies Aϕ+ϕA=2kϕ,thentheGTWconnection  ∇(k)coincideswiththeTanaka–Webster connection (see [1,2]). On theotherhand, wehave considered real hypersurfacesin G2(Cm+2)satisfying ( L(k) XT)Y=0forany vectorfieldsXandYon Min G2(Cm+2),where L(k)is the differential operatoroforderonegivenby  L(k) XY= ∇(k) XY− ∇(k) YX forany vectorfieldsXandYon M,whereTdenotesatensorfieldoftype(1,1). 546I. Jeong,J. D. Pérez,Y. J. Suh, and C.Woo Thetorsion oftheGTWconnection isgivenby  T(k)(X,Y)= F(k) X(Y)− F(k) Y(X). Theoperatordefined by T(k) X(Y)= T(k)(X,Y)iscalled thetorsion operatorassociated withX. LetSbe theRiccitensorofM.Wewill consider real hypersurfacesMin G2(Cm+2) satisfying (C-1)  L(k) XS=LXS, forany vectorfieldXon M.Thisisequivalent tothe fact T(k) XS=S T(k) X, foranyX tangent toM. On theotherhand, HopfhypersurfacesMarethosewhoseReeb vectorfieldξ= −JN isKilling or, equivalently, a principalvectorfield, verifying Aξ=αξ,wherethe smoothfunction α=g(Aξ,ξ)is saidtobe theReeb curvatureoftheReeb vector fieldξ.Thenwe can give a classification forMin G2(Cm+2)satisfying (C-1) in the particularcaseX=ξasfollows. Theorem 1.1 LetMbe a Hopfhypersurface in complextwo-planeGrassmannians G2(Cm+2),m≥3.TheRiccitensor S on M satisfies L(k) ξS=LξSif andonlyifM is locallycongruent toan open part of a tube ofsomeradius r ∈(0,π 2√2)around a totally geodesicG2(Cm+1)in G2(Cm+2). In thiscase, there aretwo kindsof focalsets in G2(Cm+2), andthe distance between them isπ 2√2.By virtueofthis Theorem,wegive anothernon-existence propertyas follows. Corollary1.2There doesnotexist anyHopfhypersurface M in G2(Cm+2),m≥3, satisfying the condition  L(k) XS=LXSforany vectorfieldXon M. In thispaper,werefer to[6,7,11,12,14,15] for Riemannian geometricstructuresof a complextwo-planeGrassmannianG2(Cm+2),m≥3. 2 Proof of Theorem Let us introduce theRiccitensorS, briefly.The curvaturetensorR(X,Y)ZofMin G2(Cm+2)canbe derived from the curvaturetensorR(X,Y)ZofG2(Cm+2).Then bycontracting andusing thegeometricstructureJJν=JνJ(ν=1,2,3),we cansee theRiccitensorSgivenby g(SX,Y)=∑4m−1 i=1g(R(ei,X)Y,ei), where{e1,...,e4m−1}denotesa basisofthetangent space TpMofM,p∈M,in G2(Cm+2)(see [12]). From the definition oftheRiccitensorSand fundamentalfor- Lie DerivativesandRicciTensor547 mulasin [12,section 2], wehave SX=4m−1 ∑ i=1 R(X,ei)ei =(4m+7)X−3η(X)ξ+hAX−A2X +3 ∑ ν=1{−3ην(X)ξν+ην(ξ)ϕνϕX−ην(ϕX)ϕνξ−η(X)ην(ξ)ξν}, (2.1) wherehdenotes thetrace ofA,thatis,h=TrA(see [10,(1.4)]). Using equation (2.1),wewill provethat theReeb vectorfieldξofMbelongseither toQorQ.Under the condition of being Hopf, weget (2.2) F(k) ξX=−kϕX. ForX=ξinto (C-1),wehave (2.3) F(k) ξ(SY)+ϕASY−S F(k) ξ(Y)−SϕAY=0 foranyYtangent toM.Taking theinnerproductof(2.3)withZ,whereZdenotesa vectorfieldtangent toM,weget g( F(k) ξ(SY),Z)+g(ϕASY,Z)−g(S F(k) ξ(Y),Z)−g(SϕAY,Z)=0. Bearing in mindthat F(k) ξis skew-symmetric andSis symmetric, wehave g(Y,−S F(k) ξ(Z)−SAϕZ+ F(k) ξ(SZ)+AϕSZ)=0. Thus,wehave−S F(k) ξ(Z)SAϕZ+ F(k) ξ(SZ)+AϕSZ=0, and, replacing YbyZ,we obtain (2.4) −S F(k) ξ(Y)−SAϕY+ F(k) ξ(SY)+AϕSY=0. Using (2.2),(2.3), and(2.4) gives us −kϕSY+ϕASY+kSϕY−SϕAY=0,(2.5) kSϕY−SAϕY−kϕSY+AϕSY=0, respectively. Bycombining these equations,wehave (2.6)S(ϕA−Aϕ)Y=(ϕA−Aϕ)SY foranyYtangent toM. Lemma2.1 LetMbe a Hopfhypersurface in G2(Cm+2),m≥3. IfMsatisfies L(k) ξS= LξS,thenξbelongs toeither the distribution Qor the distribution Q. ProofToshowthisfact,we consider that theReeb vectorfieldξsatisfies (2.7)ξ=η(X0)X0+η(ξ1)ξ1 for someunitvectors X0∈Q,ξ1∈Qandη(X0)η(ξ1) /=0. 548I. Jeong,J. D. Pérez,Y. J. Suh, and C.Woo Putting Y=ξin (2.5)and(2.6), by(2.7)andusing basic formulasin [5,Section 2], itfollows that ϕAX0=kϕX0,(2.8) AϕX0=kϕX0. On theotherhand, to provethelemma, weneed the following equation: αAϕX+αϕAX−2AϕAX+2ϕX=2 3 ∑ ν=1{−ην(X)ϕξν−ην(ϕX)ξν −ην(ξ)ϕνX+2η(X)ην(ξ)ϕξν+2ην(ϕX)ην(ξ)ξ} (2.9) ([5,Lemma A]). Putting X=X0into (2.9),wehaveαk−k2=η2(X0). Since k isnon-zeroconstant, differentiating thiswithrespect toξ,wehave ξα=−4 kη(X0){g(∇ξX0,ξ)+g(X0,∇ξξ)}=−4 kη(X0)g(∇ξX0,ξ1) =−4 kη(X0)g(X0,ϕ1Aξ)=4 kη(X0)αg(X0,ϕ1ξ)=0 wherewehaveused ∇Xξν=qν+2(X)ξν+1−qν+1(X)ξν+2+ϕνAX. Thisgivesξα=0. Dueto[4, Equation (2.10)], Aξ1=αξ1isderived from ξα=0.Equation (2.8) becomes(α−k)ϕξ1=0. AskisnonzeroconstantandϕX0nevervanishes,wehaveα=k.Thenbythe equation Yα=(ξα)η(Y)−4∑3 ν=1ην(ξ)ην(ϕY)in [5,LemmaA], we easilyobtain thatξbelongseither toQor toQ(see [10]). ThenbyLemma2.1,we candivide our consideration intotwocasesbeing thatξ belongs toeitherQorQ,respectively.Thenfirst we consider the caseξ∈Q.We can put ξ=ξ1∈Qforour convenience sake. Then[8,lemma1.2]tells us Hopfhypersurface Min G2(Cm+2)andξ∈Qgives AS=SA.Thus,(2.6) ischanged into 0=S(ϕA−Aϕ)Y−(ϕA−Aϕ)SY=SϕAY−SAϕY−ϕASY+AϕSY =SϕAY−ASϕY−ϕSAY+AϕSY=(Sϕ −ϕS)AY−A(Sϕ −ϕS)Y By virtueofLemma2.1 andthe above equations,we assert the following: Lemma2.2LetMbe a Hopfhypersurface in G2(Cm+2). IfMsatisfiesA(ϕS −Sϕ)= (ϕS −Sϕ)A andξ∈Q,thenweobtain Sϕ =ϕS. ProofSince theshapeoperatorAandthetensorϕS −Sϕ are bothsymmetricoperators and commutewitheach other, byusing themethod dueto HornandJohnson [3], there exists a common basis{Ei}i=1,...,4m−1thatgivesasimultaneous diagonalization. Since Aξ=αξand(ϕS −Sϕ)ξ=0,ξisprincipalforAandϕS −Sϕ.Wewrite Lie DerivativesandRicciTensor549 AEi=λiEiand(ϕS −Sϕ)Ei=βiEi,where eigenvaluesλiandβiarerealvalued functionsforall i∈{1,2,. . . 4m−1}. Bearing in mindthatξ=ξ1∈Q,(2.1) is simplified: (2.10)SX=(4m+7)X−7η(X)ξ−2η2(X)ξ2−2η3(X)ξ3+ϕ1ϕX+hAX−A2X. AsξisprincipalforbothAandϕS −Sϕ,weget Case1. We canrestrictX∈[ξ]⊥. Herereplacing XbyϕXin (2.10) (resp. applying ϕ to (2.10)),wehave SϕX=(4m+7)ϕX−ϕ1X+2η2(X)ξ3−2η3(X)ξ2+hAϕX−A2ϕX, ϕSX=(4m+7)ϕX−ϕ1X+2η2(X)ξ3−2η3(X)ξ2+hϕAX−ϕA2X. (2.11) Combining equationsin (2.11),weget (2.12)SϕX−ϕSX=hAϕX−A2ϕX−hϕAX+ϕA2X. Putting X=Eiinto (2.12)andusing AEi=λiEi,weobtain (2.13)(Sϕ −ϕS)Ei=hAϕEi−A2ϕEi−hλiϕEi+λ2 iϕEi. Taking theinnerproductwithEiinto (2.13),wehave βig(Ei,Ei)=hλig(ϕEi,Ei)−λ2 ig(ϕEi,Ei)=0. Since g(Ei,Ei) /=0,βi=0forall i∈1,2,...,4m−2.Thisisequivalent to (Sϕ −ϕS)Ei=0forall i∈1,2,...,4m−2. Case2.ForX∈[ξ].Thisgives(Sϕ −ϕS)ξ=0. Itfollows thatSϕX=ϕSXforany tangentvectorfieldXon M. Summing up Lemmas2.1,2.2and [12,Theorem], we conclude thatifMisaHopf hypersurface in complextwo-planeGrassmannianG2(Cm+2)satisfying (C-1)forX= ξandξα=0,thenMsatisfies the condition oftype(A)real hypersurfaces. Hereafter, let us checkwhether theRiccitensorof a modelspace oftype(A)satisfies thegiven condition (C-1)forX=ξ. First let us considerX=ξ; then (C-1)becomes (2.14) ( L(k) ξS)Y=(LξS)Y, which isequivalent to (2.15)−kϕSY−ϕASY+kSϕY−SϕAY=0. WhenξisHopfvectorfield andξ∈Q,theRiccitensorScommuteswiththe structuretensorϕand by[8,lemma1.2], MAsatisfies(2.15). IftheReeb vectorfieldξbelongs tothemaximalquaternionicsubbunbleQ,then aHopfhypersurface Min G2(Cm+2)islocallycongruent to oneoftype(B)by virtue of [6,Main Theorem]. 550I. Jeong,J. D. Pérez,Y. J. Suh, and C.Woo ForMB,(2.14) isalsoequivalent to (2.15). Sowe assumeMBsatisfies(2.15). For eacheigenspace, wehave SX= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (4m+4+hα−α2)ξifX=ξ∈Tα, (4m+4+hβ−β2)ξℓifX=ξℓ∈Tβ, (4m+8)ϕξℓifX=ϕξℓ∈Tγ, (4m+7+hλ −λ2)XifX∈Tλ, (4m+7+hµ −µ2)XifX∈Tµ. From [13], weobtain the following equations: α=−2 tan(2r),β=2cot(2r),γ=0,λ=cot(r),µ=−tan(r), λ+µ=βandh=α+3β+(4n−4)(λ+µ)=α+(4n−1)β.(2.16) Thus,weget ( L(k) ξS)Y−(LξS)Y= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0ifY=ξ∈Tα, (4−hβ+β2)(β−k)ξℓifY=ξν∈Tβ, (4−hβ+β2)ϕξℓifY=ϕξν∈Tγ, (−k+λ)(λ−µ)(h−λ−µ)ϕYifY∈Tλ, (−k+µ)(µ−λ)(h−µ−λ)ϕYifY∈Tµ. From the fourthequation of above(resp., fifth),since µ/=λ, dueto (2.16),wehave k=µorh=β(resp.,k=λorh=β). However,ifh=β,thethirdone cannothappen. Sowehavek=µ=λ.Thisgivesa contradiction. Remark2.3LetMbe a real hypersurface in complextwo-planeGrassmannian G2(Cm+2),m≥3; thenMBdoesnot satisfythegivencondition ( L(k) ξS)Y=(LξS)Y, foranyYtangent toM. Thus,wehave asserted Theorem1.1 in theintroduction. Secondly,we assumethatMAsatisfies(C-1). Putting Y=ξinto (C-1),weobtain −σϕAX+kσϕX+SϕAX−kSϕX=0, whereSξ=σξ=(4m+hα−α2)ξ. From [13], weobtain the following equation: SX= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (4m+hα−α2)ξifX=ξ∈Tα, (4m+6+hβ−β2)ξνifX=ξν∈Tβ, (4m+6+hλ −λ2)XifX∈Tλ, (4m+8)XifX∈Tµ. Lie DerivativesandRicciTensor551 ForY=ξ∈Tα,weget (2.17)( L(k) XS)ξ−(LXS)ξ= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0ifX=ξ∈Tα, (k−β)(−hα+α2+6+hβ−β2)ξ3ifX=ξ2∈Tβ, (k−β)(−hα+α2+6+hβ−β2)ξ2ifX=ξ3∈Tβ, (k−λ)(hα−α2−6−hλ +λ2)ϕXifX∈Tλ, (hα−α2−8)ϕXifX∈Tµ. From the fifth equation in (2.17),weobtain (2.18)hα−α2−8=0, and from the definition ofh,weobtain h=α+2β+(2m−2)(λ+µ). Summing theseup, by[13]wehave (2.19) (m−1)t2−(m+2)t+4=0, wheret=tan2(√2r). From thesecond equation of(2.17)and(2.18),weobtain (2.20)(k−β)(hβ−β2−2)=0. Ifwe assumethathβ−β2−2=0,thenbysumming up (2.19)with (2.20),wehave m=−1,which gives us a contradiction. Thus,k=β,sofrom the fourthequation of (2.17)and(2.18),weget hλ −λ2−2=0, whichbecomes (2.21) (2m−3)t2−4t+1=0. Combining (2.19)and(2.21) implies t=−7m+11 (m−2)(2m+1). Since m≥3andt≥0,thisgives us a contradiction. By virtueofRemark2.3,we also get the fact thatMBdoesnot satisfythegiven condition ( L(k) XS)Y=(LXS)Y.Thus,we assert Corollary1.2. AcknowledgmentThe authors wouldliketoexpress theirdeep gratitude toDr. Lee working at RIRCMforgiving us nice comments for thispaper. References [1]J. T.Cho,CR structureson real hypersurfacesofacomplexspace form. Publ. Math. Debrecen 54(1999),no. 3–4,473–487. [2] , Levi parallel hypersurfacesin a complexspace form.Tsukuba J. Math. 30(2006),no. 2, 329–343.http://dx.doi.org/10.21099/tkbjm/1496165066 [3]R.A. Hornand C.R. 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