Real hypersurfaces in the complex quadric with Reeb invariant shape operator

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Differential Geometry and its Applications

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Real hypersurfaces in the complex quadric with Reeb invariant shape operator

^✩

Young Jin Suh

KyungpookNationalUniversity,Collegeof NaturalSciences,DepartmentofMathematics,Daegu 702-701,RepublicofKorea

a r t i c l e i n f o a b s t r a c t

Article history:

Received23March2014 Availableonlinexxxx CommunicatedbyV.Cortes

Keywords:

IsometricReebflow Reebinvariant Kählerstructure Complexconjugation Complexquadric

FirstweintroducethenotionofReebinvariantshapeoperatorforrealhypersurfaces in the complex quadric Q^m = SOm+2/SOmSO2. Next we give a complete classificationofrealhypersurfacesinQ^m=SOm+2/SOmSO₂withinvariantshape operator.

© 2014ElsevierB.V.All rights reserved.

1. Introduction

WhenweconsidersomeHermitiansymmetricspacesofrank2,usuallywecangiveexamplesofRieman- nian symmetric spaces SU_m+2/S(U₂U_m) and SU_2,m/S(U₂U_m), which are said to be complex two-plane Grassmanniansand complexhyperbolic two-plane Grassmanniansrespectively(see[1,2,12,13]).Those are saidtobeHermitiansymmetricspacesandquaternionicKählersymmetricspacesequippedwiththeKähler structureJ andthequaternionicKählerstructureJonSU_2,m/S(U₂U_m).TherankofSU_2,m/S(U₂U_m) is2 and thereare exactlytwo typesofsingulartangentvectorsX of SU2,m/S(U2Um) whichare characterized bythegeometric propertiesJ X ∈JX andJ X⊥JX respectively.

As another kind of Hermitian symmetric space with rank2 of compact type different from the above ones,wecangiveanexample ofcomplexquadricQ^m=SOm+2/SOmSO2,whichisacomplexhypersurface in complexprojective space CP^m+1 (seeBerndt and Suh [3], and Smyth[10]).The complex quadricalso can be regardedas akindof real Grassmannmanifolds of compact typewith rank2 (see Kobayashiand Nomizu[6]).Accordingly,thecomplexquadricadmitsbothacomplexconjugationstructureAandaKähler structure J,whichanti-commute witheachother,thatis,AJ=−J A.Thenform≥2 thetriple(Q^m,J,g)

✩ ThisworkwassupportedbygrantProj.No.NRF-2012-R1A2A2A-01043023fromNationalResearchFoundationofKorea.

E-mailaddress:yjsuh@knu.ac.kr.

http://dx.doi.org/10.1016/j.difgeo.2014.11.003 0926-2245/© 2014ElsevierB.V.All rights reserved.

isaHermitian symmetricspace ofcompacttype withrank2andits maximalsectionalcurvature isequal to4 (seeKlein[5]andReckziegel[9]).

Forthe complexprojective spaceCP^m afullclassification wasobtainedby Okumurain[7].He proved thattheReebflow onareal hypersurfaceinCP^m=SUm+1/S(UmU1) is isometricif andonlyif M is an open partofatubearound atotally geodesicCP^k ⊂CP^m forsomek∈ {0,. . . ,m−1}. Forthe complex 2-planeGrassmannian G₂(C^m+2) =SU_m+2/S(U_mU₂) the classification wasobtained byBerndt and Suh in[1].TheReebflowonarealhypersurfaceinG2(C^m+2) isisometric ifandonlyifM isanopen partofa tubearoundatotallygeodesicG2(C^m+1)⊂G2(C^m+2). But,whenwe consideranisometric Reebflow for real hypersurfacesinthecomplexquadric Q^m=SOm+2/SOmSO2,the resultisquitedifferent from CP^m andG₂(C^m+2).InviewoftheprevioustworesultsinCP^m andG₂(C^m+2) anaturalexpectationmightbe that theclassification involves at least the totally geodesic Q^m−1 ⊂ Q^m. But, surprisingly, in [2] Berndt andSuhhaveprovedthefollowingresult:

Theorem 1.1. Let M be a real hypersurface of the complex quadric Q^m, m ≥ 3. The Reeb flow on M is isometric if andonly ifm iseven, saym= 2k,andM isan open partof atube aroundatotally geodesic CP^k⊂Q^2k.

InapaperduetoPérez,SantosandSuh[8],wehaveconsideredanotionofLieξ-parallelstructureJacobi operator,LξRξ = 0,forreal hypersurfacesincomplexprojective spaceCP^m,andinapaper[11], Suhhas givenacharacterizationofatubeofradiusr,0< r <^π₂,overatotallygeodesicG2(C^m+1) inG2(C^m+2) in termsofLieξ-parallel shapeoperator.

In this paper we consider anotion of Lie parallel shape operator S for real hypersurfaces in complex quadricQ^m alongthedirectionoftheReebvectorfieldξ,thatis,LξS= 0.Inthiscasetheshapeoperator S ofMinQ^m issaidtobeReebinvariant.Motivatedbytheresultsmentionedaboveandusingthenotion of isometric Reeb flow in Theorem 1.1, we give a new characterization of real hypersurfaces in complex quadricQ^mwithReebinvariant shapeoperatoras follows:

MainTheorem.LetM be arealhypersurfaceinthecomplex quadricQ^m,m≥3withReebinvariantshape operator. Then m = 2k, and M is locally congruent to a tube over a totally geodesic complex projective spaceCP^k in Q^2k.

2. Thecomplexquadric

Formoredetailsinthissectionwereferto[3–6,9].ThecomplexquadricQ^misthecomplexhypersurface in CP^m+1 which is defined by the equation z²₁ +. . .+z_m+2² = 0, where z1,. . . ,zm+2 are homogeneous coordinatesonCP^m+1.Weequip Q^m withtheRiemannianmetricwhichisinducedfromtheFubiniStudy metriconCP^m+1withconstantholomorphicsectionalcurvature4.TheKählerstructureonCP^m+1induces canonicallyaKählerstructure (J,g) onthecomplexquadric.Foreachz∈Q^mwe identifyT_zCP^m+1 with theorthogonalcomplementC^m+2CzofCz inC^m+2 (seeKobayashiandNomizu [6]).Thetangentspace TzQ^mcanthen beidentifiedcanonicallywiththeorthogonalcomplementC^m+2(Cz⊕CN) ofCz⊕CN inC^m+2,whereN ∈νzQ^misanormalvectorofQ^minCP^m+1 atthepointz.

The complex projective space CP^m+1 is a Hermitian symmetric space of the special unitary group SU_m+2, namely CP^m+1 = SU_m+2/S(U_m+1U₁). We denote by o = [0,. . . ,0,1]∈ CP^m+1 the fixed point oftheactionofthestabilizer S(Um+1U1).ThespecialorthogonalgroupSOm+2⊂SUm+2 actsonCP^m+1 withcohomogeneityone.TheorbitcontainingoisatotallygeodesicrealprojectivespaceRP^m+1⊂CP^m+1. ThesecondsingularorbitofthisactionisthecomplexquadricQ^m=SOm+2/SOmSO2.Thishomogeneous space modelleadsto thegeometric interpretationof thecomplexquadricQ^mas theGrassmannmanifold G⁺₂(R^m+2) oforiented 2-planes in R^m+2. It also givesamodel ofQ^m as a Hermitian symmetric spaceof

rank2.ThecomplexquadricQ¹isisometrictoasphereS²withconstantcurvature,andQ²isisometricto the Riemannianproduct of two 2-spheres withconstant curvature. Forthis reasonwe will assume m≥3 from nowon.

For aunitnormal vectorN ofQ^m atapointz ∈Q^m wedenote byA=A_N theshapeoperatorof Q^m inCP^m+1 withrespect toN.Theshapeoperatorisaninvolution onthetangentspaceT_zQ^mand

T_zQ^m=V(A)⊕J V(A),

whereV(A) isthe+1-eigenspaceandJ V(A) isthe(−1)-eigenspaceofAN.Geometricallythismeansthatthe shapeoperatorA_N definesarealstructureonthecomplexvectorspaceT_zQ^m,orequivalently,isacomplex conjugation onT_zQ^m.SincetherealcodimensionofQ^minCP^m+1 is2,thisinducesanS¹-subbundleAof theendomorphismbundle End(T Q^m) consistingof complexconjugations.

Thereisageometricinterpretationoftheseconjugations.ThecomplexquadricQ^mcanbeviewedasthe complexificationofthem-dimensionalsphereS^m.Througheachpointz∈Q^mthereexistsaone-parameter family of realforms of Q^m whichare isometric to thesphere S^m. These realforms are congruentto each other underaction of thecenterSO2 of theisotropysubgroup of SOm+2 at z. Theisometric reflection of Q^m insucharealform S^misanisometry, andthedifferentialatz ofsuchareflectionisaconjugationon TzQ^m.InthiswaythefamilyAofconjugationsonTzQ^mcorrespondstothefamilyofrealformsS^mofQ^m containing z, and the subspacesV(A) ⊂TzQ^m correspond to the tangentspaces TzS^m of thereal forms S^mofQ^m.

The Gauss equation for Q^m ⊂CP^m+1 implies thatthe Riemanniancurvature tensor R of Q^m canbe described intermsofthecomplexstructureJ andthecomplexconjugationA∈A:

R(X, Y)Z =g(Y, Z)X−g(X, Z)Y +g(J Y, Z)J X−g(J X, Z)J Y −2g(J X, Y)J Z +g(AY, Z)AX−g(AX, Z)AY +g(J AY, Z)J AX−g(J AX, Z)J AY.

Note thatJ andeachcomplexconjugationAanti-commute,thatis,AJ=−J AforeachA∈A.

Recall that a nonzero tangent vector W ∈ TzQ^m is called singular if it is tangent to more than one maximal flatinQ^m.There aretwotypesofsingulartangentvectorsforthecomplexquadricQ^m:

1. Ifthere exists aconjugationA∈Asuch thatW ∈V(A), thenW issingular. Suchasingulartangent vectoriscalled A-principal.

2. If there exist aconjugation A∈A and orthonormalvectors X,Y ∈ V(A) such thatW/||W||= (X+ J Y)/√

2,thenW issingular.Suchasingulartangentvectoriscalled A-isotropic.

For every unit tangent vector W ∈ T_zQ^m there exist a conjugation A ∈ A and orthonormal vectors X,Y ∈V(A) suchthat

W = cos(t)X+ sin(t)J Y

forsomet∈[0,π/4].Thesingulartangentvectorscorrespondtothevaluest= 0 andt=π/4.If0< t< π/4 thentheuniquemaximalflatcontainingW isRX⊕RJ Y.Laterwewillneedtheeigenvaluesandeigenspaces of theJacobioperatorR_W =R(·,W)W forasingularunittangentvectorW.

1. IfW is anA-principalsingularunittangentvectorwithrespect toA∈A, thentheeigenvaluesofRW

are0 and 2 andthecorresponding eigenspaces areRW ⊕J(V(A)RW) and(V(A)RW)⊕RJ W, respectively.

2. If W isanA-isotropic singularunittangentvectorwith respect to A∈A andX,Y ∈V(A),then the eigenvalues of R_W are 0, 1 and 4 and thecorresponding eigenspaces are RW ⊕C(J X+Y), T_zQ^m (CX⊕CY) andRJ W,respectively.

3. ThetotallygeodesicCP^k ⊂Q^2k

Wenowassumethatm iseven,saym= 2k.Themap

CP^k →Q^2k⊂CP^2k+1, [z1, . . . , zk+1]→[z1, . . . , zk+1, iz1, . . . , izk+1]

provides an embeddingof CP^k into Q^2k as atotally geodesiccomplex submanifold.We define acomplex structurej onC^2k+2 by

j(z1, . . . , zk+1, zk+2, . . . , z2k+2) = (−zk+2, . . . ,−z2k+2, z1, . . . , zk+1).

Notethatij=ji.WecanthenidentifyC^2k+2 withC^k+1⊕jC^k+1 andget T_zCP^k =

X+ijX|X ∈C^k+1Cz .

NowconsiderthestandardembeddingofUk+1into SO2k+2whichisdeterminedbytheLiealgebraembed- ding

uk+1→so2k+2, C+iD→

C −D

D C

,

whereC,D∈M_k+1,k+1(R).TheactionofU_k+1 onQ^2kisofcohomogeneityoneandCP^kistheorbitofthis actioncontainingthepointz= [1,0,. . . ,0,i,0,. . . ,0]∈Q^2k,where theisitsinthe(k+ 2)-ndcomponent.

WenowfixaunitnormalvectorN ofQ^2k atzanddenotethecorrespondingcomplexconjugationAN∈A byA.Thenwe canwritealternatively

T_zCP^k=

X+ijX|X ∈V(A) .

NotethatthecomplexstructureionC^2m+2correspondstothecomplexstructureonTzQ^2mviatheobvious identifications.

Weare nowgoing tocalculatetheprincipalcurvatures andprincipalcurvature spacesofthe tubewith radiusr aroundCP^k inQ^2k.Forthis weusethestandardJacobifieldmethodas describedinSection 8.2 of[4].LetN = (X+J Y)/√

2 beaunitnormalvectorofCP^kinQ^2k,whereX,Y ∈V(A) areorthonormal.

The normalJacobi operatorR_N leavesthe tangentspace T_zCP^k and the normal spaceν_zCP^k invariant.

WhenrestrictedtoTzCP^k,theeigenvaluesofRNare0 and1 withcorrespondingeigenspacesC(J X+Y) and TzCP^kC(J X+Y).Thecorrespondingprincipalcurvaturesonthetubeofradiusrare0 andtan(r),andthe correspondingprincipalcurvaturespacesaretheparalleltranslatesofC(J X+Y) andTzCP^kC(J X+Y) along thegeodesicγ inQ^2k withγ(0)=z andγ(0)˙ =N from γ(0) toγ(r).We denote thelatterparallel translatebyW₁. Whenrestrictedto ν_zCP^kRN,the eigenvaluesofR_N are 1 and4 with corresponding eigenspacesνzCP^kCNandRJ N.WedenotethefirstparalleltranslatebyW2.Thecorrespondingprincipal curvatures onthetubeofradius rare −cot(r) and−2cot(2r),andthe correspondingprincipalcurvature spaces are the parallel translates of νzCP^k CN and RJ N along γ from γ(0) to γ(r). This shows in particularthatthetubeisaHopfhypersurface.

For 0 < r < π/2 this process leads to real hypersurfaces in Q^2k, whereas for r = π/2 we get an- other totally geodesic CP^k ⊂ Q^2k. The two totally geodesic complex projective spaces are precisely the

two singular orbits of the U_k+1-action on Q^2k, and the tubes of radius 0 < r < π/2 are the prin- cipal orbits of this action. Using the homogeneity of the tubes we can now conclude that the tube M of radius 0 < r < π/2 has four distinct constant principal curvatures and the property that the shape operator leaves invariant the maximal complex subbundle C of T M. Moreover, all principal cur- vature spaces in C are J-invariant. Summing upall the properties mentioned above, we have the follow- ing

Proposition3.1. (See[3].)LetM bethetubeof radius0< r < π/2aroundthetotallygeodesicCP^k inQ^2k. Then thefollowingstatementshold:

1. M isaHopfhypersurface.

2. Every unit normal vector N of M isA-isotropic and therefore can be written in the form N = (X+ J Y)/√

2withsome orthonormalvectors X,Y ∈V(A)andA∈A.

3. The principalcurvaturesandcorresponding principalcurvature spacesof M are

principal curvature eigenspace multiplicity

0 C(J X+Y) 2

tan(r) W1 2k−2

−cot(r) W2 2k−2

−2 cot(2r) RJ N 1

4. Each of thetwofocal sets ofM isatotally geodesicCP^k⊂Q^2k. 5. The Reebflow of M isan isometric flow.

6. The shapeoperator S andthestructuretensor fieldφsatisfy Sφ=φS.

7. M is a homogeneous hypersurface of Q^2k. More precisely, it is an orbit of the U_k+1-action on Q^2k isomorphic toU_k+1/U_k−1U₁,an S^2k−1-bundle overCP^k.

4. Somegeneralequations

LetMbearealhypersurfaceinQ^manddenoteby(φ,ξ,η,g) theinducedalmostcontactmetricstructure.

Note thatξ =−J N, where N is a (local) unitnormal vector fieldof M. Thetangent bundle T M of M splits orthogonally into T M = C ⊕Rξ, where C = ker(η) is the maximal complex subbundle of T M. The structure tensor field φ restricted to C coincides with the complex structure J restricted to C, and φξ= 0.

Wenow assumethatM isaHopfhypersurface.Thenwehave Sξ=αξ

withthesmoothfunctionα=g(Sξ,ξ) onM.Whenweconsider atransformJ X oftheKähler structureJ onQ^m foranyvectorfieldX onM inQ^m,wemayput

J X =φX+η(X)N

foraunitnormalN to M.Thenwenowconsider theCodazziequation g

(∇XS)Y −(∇YS)X, Z

=η(X)g(φY, Z)−η(Y)g(φX, Z)−2η(Z)g(φX, Y) +g(X, AN)g(AY, Z)−g(Y, AN)g(AX, Z) +g(X, Aξ)g(J AY, Z)−g(Y, Aξ)g(J AX, Z).

PuttingZ =ξweget g

(∇XS)Y −(∇YS)X, ξ

=−2g(φX, Y)

+g(X, AN)g(Y, Aξ)−g(Y, AN)g(X, Aξ)

−g(X, Aξ)g(J Y, Aξ) +g(Y, Aξ)g(J X, Aξ).

Ontheotherhand,wehave g

(∇XS)Y −(∇YS)X, ξ

=g

(∇XS)ξ, Y

−g

(∇YS)ξ, X

= (Xα)η(Y)−(Y α)η(X) +αg

(Sφ+φS)X, Y

−2g(SφSX, Y).

Comparingtheprevioustwo equationsandputtingX =ξyields

Y α= (ξα)η(Y)−2g(ξ, AN)g(Y, Aξ) + 2g(Y, AN)g(ξ, Aξ).

Reinsertingthisintothepreviousequationyields g

(∇XS)Y −(∇YS)X, ξ

=−2g(ξ, AN)g(X, Aξ)η(Y) + 2g(X, AN)g(ξ, Aξ)η(Y) + 2g(ξ, AN)g(Y, Aξ)η(X)−2g(Y, AN)g(ξ, Aξ)η(X) +αg

(φS+Sφ)X, Y

−2g(SφSX, Y).

Altogetherthisimplies

0 = 2g(SφSX, Y)−αg

(φS+Sφ)X, Y

−2g(φX, Y) +g(X, AN)g(Y, Aξ)−g(Y, AN)g(X, Aξ)

−g(X, Aξ)g(J Y, Aξ) +g(Y, Aξ)g(J X, Aξ)

+ 2g(ξ, AN)g(X, Aξ)η(Y)−2g(X, AN)g(ξ, Aξ)η(Y)

−2g(ξ, AN)g(Y, Aξ)η(X) + 2g(Y, AN)g(ξ, Aξ)η(X).

Ateachpoint z∈M wecanchooseA∈Az suchthat

N = cos(t)Z1+ sin(t)J Z2

for some orthonormal vectors Z1,Z2 ∈ V(A) and 0 ≤t ≤ ^π₄ (see Proposition 3 in [9]). Note that t is a functiononM.Firstofall,sinceξ=−J N,wehave

N = cos(t)Z1+ sin(t)J Z2, AN = cos(t)Z1−sin(t)J Z2, ξ= sin(t)Z₂−cos(t)J Z₁, Aξ= sin(t)Z₂+ cos(t)J Z₁.

This impliesg(ξ,AN)= 0 andhence

0 = 2g(SφSX, Y)−αg

(φS+Sφ)X, Y

−2g(φX, Y) +g(X, AN)g(Y, Aξ)−g(Y, AN)g(X, Aξ)

−g(X, Aξ)g(J Y, Aξ) +g(Y, Aξ)g(J X, Aξ)

−2g(X, AN)g(ξ, Aξ)η(Y) + 2g(Y, AN)g(ξ, Aξ)η(X).

5. ProofofMainTheorem

BeforegoingtoproveourMainTheorem,firstletusshow thattheshapeoperatorofMwhichbecomes thetubeofradiusroveracomplexprojectivespaceCP^k inQ^2k isReebinvariantornot; thatis,LξS = 0 ornot.Infact,theLiederivativevanishingalongtheReebvectorfieldisgivenasfollows:

(LξS)X =Lξ(SX)−SLξX

= (∇ξS)X−φS²X+SφSX

= 0

forany vectorfieldX onM inQ^2k.Thenthisisequivalent tothefollowing (∇ξS)X =φS²X−SφSX.

Inorder to dothis,letus mentionthattheshapeoperatorS ofthe tubeover CP^k commuteswiththe structure tensorφ,thatis,Sφ=φS asinProposition 3.1.Sotherightsideoftheaboveequationvanishes.

Now letus check whetherthe leftside ∇ξS = 0 ornot. Thenby apaperdue to Berndtand Suh (see[2], page1350050-14),theexpressionofcovariantderivativefortheshapeoperatorofMincomplexquadricQ^m becomes

(∇XS)Y =

dα(X)η(Y) +g

αSφ−S²φ X, Y

+δη(Y)ρ(X) +δg(BX, φY) +η(BX)ρ(Y)

ξ +

η(Y)ρ(X) +g(BX, φY)

Bξ+g(BX, Y)φBξ

−ρ(Y)BX−η(Y)φX−η(BY)φBX forany vectorfieldsX andY onM inQ^m,wherewehaveput

AY =BY +ρ(Y)N, ρ(Y) =g(AY, N)

foracomplexconjugationA∈A.PuttingX =ξandusingαconstantandρ(ξ)= 0 fortheA-isotropicunit normalvectorfieldN ofM inQ^2k,wehave

(∇ξS)Y =δg(Bξ, φY) +η(Bξ)ρ(Y)}ξ +

η(Y)ρ(ξ) +g(Bξ, φY)

Bξ+g(Bξ, Y)φBξ

−ρ(Y)Bξ−η(Y)φξ−η(BY)φBξ

=

g(Bξ, φY)−ρ(Y) Bξ

=

−g(φBξ, Y) +g(Y, φBξ) Bξ= 0,

whereinthethirdequalitywehaveused

ρ(Y) =g(AY, N) =g(Y, AN)

=g(Y, AJ ξ)

=−g(Y, J Aξ) =−g(Y, J Bξ)

=−g(Y, φBξ).

Asmentionedabove,areal hypersurfaceM inQ^2k withcommutingshapeoperator,thatis, Sφ=φS,has aparallelshapeoperatoralongtheReebdirection∇ξS = 0.Accordingly,weknowthattheshapeoperator ofthetubeoverCP^k isReebinvariant,thatis,LξS= 0.

Nowconversely,letusproveourMainTheoremintheintroduction.Letus assumeLξS = 0.

On the other hand, the equation of Codazzi in Section 4 can be written as follows (see Berndt and Suh[2]):

(∇XS)Y −(∇YS)X =η(X)φY −η(Y)φX−2g(φX, Y)ξ +ρ(X)BY −ρ(Y)BX

+η(BX)φBY −η(BX)ρ(Y)ξ

−η(BY)φBX+η(BY)ρ(X)ξ,

where we haveput AY =BY +ρ(Y)N, and ρ(X)= g(AX,N) = g(J X,Aξ) for aunit normal N of M inQ^m.Thenweassertthefollowing

Proposition 5.1. Let M be a real hypersurface in complex quadrics Q^m, m ≥ 3. If the Reeb flow on M satisfiesLξS = 0,thentheshape operatorS commutes with thestructuretensor φ,thatis, Sφ=φS.

Proof. Firstnotethat

(LξS)X =Lξ(SX)−ALξX

=∇ξ(SX)− ∇SXξ−S(∇ξX− ∇Xξ)

= (∇ξS)X− ∇SXξ+S∇Xξ

= (∇ξS)X−φS²X+SφSX

foranyvectorfieldX onM.ThentheassumptionLξS= 0 holdsifandonlyif(∇ξS)X =φS²X−SφSX.

Now, let us take an orthonormal basis {e1,e2,· · ·,e2m−1} for the tangent space TxM, x ∈ M, of a real hypersurfaceM inQ^m.ThenbytheequationofCodazziwemayput

(∇eiS)Y −(∇YS)e_i=η(e_i)φY −η(Y)φe_i−2g(φe_i, Y)ξ

+ρ(ei)BY −ρ(Y)Bei+η(Bei)φBY −η(BY)ρ(ei)ξ

−η(BY)φBei+η(BY)ρ(ei)ξ, (5.1)

from which,togetherwiththefundamentalformulaswhichis introducedinSection2itfollowsthat

2m−1

i=1

g

(∇eiS)Y, φe_i

=−(2m−2)η(Y)

+

2m−1

i=1

ρ(ei)g(BY, φei)−

2m−1

i=1

ρ(Y)g(Bei, φei)

+

2m−1

i=1

η(Bei)g(φBY, φei)−

2m−1

i=1

η(BY)ρ(ei)g(ξ, φei)

−η(BY)

2m−1

i=1

g(φBe_i, φe_i). (5.2)

NowletusdenotebyU thevector∇ξξ=φSξ.Thenusingtheequation(∇Xφ)Y =η(Y)AX−g(SX,Y)ξ, itsderivativecanbegivenby

∇eiU = (∇eiφ)Sξ+φ(∇eiS)ξ+φS∇eiξ

=η(Sξ)Sei−g(Sei, Sξ)ξ+φ(∇eiS)ξ+φSφSei. Then itsdivergenceisgivenby

divU =

2m−1

i=1

g(∇eiU, ei)

=hη(Sξ)−η S²ξ

−

2m−1

i=1

g

(∇eiS)ξ, φei

−

2m−1

i=1

g(φSei, Sφei), (5.3) where hdenotes thetraceoftheshapeoperatorS ofM inQ^m.

Now wecalculatethesquarednormofthetensorSφ−φS as follows:

φS−Sφ²=

i

g

(φS−Sφ)ei,(φS−Sφ)ei

=

i,j

g

(φS−Sφ)e_i, e_j g

(φS−Sφ)e_i, e_j

=

i,j

g(φSej, ei) +g(φSei, ej)

g(φSej, ei) +g(φSei, ej)

= 2

i,j

g(φSe_j, e_i)g(φSe_j, e_i) + 2

i,jg(φSe_j, e_i)g(φSe_i, e_j)

= 2

j

g(φSej, φSej)−2

j

g(φSej, Sφej)

=−2

j

g

Se_j, φ²Se_j

−2

j

g(φSe_j, Sφe_j)

= 2 TrS²−2η S²ξ

+ 2 divU −2η(Sξ) TrS + 2η

S²ξ

+ 2

j

g

(∇ejS)ξ, φej

, (5.4)

where _i (respectively, _i,j)denotes thesummationfrom i= 1 to 2m−1 (respectively, from i,j= 1 to 2m−1) andinthefinal equalitywehaveused (5.3).

Fromthis, togetherwiththeformula(5.2),itfollowsthat divU = 1

2φS−Sφ²−TrS² +hη(Sξ)−

i

g

(∇eiS)ξ, φe_i

. (5.5)

By(5.5)andtheassumptionofLξS= 0,letusshowthatthestructuretensorφcommuteswiththeshape operatorS,thatis, φS=Sφ.

Ontheotherhand,weknowthatLξS = 0 isequivalentto (∇ξS)X =φS²X−SφSX.

ThenbytheequationofCodazzi,we knowthat

(∇XS)ξ= (∇ξS)X−φX+ρ(X)Bξ−ρ(ξ)BX+η(BX)φBξ

−η(BX)ρ(ξ)ξ−η(Bξ)φBX+η(Bξ)ρ(X)ξ

=φS²X−SφSX−φX+ρ(X)Bξ

+η(BX)φBξ−η(Bξ)φBX+η(Bξ)ρ(X)ξ. (5.6) Nowletustakeaninner product(5.6)withtheReebvectorfieldξ. Thenwehave

g

(∇XS)ξ, ξ

=−g(SφSX, ξ) +ρ(X)g(Bξ, ξ) +η(Bξ)ρ(X)

=g(SX, U) + 2g(Bξ, ξ)ρ(X). (5.7)

Ontheotherhand,bythealmost contactstructureφwehave

φU =φ²Sξ=−Sξ+η(Sξ)ξ=−Sξ+αξ,

wherethefunctionαdenotesη(Sξ).Fromthis,differentiatingandusingtheformula(∇Xφ)Y =η(Y)AX− g(AX,Y)ξgives

−g(SX, U)ξ+φ∇XU = (∇Xφ)U+φ∇XU

=−(∇XS)ξ−S∇Xξ+ (Xα)ξ+α∇Xξ. (5.8) Fromthis, takinganinnerproduct withξ,itfollows that

−g(SX, U) =−g

(∇XS)ξ, ξ

+g(SX, U) +Xα.

Then,together with(5.7),itfollowsthat

g(SX, U)−2g(Bξ, ξ)ρ(X) +Xα= 0. (5.9) Substituting(5.6)and(5.9)into(5.8),wehavethefollowing

φ∇XU =φX−φS²X+αφSX−ρ(X)Bξ

−η(BX)φBξ+η(Bξ)φBX−η(Bξ)ρ(X)ξ.

Now summingupwithrespectto1,· · ·,m−1 for anorthonormalbasisof T_xM,x∈M,wehave

i

g(φ∇eiU, φe_i) = (2m−2)−

TrS²−η S²ξ

+α{TrS−α}

−

i

ρ(ei)g(Bξ, φei)−

i

η(Bei)g(φBξ, φei)

+

i

η(Bξ)g(φBe_i, φe_i). (5.10)

Ontheother hand,weknow that

g(φ∇eiU, φe_i) =−g

∇eiU,−e_i+η(e_i)ξ

=

i

g(∇eiU, ei)−g(∇ξU, ξ)

= divU+U²

= divU+η S²ξ

−η(Sξ)², (5.11)

where U²=g(φSξ,φSξ)=η(S²ξ)−η(Sξ)².

Summingup(5.10)with(5.11),wegetthefollowing divU = (2m−2)−TrS²+αTrS

−

i

ρ(e_i)g(Bξ, φe_i)−

i

η(Be_i)g(φBξ, φe_i)

+

i

η(Bξ)g(φBei, φei), (5.12)

where wehavedenotedthefunctionη(Sξ) byα=η(Sξ).

Ontheotherhand,(5.2)forY =ξgivesthefollowing

i

g

(∇eiS)ξ, φe_i

=−(2m−2)

+

i

ρ(ei)g(Bξ, φei)

+

i

η(Be_i)g(φBξ, φe_i)−η(Bξ)

i

g(φBe_i, φe_i), (5.13) where wehaveused _iρ(ξ)g(Be_i,φe_i)= 0 and _iη(Bξ)ρ(e_i)g(ξ,φe_i)= 0.Fromthis, togetherwith(5.5), (5.10) and(5.13), itfollows that

divU =1

2 φS−Sφ²−TrS²+hη(Sξ)−

i

g

(∇eiS)ξ, φei

=1

2 φS−Sφ²−TrS²+hη(Sξ) + (2m−2)−

i

ρ(ei)g(Bξ, φei)−

i

η(Bei)g(φBξ, φei) +η(Bξ)

i

g(φBe_i, φe_i)

= (2m−2)−TrS²+hη(Sξ)

−

i

ρ(ei)g(Bξ, φei)−

i

η(Bei)g(φBξ, φei)

+

i

η(Bξ)g(φBei, φei), (5.14)

where wehaveused (5.13) inthesecond equalityand (5.12)inthe thirdequalityrespectively.Then com- paringthe both sides inthe thirdequality of(5.14) gives φS−Sφ² = 0, thatis, theshape operatorS commutes with the structure tensor φ. This means thatthe Reeb flow on M is an isometric flow, which givesacomplete proofofourProposition. 2

Byvirtueofthisproposition,togetherwithTheorem 1.1,wegiveacompleteproofofourMainTheorem intheintroduction.

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