The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

arXiv:0707.1860v1 [math.DG] 12 Jul 2007

TheGauss-Bonnet-GrotemeyerTheoreminspacesof

constantcurvature

EricL.GrinbergandHaizhongLi

Abstract

In1963,K.P.GrotemeyerprovedaninterestingvariantoftheGauss-BonnetThe-orem.LetMbeanorientedclosedsurfaceintheEuclideanspaceR3withEulercharacteristicχ(M),GausscurvatureGandunitnormalvector eld n.Grote-meyer’sidentityreplacestheGauss-BonnetGbythenormalmoment integrand22( a· n)G,whereaisa xedunitvector:M( a· n)Gdv=2π

3

χ(M),(1.1)

where a· ndenotestheinnerproductof aand n,χ(M)istheEulercharacteristicofM.Remark1.1Let{E1,E2,E3}bea xedorthogonalframeinR3andchoose a=Ei.Wehave

2π

(Ei· n)2Gdv=

M

TheprojectispartiallysupportedbythegrantNo.10531090ofNSFC.

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Notingthat(Ei· n)(Ei· n)= n· n=1,weobtainthefollowingGauss-Bonnetformulaviasummationof(1.2)overifrom1to3:

i

Corollary1(Gauss-BonnetTheorem).UnderthesamehypothesisofTheorem1,wehave

Gdv=2πχ(M).(1.3)

M

ThuswecanconsiderGrotemeyer’sTheorem1asanextendedformoftheGauss-Bonnet

Theorem.

LetnbeevenandletNn+1(k)bean(n+1)-dimensionalsimplyconnectedRiemannianmanifoldofconstantsectionalcurvaturek.Thatis,Nn+1(k)=Rn+1ifk=0;Nn+1(k)=Sn+1(1k),an(n+1)-dimensionalspherespacewithradius1kifk>0;Nn+1(k)=

1

),an(n+1)-dimensionalhyperbolicspacewith,asBolyaiwouldsay,radiusHn+1( k√

kifk<0.WewilloftencallNn+1(k)aspaceform.WewillviewNn+1(k)as

standardlyimbeddedinanappropriatelinearspaceLn+1(k)(Rn+2ifk>0,Rn+1,1ifk<0andRn+1ifk=0).

Thiswillenableustode nefunctionsonMsuchas( a· n),where aisa xedvectorintheambientlinearspace, nisanormalvector eldonM,and(·)denotestheinnerproductontheambientlinearspace.ThegeneralizedGrotemeyerTheoremwehaveinmindcanbestatedasasfollows:

Theorem2Letneven,n≥2.Let x:M→Nn+1(k)beanimmersedn-dimensionalorientedclosedhypersurfaceinthe(n+1)-dimensionalspaceformNn+1(k),withEulercharacteristicχ(M),Gauss-KroneckercurvatureGandunitnormalvector eld n.Assume

n+1

thatN(k)isstandardlyimbeddedinthelinearspaceLn+1(k).Thenforany xedunitvector ainLn+1(k)wehave

1i2

χ(M) icikMKn ( a· n)Gdv= 2idv]2M(1.4)k2

+( a· x)Gdv,n+1M

wheretheciareconstantsthatdependonlyonthedimensionnandKiisthei-thmean

curvatureofM.

Inthecasen=2intheTheoremabove,weobtain

Corollary2LetMbeanorientedclosedsurfaceinthe3-dimensionalspaceformN3(k)withextrinsiccurvatureGandunitnormalvector eld n.Thenforany xedunitvector ainthelinearspaceL3(k)wehave

vol(M)( a· n)2Gdv=2π

3M (1.5)k2

+( a· x)Gdv,3MwhereK1isthemeancurvatureofMandχ(M)istheEulercharacteristicofM.

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Remark1.2OurCorollaryreducestoGrotemeyer’soriginaltheoreminthecasek=0.Remark1.3Inthecasek=0andn≥3,Theorem2wasprovedbyB.-Y.Chenin[Ch]byadi erentmethod.

Remark1.4.WecanrecoverthestandardGauss-BonnetTheoremfromourTheoremasfollows.LetmbethedimensionofthelinearspaceLn+1(k).(Thusm=n+1inthe atcase,m=n+2inthepositiveandnegativelycurvedcases.)Let{E1,···,Em}bea xedorthonormalframeinLn+1(k);choose a=Ei.Then

1i2

x(M) ckKn 2idv](E· n)Gdv=iii2MM

(1.6)+k

G(Ei·x)2dv,(i=1,2,...)n+1M

Noting(Ei· n)(Ei· n)= n· n=1andi(Ei· n)(Ei·x)= n·x=0,weobtainthe

i

followingGauss-Bonnetformulabysummingof(1.6)overallappropriatei:

Corollary3(Gauss-BonnetTheorem).UnderthesamehypothesisofTheorem2,wehave

volSn(1)

Gdv=

M

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Forthefollowingcriterionforselfadjointnessoftheoftheoperator2seeCheng-Yau[CY]orLi[L1],[L2].Proposition 2.1LetMbeaclosedorientableRiemannianmanifoldwithsymmetrictensorφ=φijθiθj.Then2isaselfadjointoperatorifandonlyif

i,j

n j=1

where{θij}istheLevi-Civitaconnectionofg.

φij,j=0,1≤i≤n.(2.3)

Hereφij,kisthederivativeofthetensorφijinthedirectionek.

Remark2.1Wecall2theCheng-Yauoperator.ItwasintroducedbyS.Y.Chengand

S.T.Yauin1977[CY].Ifφ=φijθiθjsatis estheCheng-Yaucondition(2.3),then

i,j

2f=

i,j

φijfij=

i,j

(φijfi)j=div(φ f).

Letx:M→Nn+1(k)beann-dimensionalclosedhypersurfaceinan(n+1)-dimensionalspaceformofconstantsectionalcurvaturek.Let(hij)bethecomponentsofthesecondfundamentalformofM.WerecalltheReillyoperator,whichisasecondorderdi erentialoperatorLr:C∞(M)→C∞(M)de nedby

r

Lrf=Tijfij,f∈C∞(M),(2.4)

i,j

r

whereTijisgivenby

Tij

=δij,

rTij

=Krδij

k

r 1hikTkj,

r=1,2,···,n.(2.5)

(SeeReilly[Re],Rosenberg[Ro]orBarbosa-Colares[BC].)

DenotetherthmeancurvatureofMby

B=(hij)=(kiδij).ki1···kir,Kr=

i1<···<ir

(2.6)

WenotethattheGauss-KroneckercurvatureofMisG≡Kn.

De nition2.1([Re])Ther-thNewtontransformation,r∈{0,1,···,n}isthelinear

transformation

Tr=KrI Kr 1B+···+( 1)rBr,(2.7)i.e.,

r

Tij

=Krδij Kr 1hij+···+( 1)

r

j1,···,jr

hij1hj1j2···hjrj.

(2.7)′

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Thenwehave(seeReilly[Re])

Kr=

1

IfI≡i1,···,iqde ne

1,JδI= 1,

0,

andJ≡j1,···,jqaremulti-indicesofintegersbetween1andn,ifi1,···,iqaredistinctandJisanevenpermutationofIifi1,···,iqaredistinctandJisanoddpermutationofIotherwise.

r!

j1···jrjδihi1j1···hirjr.1···iri

(2.9)

Proposition2.3eachr,wehave Forr

(1)divTr=Tij,j=0,

j

(2)Newton’sformula:trace(BTr)=(r+1)Kr+1,(3)trace(Tr)=(n r)Kr

Proposition2.4Let x:M→Nn+1(k)beann-dimensionalhypersurfacewithunitnormalvector eld n.Thenwehave

xi=ei, ni= hijej,xij=hij n kxδij.(2.10)

j

Lrx=(r+1)Kr+1 n (n r)kKrx,

Proof.Let abea xedvectorinLn(k).Write

f= n· a,

Then(2.11)isequivalentto

Lrg=(r+1)Kr+1f (n r)kKrg.

g= x· a.

(2.11)

(2.12)

(2.11)′

Choosinganorthonormalframe{e1,···,en, n}andtheirdualframe{θ1,···,θn,θn+1}

alongMinNn+1(k),wehavethestructureequations

dx=θiei,dei=θijej+hijθj n kxθi,d n= hijθjei.(2.13)

i

j

j

i,j

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Herewehavesometimesabbreviated xasmerelyx,forsimplicity.Byuseof(2.13)andthroughadirectcalculationweget

gi=ei· a,

gij=fhij kgδij.

(2.14)

Byuseofproposition2.3and(2.14),weget

rrr

Lrg=Tijgij=Tijhijf kgTijδij=(r+1)Kr+1f k(n r)gKr.

i,j

ij

i,j

Thuswehaveproved(2.11)′,whichisequivalentto(2.11).Similarly,fromde nitionsoffi,wegetbyuseof(2.13)

fi= hij(ej· a).

j

(2.15)

Because aisarbitrary,wehaveproved(2.10)from(2.14)and(2.15).

Proposition2.5LetMbeann-dimensionalorientedclosedhypersurfacein(n+1)-dimensionalspaceformNn+1(k).ThenforanysmoothfunctionsfandgonMwehave

gLn 1fdv=fLn 1gdv,Ln 1fdv=0.(2.16)

M

M

M

Proof.Choosingr=n 1in(1)ofproposition2.3,andusingthecriteriorfrompropostion

2.1,weknowthattheoperatorLn 1isaselfadjointoperator.Thusweobtain(2.16).Proposition2.6LetMbeann-dimensionalhypersurfacein(n+1)-dimensionalspaceformNn+1(k).Thenwehave

n 1hikTkj=0.(2.17)Gδij

k

Proof.Choosingr=n 1in(2.5)andnotingthatG=Kn,wehave

n 1n

hikTkjTij=Gδij .

k

j1···jnjn

Fromthede nitionofTijin(2.9)andthede nitionofδi,wehave1···ini

n

Tij=0.

(2.18)

(2.19)

Now(2.17)followsfrom(2.18)and(2.19).

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

3.ProofofTheorem2

Proposition3.1Letx:M→Nn+1(k)beann-dimensionalorientedclosedhypersurfacein(n+1)-dimensionalspaceformNn+1(k).AssumeMhasGauss-KroneckercurvatureG=Knandaunitnormalvector n.Thenforany xedunitvector ainLn+1(k),wehave

m+1

0=(n+m) M( a· n)Gdv mM( a· n )m 1Gdv

(3.1)

kM( a· n)m( a· x)Kn 1dv+mkM( a· n)m 1( a· x)2Gdv,

whereKn 1isthe(n 1)-thmeancurvatureofM.Proof.Write

f=qmx,

q= a· n.(3.2)

Byde nitionofthe rstderivativeandthesecondderivativeoff(see(2.2)),wehave

fi=(qm)ix+qmxi,

fij=(qm)ijx+(qm)ixj+(qm)jxi+qmxij.

Byde nitionofoperatorLn 1,wehave

Ln 1(f)=xLn 1(q)+2

Letr=n 1in(2.11).Wehave

Ln 1x=nG n kKn 1x.

(3.6)

m

(3.3)(3.4)

i,j

n 1m

Tij(q)ixj+qmLn 1x.

(3.5)

thatis,weobtainform=1,2,3,···

ByProposition2.5,(3.6),(2.10)andproposition2.6,wegetbyintegrating(3.5)overM

n 1m

(q)ixjdv0=2Mqm(Ln 1x)dv+2MTij

i,j m n 1m 1

Tijmq[ hik( a·ek)]ejdv=2Mq(nG n kKn 1x)dv+2M

(3.7) m i,j,km 1

=2Mq(nG n kKn 1x)dv 2mMqG( a·ej)ejdv

m m 1j

=2Mq(nG n kKn 1x)dv 2mMqG[ a ( a· n) n k( a· x) x]dv,

qm 1G advndv m0=(n +m)MqG M mm 1

kMqKn 1xdv+mkMq(x· a)Gxdv.

m

(3.8)

Takingthescalarproductof awithbothsidesof(3.8),wegetProposition3.1.

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Remark3.1Equation(3.1)wasprovedbyBang-YenCheninthecasek=0byadi erentmethod.

ProofofTheorem2Choosingm=1inProposition3.1,wehave

2

(n+1)M( a· n)Gdv=a· n)( a· x)Kn 1dv MGdv+k2M(

kMG( a· x)dv.

(3.9)

BecauseMisaclosedhypersurfaceinNn+1(k),theGauss-BonnetTheoremstatesinthis

casethat

volSn(1)

Gdv=

M

n+m 1

[qm 2Gdv kqm 2(x· a)2Gdv+

M

M

k

[volS

(n+m 1)(n+m 3)···(n+1)+k

qm 2+n+m 1[

1

n(1)

(m 1)(m 3)(n+m 1)(n+m 3)

M

qm 3+···+

(n+m 1)(n+m 3)···(n+1)

(m 1)(m 3)···3

]( x· a)2Gdv

+k

M

[

n+m 1

1

qm 2+

(m 1)(m 3)(n+m 1)(n+m 3)

qm 3+···+

(n+m 1)(n+m 3)···(n+2)

(m 1)(m 3)···2

q]( x· a)2Gdv

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Note:Inthecasek=0,Proposition3.2wasprovedbyBang-YenChen;seeTheorem2in[Ch].

Acknowledgements.TheauthorsbeganthisresearchworkswhenH.Livisiteddepart-mentofmathematicsandstatisticsinUniversityofNewHampshireonJulyof2006.H.LiwouldliketothankE.L.G.anddepartmentfacultymembersfortheirhospitalityandthehelptheyextendedtohimforhisacademicvisit.

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The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

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EricL.Grinberg

DepartmentofMathematics&StatisticsUniversityofNewHampshireDurham,NH03824

UnitedStatesofAmericaEmail:grinberg@unh.eduHaizhongLi

DepartmentofMathematicalSciencesTsinghuaUniversity100084,Beijing

People’sRepublicofChina

Email:hli@

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