CONFORMALLY FLAT AFFINE HYPERSURFACES WITH SEMI-PARALLEL CUBIC FORM?

(徐輝陽(yáng)) (李策策)

School of Mathematics and Statistics, Henan University of Science and Technology,Luoyang 471023, China

E-mail: xuhuiyang@haust.edu.cn; ceceli@haust.edu.cn

Abstract In this paper,we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric.As a main result,we classify these hypersurfaces as not being of a flat affine metric.In particular,2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.

Key words affine hypersurface;semi-parallel cubic form;Levi-Civita connection;conformally flat;warped product

1 Introduction

The classical equiaffine differential geometry is mainly concerned with the geometric properties and invariants of hypersurfaces in the affine space which are invariant under unimodular affine transformations.Let Rn+1be the (n+1)-dimensional real unimodular affine space.For any non-degenerate hypersurface immersion of Rn+1,it is well known how to induce the affine connection?,the affine shape operatorSwhose eigenvalues are called affine principal curvatures,and the affine metrich.The classical Pick-Berwald theorem states that the cubic formC:=?hvanishes if and only if the hypersurface is a non-degenerate hyperquadric.In that sense,the cubic form plays the role of the second fundamental form for submanifolds of real space forms.

In recent decades,in a manner similar to that for the Pick-Berwald theorem,geometric conditions on the cubic form have been used to classify natural classes of affine hypersurfaces;see e.g.,[4,6,7,12,14–17,27,29,30].Among these,one of the most interesting developments may be the classification of locally strongly convex affine hypersurfaces withC=0,whereis the Levi-Civita connection of the affine metric.In this regard,Dillen,Vrancken,et al.obtained the classifications for lower dimensions in [10,13,18,25],and Hu,Li and Vrancken completed the classification for all dimensions as follows:

Theorem 1.1([20]) LetMbe ann-dimensional (n ≥2) locally strongly convex affine hypersurface in Rn+1withC=0.ThenMis either a hyperquadric (i.e.,C=0) or a hyperbolic affine hypersphere withC≠0.In the latter case,either

(i)Mis obtained as the Calabi product of a lower dimensional hyperbolic affine hypersphere with parallel cubic form and a point,or

(ii)Mis obtained as the Calabi product of two lower dimensional hyperbolic affine hyperspheres with parallel cubic form,or

(iii)n=m(m+1)-1,m ≥3,(M,h)is isometric to SL(m,R)/SO(m),andMis affinely equivalent to the standard embedding SL(m,R)/SO(m)Rn+1,or

(iv)n=m2-1,m ≥3,(M,h) is isometric to SL(m,C)/SU(m),andMis affinely equivalent to the standard embedding SL(m,C)/SU(m)Rn+1,or

(v)n=2m2-m-1,m ≥3,(M,h) is isometric to SU?（2m）/Sp(m),andMis affinely equivalent to the standard embedding SU?（2m）/Sp(m)Rn+1,or

(vi)n=26,(M,h)is isometric to E6(-26)/F4,andMis affinely equivalent to the standard embedding E6(-26)/F4R27.

As in[20,21],we say that an affine hypersurface has the semi-parallel(resp.parallel)cubic form relative to the Levi-Civita connection of the affine metric if·C=0 (resp.C=0),whereis the curvature tensor of the affine metric,and the tensor·Cis defined by

for tangent vector fieldsX,Y.Obviously,the parallelism of cubic form implies its semiparallelism;the converse is not true,and we refer to Remark 3.2 for the counter-examples.

As a generalization of Theorem 1.1,one may naturally posit the following problem:

Problem 1.2Classify all then-dimensional locally strongly convex affine hypersurfaces with·C=0.

Related to Problem 1.2,Hu and Xing [21]showed that such a surface is either a quadric or a flat affine surface.Recently,the present authors and Xing[24]gave an answer to Problem 1.2 in two cases for such hypersurfaces with at most one affine principal curvature of multiplicity one: one with no affine hyperspheres,the other with affine hyperspheres of constant scalar curvature.Based on the latter,the following conjecture was posed:

Conjecture 1.3For any locally strongly convex affine hypersphere of dimensionn,the semi-parallelism and parallelism of the cubic form are equivalent,i.e.,·C=0 iff=0.

In this paper,we continue to study locally strongly convex affine hypersurfaces with·C=0,and pay attention to the case where the Weyl curvature tensor vanishes identically.First,by investigating such affine hyperspheres,we can confirm Conjecture 1.3 for this case.

Theorem 1.4LetMn,n ≥3 be a locally strongly convex affine hypersphere in Rn+1with·C=0 and the vanishing Weyl curvature tensor.ThenMnis affinely equivalent to either a hyperquadric,or the flat and hyperbolic affine hypersphere

or the hyperbolic affine hypersphere with quasi-Einstein affine metric

where (x1,···,xn+1) are the standard coordinates of Rn+1.

From Theorem 1.4 and the fact that the Weyl curvature tensor vanishes automatically forn=3,we immediately obtain

Corollary 1.5LetM3be a locally strongly convex affine hypersphere in R4with·C=0.ThenM3is affinely equivalent to either a hyperquadric,or one of the two hyperbolic affine hyperspheres

where (x1,···,x4) are the standard coordinates of R4.

Remark 1.6Conjecture 1.3 is true forn=2,3,4,by the following three facts:

(1) By Theorem 1.1 and Corollary 2.1 in [9],Theorem 1.4 confirms Conjecture 1.3 if eithern=3,orMnis conformally flat forn ≥4.

(2)It has been shown in[24]that Conjecture 1.3 is true if eithern=2,orMnis a constant scalar curvature forn ≥3.

(3) It was shown in Lemma 3.1 of [13]that if·C=0 andn=4,then the Pick invariantJcan only take four constant values at any point,which implies thatJ,and thus the scalar curvature are constant.This,together with (2),confirms Conjecture 1.3 forn=4.

Second,by removing the restriction of affine hyperspheres,we give an answer to Problem 1.2 under the assumption that the Weyl curvature tensor vanishes identically.

Theorem 1.7LetMn,n ≥3 be a locally strongly convex affine hypersurface in Rn+1with·C=0 and the vanishing Weyl curvature tensor.Denote bymthe number of distinct eigenvalues of its Schouten tensor.Thenm ≤2,and one of the two cases occurs:

(i)m=1,Mnis either a hyperquadric,or a flat affine hypersurface;

(ii)m=2,Mnis affinely equivalent to either (1.3),or one of the six quasi-umbilical affine hypersurfaces with the quasi-Einstein affine metric of nonzero scalar curvature,as is explicitly described in Theorem 4.3.

Remark 1.8m=1 in case(i)means that the affine metric is a constant sectional curvature.Related to this,we prove in Proposition 3.1 that any non-degenerate affine hypersurface satisfies that·C=0,and that the affine metric is a constant sectional curvature if and only if it is either a hyperquadric or a flat affine hypersurface.

Remark 1.9The case (i) is partially classified in Remark 4.1,though its complete classification is still complicated and not solved.In fact,Anti?-Li-Vrancken-Wang [3]recently classified the locally strongly convex affine hypersurfaces with constant sectional curvature for when the hypersurface admits at most one affine principal curvature of multiplicity one.Even for affine surfaces with a flat affine metric,the classification problem is still open.

Remark 1.10The examples in case (ii) are of the generalized Calabi compositions of a hyperquadric and a point in some special forms[1].The construction method of such examples initially originates from Calabi [8],and has been extended and characterized by Anti?,Dillen,Vrancken,Hu,Li,et al.in [1,2,11,19,22].

The rest of this paper is organized as follows.In Section 2,we briefly review the local theory of equiaffine hypersurfaces,and some notions and results of conformally flat manifolds and warped product manifolds.In Section 3,we study the properties of the hypersurfaces involving the eigenvalues and eigenvalue distributions of the Schouten tensor,the difference tensor and the affine principal curvatures,and present the proof of Theorem 1.4.Based on these properties and known results,in Section 4 we discuss all of the possibilities of the immersion,and complete the proof of Theorem 1.7.

2 Preliminaries

In this section,we briefly review the local theory of equiaffine hypersurfaces.For more details,we refer to the monographs [23,26].

Let Rn+1denote the standard (n+1)-dimensional real unimodular affine space that is endowed with its usual flat connectionDand a parallel volume formω,given by the determinant.LetF:Mn →Rn+1be an oriented non-degenerate hypersurface immersion.On such a hypersurface,up to a sign,there exists a unique transversal vector fieldξ,called the affine normal.A non-degenerate hypersurface equipped with the affine normal is called an(equi)affine hypersurface,or a Blaschke hypersurface.Denote always byX,Y,Z,Uthe tangent vector fields onMn.By the affine normal we can write that

which induce onMnthe affine connection?,a semi-Riemannian metrichcalled the affine metric,the affine shape operatorS,and the cubic formC:=?h.An affine hypersurface is said to be locally strongly convex ifhis definite,and we always chooseξ,up to a sign,such thathis positive definite.We call a locally strongly convex affine hypersurface quasi-umbilical (resp.quasi-Einstein) if it admits exactly two distinct eigenvalues ofS(resp.Ricci tensor ofh),one of which is simple.

We also writeKXYandKX=?X-.Since both?andhave zero torsion,Kis symmetric inXandY.It is also related to the totally symmetric cubic formCby

which implies that the operatorKXis symmetric relative toh.Moreover,Ksatisfies the apolarity condition,namely,trKX=0 for allX.

Here,by definition,[KX,KY]Z=KXKY Z-KY KXZ,and

Contracting the Gauss equation (2.5) twice we have that

Mnis called an affine hypersphere ifS=H id.Then it follows from(2.7)thatHis constant ifn ≥2.Mnis said to be a proper (resp.improper) affine hypersphere ifHis nonzero (resp.zero).Moreover,a locally strongly convex affine hypersphere is said to be parabolic,elliptic or hyperbolic according to whetherH=0,H>0 orH<0,respectively.For affine hyperspheres,the Gauss and Codazzi equations reduce to

We collect the following two results for later use:

Theorem 2.1(cf.Theorem 1.1 and Corollary 2.1 of [9]) LetMn,n ≥3 be a locally strongly convex affine hypersphere in Rn+1with the constant scalar curvaturer.ThenJr ≤0 and the traceless Ricci tensorsatisfies that

where‖·‖hdenotes the tensorial norm with respect toh.This equality sign holds identically if and only ifMnhas the parallel cubic form and the vanishing Weyl curvature tensor,and if and only if one of three cases occurs:

(i)J=0,Mnis affinely equivalent to a hyperquadric;

(ii)J≠0,r=0,Mnis affinely equivalent to (1.2);

(iii)J≠0,r<0,Mnis affinely equivalent to (1.3).

Theorem 2.2(cf.Theorem 1 of[1]) LetMm+1,m ≥2 be a locally strongly convex affine hypersurface of the affine space Rm+2such that its tangent bundle is an orthogonal sum with respect to the affine metrichof two distributions: a one-dimensional distributionD1spanned by a unit vector fieldT,and anm-dimensional distributionD2such that

Then eitherMm+1is an affine hypersphere such thatKT=0,or it is affinely congruent to one of the following immersions:

(1)f(t,x1,···,xm)=(γ1(t),γ2(t)g2(x1,···,xm)) forγ1,γ2such that

(2)f(t,x1,···,xm)=γ1(t)C(x1,···,xm)+γ2(t)em+1forγ1,γ2such that

(3)f(t,x1,···,xm)=C(x1,···,xm)+γ2(t)em+1+γ1(t)em+2forγ1,γ2such that

Hereg2:Rm →Rm+1is a proper affine hypersphere centered at the origin with the affine mean curvature?,andC:Rm →Rm+2is an improper affine hypersphere given byC(x1,···,xm)=(x1,···,xm,p(x1,···,xm),1) with the affine normalem+1.

Finally,we review some notions and results regarding conformally flat manifolds and warped product manifolds.The Schouten tensor of (1,1)-type,on a Riemannian manifold (M,h) of dimensionn ≥3,is a self-adjoint operator relative tohdefined by

whereQ,Iandrare the Ricci operator,identity operator and scalar curvature,respectively.Note that (M,h) is conformally flat,which means that a neighborhood of each point can be conformally immersed into the Euclidean space Rn.Forn ≥4,(M,h)is conformally flat if and only if the Weyl curvature tensor defined below vanishes:

Forn=3,it is well known thatW=0 identically,and that (M,h) is conformally flat if and only if the Schouten tensor is a Codazzi tensor.IfW=0,the Riemannian curvature tensor can be rewritten by

(M,h)of dimensionn ≥3 is of constant sectional curvature if and only ifW=0 and its metric is Einstein,where the Einstein metric means that the eigenvalues of the operatorPorQare single.

For Riemannian manifolds(B,gB),(M1,g1)and a positive functionf:B →R,the product manifoldM:=B×M1,equipped with the metrich=gB ⊕f2g1,is called a warped product manifold with the warped functionf,denoted byB×f M1.IfEandU,Vare independent vector fields ofBandM1,respectively,then(cf.(2)of[5])the sectional curvatures ofMsatisfy that

Theorem 2.3(cf.Theorem 3.7 of [5]) LetM=B×f M1be a warped product with dimB=1 and dimM1≥2.ThenMis conformally flat if and only if,up to a reparametrization ofB,(M1,g1) is a space of constant curvature andfis any positive function.

3 Conformally Flat Affine Hypersurfaces with ·C=0

From this section on,when we say that an affine hypersurface has the semi-parallel cubic form,this always means that·C=0,or equivalently,that·K=0.Then,by the Ricci identity ofK,we have that

In fact,by (2.4) and the Ricci identities ofCandK,the equivalence above follows from the following formula:

First,under the assumption that the affine metric is of constant sectional curvature,we extend the result of Theorem 6.2 in [21],stating that a locally strongly convex affine surface has the semi-parallel cubic form if and only if it is either a quadric or a flat affine surface,from the surface to the non-degenerate affine hypersurface,as follows:

Proposition 3.1LetMn,n ≥2 be a non-degenerate affine hypersurface of Rn+1.ThenMnhas the semi-parallel cubic form and an affine metric of constant sectional curvature if and only ifMnis either a hyperquadric or a flat affine hypersurface.

ProofThe “if part” follows from (3.1) and the Pick-Berwald theorem.

Now,we prove the “only if part”.Assume that the affine metric is of constant sectional curvaturec,then

Let us choose an orthonormal basis{e1,···,en} such thath(ei,ej)=?iδijand?i=±1.Then,from (3.1),we get that

which,together with (3.3),implies that,fori≠j,

Therefore,eitherc=0,orc≠ 0 andKeiei=cieifor alli.In the latter case,the apolarity condition shows thatcj=trKej=0 for eachj,and thusKeiei=0 for alli.AsKis symmetric,we have thatK=0 identically.By the Pick-Berwald theorem,the conclusion follows.

Remark 3.2To see the examples whose cubic forms are semi-parallel but not parallel,we refer to Remark 6.2 in [21]for flat surfaces,and Theorem 4.1 in [3]for the flat quasi-umbilical affine hypersurfaces.

From now on,letF:Mn →Rn+1,n ≥3 be a locally strongly convex affine hypersurface with·C=0 and the vanishing Weyl curvature tensor.Then we have(3.1)and(2.12).Denote by{e1,···,en} the local orthonormal frame ofMn,whereeiare the eigenvector fields of the Schouten tensorPwith corresponding eigenvaluesνi,i=1,···,n.Then,we see from (2.12)that

for any tangent vectorZ.TakingX=ei,Y=ej,u=ek,Z=e?in (3.1),we have that

Fork=?=i≠jin (3.5),we obtain that

Taking the inner product of (3.6) withei,we deduce that

Interchanging the roles ofeiandej,similarly we have that

Taking the inner product of (3.6) withej,by (3.8),we see that

Together with (3.7),we obtain from (3.6) that

On the other hand,settingk=i,?=jin (3.5),and by (3.7) and (3.8),we deduce that

For more information,we denote byν1,···,νmthemdistinct eigenvalues forPof multiplicities(n1,n2,···,nm).Let D(νi)be the eigenvalue distribution of eigenvalueνifori=1,···,m.Then,we can prove the following two lemmas:

Lemma 3.3It holds that

(i) ifνi≠0 andni ≥2,thenh(K(u,v),w)=0,?u,v,w ∈D(νi);

ProofIfνi≠ 0 andni ≥2,then,by takingνj=νiandei=uin (3.9),we haveh(K(u,u),u)=0 for any unit vectoru ∈D(νi).Then,the conclusion (i) follows from the symmetric property ofK.

Then,by (3.9),we have thath(K(u,u),u)=0 for any vectoru ∈D(νi),which,together with the symmetric property ofK,implies thatK(u,u)?D(νi).Furthermore,it follows from(3.10)and (3.11) that

for any unit vectorsu ∈D(νi),v ∈D(νj).Ifm=2,then (3.13) immediately implies thatK(u,u)=K(v,v)=0.Ifm ≥3,for arbitraryνk,different fromνiandνj,then by (3.12),eitherνk+νi≠ 0 orνk+νj≠ 0 holds.In either case,by (3.7) and the arbitrariness ofνkwe see from (3.13) thatK(u,u)=K(v,v)=0.Therefore,by the symmetric property ofKwe have (ii).

Lemma 3.4The numbermof distinct eigenvalues of the Schouten tensorPis at most 2.In particular,ifm=2,denoting byν1,ν2the two distinct eigenvalues ofP,then one of their multiplicity must be 1,and

ProofFirst,form ≥2,we claim that there must exist two distinct eigenvalues,νiandνj,ofPsuch thatνi+νj=0.Otherwise,we have thatfor anyνi≠νj,which,together with Lemma 3.3 (ii),implies thatK=0 identically.Then,the Pick-Berwald theorem implies that the hypersurface is a hyperquadric,and is thus of constant sectional curvature.This means thatm=1,which stands in contradiction tom ≥2.

By the claim above,form ≥2,we denote byν1,ν2the two distinct eigenvalues ofPsuch thatν1+ν2=0,and thusν1ν2≠0.

Next,we prove thatm ≤2.Otherwise,ifm ≥3,then,for arbitraryνi,differently that forν1andν2,we have that (νi+ν1)(νi+ν2)≠0,and thus,

It follows from Lemma 3.3 (ii) that

for any vectorsu ∈D(ν1),v ∈D(ν2),w ∈D(νi).Therefore,the arbitrariness ofνiimplies thatK(u,v)=0.Together with (3.15) and the symmetric property ofK,we have thatK=0 identically.As before,it follows from the Pick-Berwald theorem thatm=1,which stands in contradiction tom ≥3.Hencem ≤2.

Finally,form=2,letν1,ν2be the two distinct eigenvalues ofPwith multiplicities(n1,n2).From the analysis above,we have (3.14).It is sufficient to prove that eithern1=1 orn2=1 holds.On the contrary,assume thatn1≥2 andn2≥2.Denote by,···,the orthonormal eigenvector fields ofP,which span D(νi) fori=1,2.Then,asνi≠ 0 andni ≥2,by Lemma 3.3 (i) and (3.11),we have that

fori=1,2.It follows from the apolarity condition that

for anyu ∈D(ν1).Therefore,h(Kuv,v)=0 for anyu ∈D(ν1),v ∈D(ν2).Similarly,we have thath(Kvu,u)=0.Then,by the symmetric property ofK,we have thatK(u,v)=0.Together with (3.16),we further obtain thatK=0 identically.As before,this stands in contradiction to the fact thatm=2.Lemma 3.4 has been proven.

Next,based on Lemma 3.4,we pay attention to the case ofm=2.We always denote byν1,ν2the two distinct eigenvalues ofPwith multiplicities 1,n-1,respectively.LetTbe the unit eigenvector field of the eigenvalueν1,and let{X1,···,Xn-1} be any orthonormal frame of D(ν2).By (3.14),(3.10),(3.11) and Lemma 3.3 (i),we see that

It follows from the apolarity condition that

Summing the above,asm=2 and thusK≠0,we can assume that

Combining the last formula with (2.11) and the fact that trQ=r,we obtain that

which implies that the affine metric is quasi-Einstein with nonzero scalar curvature.

Now,we are ready to prove the following:

Lemma 3.5Ifm=2,then the numberσof distinct affine principal curvatures is at most 2.Moreover,there exists a local orthonormal frame,still denoted by{T,X1,···,Xn-1},such that there hold (3.18),(3.19) and

ProofFrom the analysis above we still have the freedom to rechoose the orthonormal frame of D(ν2) such that (3.18) and (3.19) hold.Therefore,we can reselect the orthonormal frame of D(ν2) if necessary,still denoted by{X1,···,Xn-1},such that

Then,it follows from (3.18),(3.4) and the Gauss equation (2.5) that

which,together with (3.21),implies that both D(ν1) and D(ν2) are the invariant subspaces ofS.Therefore,we can rechoose the local orthonormal frame on D(ν2),still denoted by{X1,···,Xn-1},such that

Then,we see from (3.18),(3.4) and the Gauss equation (2.5) that

which imply that

Therefore,we have thatσ ≤2 and (3.20).

Finally,we conclude this section by proving Theorem 1.4.

Completion of Proof Theorem 1.4LetF:Mn →Rn+1,n ≥3 be a locally strongly convex affine hypersphere with·C=0 and the vanishing Weyl curvature tensor.Then,we see from Lemma 3.4 that the numbermof distinct eigenvalues of the Schouten tensorPis at most 2.

Ifm=1,byW=0 we see from (2.12) thatMnis of constant sectional curvature.Then,it follows from Proposition 3.1 and the Main theorem of [28]thatMnis affinely equivalent to either a hyperquadric,or the flat and hyperbolic affine hypersphere (1.2).

Ifm=2,based on Lemma 3.5,by the fact thatH=μ1=μ2is constant,we see from(3.20),(3.19) and (2.8) that

and thusris a negative constant.Defining the traceless Ricci tensorwe deduce from (3.19) and (3.22) that all of the components ofvanish except for

Combining this with (3.22),we have that

where‖·‖hdenotes the tensorial norm with respect toh.

In summary,Mnis a hyperbolic affine hypersphere with constant negative scalar curvature andJ≠ 0,which further satisfies the formula (3.23).Then,it follows from Theorem 2.1 thatMnis affinely equivalent to the hyperbolic affine hypersphere (1.3).

4 Proof of Theorem 1.7

LetMn,n ≥3 be a locally strongly convex affine hypersurface in Rn+1with·C=0 and the vanishing Weyl curvature tensor.Denote bym(resp.σ)the number of distinct eigenvalues of its Schouten tensor (resp.affine shape operator).Then,by Lemma 3.4 we have thatm ≤2.

Ifm=1,byW=0,we see from (2.12) thatMnis of constant sectional curvature.Then,it follows from Proposition 3.1 thatMnis either a hyperquadric,or a flat affine hypersurface.

Remark 4.1IfMnis a flat affine hypersurface withC≠ 0,it follows from Theorems 1.1 and 1.2 of [3]and Theorem 1.4 thatMnis affinely equivalent to either (1.2) ifσ=1,or one of the three flat quasi-umbilical affine hypersurfaces (1),(2) or (7) explicitly described in Theorem 4.1 of [3]ifσ=2.However,forσ ≥3,Mnis a flat affine hypersurface with at least two affine principal curvatures of multiplicity one,whose classification is still complicated and has not been involved until now.

Ifm=2,by Lemma 3.5,we have thatσ ≤2.Forσ=1,i.e.,Mnis an affine hypersphere,Theorem 1.4 shows thatMnis affinely equivalent to (1.3).

For the last case,m=σ=2,we see from Lemma 3.5 thatMnis a quasi-umbilical affine hypersurface with a quasi-Einstein affine metric of nonzero scalar curvature.Moreover,for the local orthonormal frame{T,X1,···,Xn-1},it holds that

By (3.18) and (3.19),we further obtain from (2.12) that

Remark 4.2Forσ=m=2,Mnis affinely equivalent to one of the three classes of immersions in Theorem 2.2.In fact,it follows from (4.1) thatMnsatisfies the conditions of Theorem 2.2.In the proof of Theorem 2.2 in [1],it was shown in Lemma 3 that ifλ2=0,thenKT=0 andMnis an affine hypersphere.In our situation,byMnnot an affine hypersphere,we can exclude this possibility in Theorem 2.2.

Next,we give more information about the three classes of warped product immersions in Theorem 2.2.Denote by D(μ2)=span{X1,···,Xn-1} the eigenvalue distribution ofScorresponding toμ2.In the proof of Theorem 2.2 in [1],by (4.2) it was shown that

and thatMnis locally a warped product R×fM2,where the warping functionfis determined byα=-T(lnf),and R andM2are,respectively,integral manifolds of the distributions span{T}and D(μ2),andM2is an affine hypersphere.The same proof implies that the projection of the difference tensor on D(μ2) is the difference tensor of the componentsM2given by

Then,it follows from (4.1) thatL2=0.HenceM2is a hyperquadric with constant sectional curvaturec.

By the warped product structure,if not otherwise stated,we always take the local coordinates{t,x1,···,xn-1} onMnsuch thatThen all functionsμi,λi,α,randfdepend only ont.Denoting that?t()=(·)′,we have thatα=-f′/f.It follows from (2.13),(4.3) andα′=α2in (4.4) that

Furthermore,we can get,from the equations above forfandα,that,up to a translation and a direction of the parametrict,

where locally we take thatt>0.Together with(4.2)and(4.4)we can check thatiff=1,andiff=t.It follows from (4.2) and (4.6) that

where the scalar curvatureris nonzero in either case.

The Pick-Berwald theorem states that the cubic form or difference tensor vanishes if and only if the hypersurface is a non-degenerate hyperquadric.For the locally strongly convex case,for later use,we recall from [23]pages 104–105 the following three hyperquadrics revised for dimensionn-1:

(1) The ellipsoid described in Rnby

This has constant sectional curvaturecand the affine normal-cF1,where in local coordinates we denote this immersion byF1(x1,···,xn-1).

(2) The hyperboloid described in Rnby

This has constant sectional curvaturecand the affine normal-cF2,where in local coordinates we denote this immersion byF2(x1,···,xn-1).

(3) The elliptic paraboloid described in Rnby

The affine metric is flat.This is the only parabolic affine hypersphere with constant sectional curvature.

Finally,armed with the above preparations,by Remark 4.2 and the computations carried out in[3]for the three classes of immersions in Theorem 2.2,we can prove the following theorem,which,together with the previous cases,completes the proof of Theorem 1.7:

Theorem 4.3LetMn,n ≥3 be a locally strongly convex affine hypersurface in Rn+1with·C=0 and the vanishing Weyl curvature tensor.Assume that both the numbers of distinct eigenvalues of its Schouten tensor and affine shape operator are 2.ThenMnis a quasi-umbilical affine hypersurface with a quasi-Einstein metric of nonzero scalar curvaturer.Moreover,(Mn,h) is locally isometric to the warped product R+×f M2,where the warped function isf(t)=1 ort,M2is of constant sectional curvaturec,andMnis affinely equivalent to one of the following six hypersurfaces:

(1) The immersion (γ1(t),γ2(t)F1(x1,···,xn-1)),where

(a)f(t)=1,andis positive constant,

(b)γ1andγ2are determined by

wherec1,c2are constants such thatγ2>0.

(2) The immersion (γ1(t),γ2(t)F2(x1,···,xn-1)),where

(a)f(t)=1,andis negative constant,

(b)γ1andγ2are determined by

wherec1,c2are constants such thatγ2>0.

(3) The immersion (γ1(t),γ2(t)F1(x1,···,xn-1)),where

(a)f(t)=t,andis positive constant,

(b)γ1andγ2are determined by

wherek(t) is a positive solution to the linear differential equation

(4) The immersion (γ1(t),γ2(t)F2(x1,···,xn-1)),where

(a)f(t)=t,andis negative constant,

(b)γ1andγ2are determined by

wherek(t) is a positive solution to the linear differential equation

(b)γ1andγ2are determined by

wherec1is a constant.

Remark 4.4By Theorem 2.3,the warped product structure in Theorem 4.3 implies that all of the examples above,especially for the 3-dimensional case,are conformally flat.

ProofBased on the previous analysis,we will discuss the three classes of immersions in Theorem 2.2 by takingm=n-1.

Case 1It was shown in [1]that if

thenMnis affinely equivalent to the immersion(γ1(t),γ2(t)g2(x1,···,xn-1)),whereg2:M2→Rnis a proper affine hypersphere with the difference tensorL2defined by (4.5).It follows from the analysis above and (4.7)-(4.9) thatL2=0,andg2has constant sectional curvaturec≠ 0,the affine mean curvaturecand the affine normal-cg2.Thusg2=F1ifc>0,org2=F2ifc<0.Asr≠ 0,(4.8) implies thatc≠ 1 iff=t.Moreover,by the computations of the immersion on pages 292–294 of [3],we take thatλ=-cin (4.3) of [3],and deduce that

wherek(t),and thusγ2are positive functions.

Iff=1,we see from (4.10) thatandγ2=k(t)1/(n+1),where

Solving this equation,we obtain that

where the constantsc1,c2are chosen such thatγ2>0.We obtain the immersion (1) ifc>0,and the the immersion (2) ifc<0.

Iff=t,we see from (4.2),(4.8) and (4.10) thatand

In particular,ifk(t) is a power function oft,we can solve this equation to obtain that

wherec1is a positive constant,the constantsc2,c3are chosen such thatγ2>0,andτ1,τ2are the solutions of the quadric equationτ2-(n+2)τ+(n+1)c=0.We obtain the immersion(3) ifc>0,and the immersion (4) ifc<0.

Case 2It was shown in [1]that if

thenMnis affinely equivalent to the immersion

whereg(x1,···,xn-1) is a convex function whose graph immersion is a parabolic affine hypersphere with the difference tensorL2defined by (4.5).It follows from the first equation of (4.2),(4.7),(4.8) and (4.14) thatL2=0,f=t,c=0,and that this parabolic affine hypersphere has a flat affine metric,and thus

Moreover,we recall from the computations of the hypersurfaces on page 294 of [3]thatγ1,γ2satisfy that

which yields that

Here,by applying equiaffine transformations,we may set thatc2=-(n+1)/(n+2)2andc3=0.We have the immersion (5).

Case 3It was shown in [1]that if

thenMnis affinely equivalent to the immersion (x1,···,xn-1,g(x1,···,xn-1)+γ1(t),γ2(t)),whereg(x1,···,xn-1) is a convex function whose graph immersion is a parabolic affine hypersphere with the difference tensorL2defined by (4.5).It follows from the analysis above,λ2≠0 in (4.2),(4.7),(4.8) and (4.16),thatL2=0,μ2=0,λ2=α=-,f=tandc=0.As before,

Moreover,we see from the computations of the hypersurfaces on page 295 of [3]thatγ1,γ2satisfy that

Then,takingf=tinto this equation,we deduce that

where?∈{-1,1}.Then,we can directly solve these equations to obtain that

By applying a translation and a reflection in Rn+1,we may assume thatc1=c3=0 andγ2>0,i.e.,?=1.Also,by possibly applying an equiaffine transformation,we may set thatc2=0.Hence,we obtain the immersion (6).

Conflict of InterestThe authors declare no conflict of interest.