We study the space-like hypersurfaces with vanishing conformal form in the conformal geometry, and classify the the Einstein space-like hypersurfaces in the conformal spac

Huizhang, Y.

Xia, S.

English

Università del Salento: ESE - Salento University Publishing

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 32 (2012) no. 2, 63–71. doi:10.1285/i15900932v32n2p63Space-like hypersurfaces with vanishingconformal forms in the conformal spaceYang Huizhang iDepartment of Mathematics, University of Honghe,Mengzi, Yunnan, 661100 P.R.Chinayanghuizhangyn@163.comSun XiaDepartment of Arts and Science, University of Yunnan Radio & TV ,Kunming, Yunnan, 650223 P.R.Chinasunx@sina.comReceived: 14.4.2010; accepted: 7.3.2012.Abstract. We study the space-like hypersurfaces with vanishing conformal form in the con-formal geometry, and classify the the Einstein space-like hypersurfaces in the conformal space.Keywords: space-like hypersurfaces; conformal space; conformal form.MSC 2000 classification: 53A30; 53C21; 53C40IntroductionWe define the pseudo-Euclidean inner product 〈·, ·〉s in Rn+p as〈X,Y 〉s = −s∑i=1xiyi +n+p∑i=s+1xiyi, X = (xi), Y = (yi) ∈ Rn+p,Let RPn+2 be a real projective space. < ·, · >2 is the pseudo-Euclidean innerproduct in Rn+3. The quadratic surfaceQn+11 = {[ξ] ∈ RPn+2|〈ξ, ξ〉2 = 0}in RPn+2 is called the conformal space.Suppose that x : Mn→Qn+11 is a space-like hypersurface in the conformalspace Qn+11 , {ei} is a local orthonormal frame of Mn for the standard metricI = dx · dx with dual basis {θi}. Then we define the first fundamental form I,the second fundamental form II and the mean curvature of x asI = 〈dx, dx〉1 =∑iθi ⊗ θi; II =∑ijhijθi ⊗ θj ; H = 1n∑ihii.iThis work is partially supported by the Science Foundation of Honghe Univer-sity(10XJY121) of Chinahttp://siba-ese.unisalento.it/ c© 2012 Universita` del Salento64 AnghelIn [5], Nie and Wu classified the hypersurfaces with parallel conformal secondfundamental form.They obtainedTheorem 1. [5] Let x :Mn→Qn+11 be a space-like hypersurface with parallelconformal second fundamental form, then Mn is conformally equivalent to anopen part of one of the following hypersurfaces in Qn+11 :1) Sk(a)×Hn−k(b) ⊂ Sn+112) Hk(a)×Hn−k(b) ⊂ Hn+11 ;3) Hk(a)×Rn−k ⊂ Rn+11 ;4) WP (p, q, a) ⊂ Rn+11 , WP (p, q, a) is the warped product embedding u :Sp(a)×Hq(b)×R+ ×Rn−p−q−1 ⊂ Rn+21 → Rn+11 , a > 1, b =√a2 − 1, which isgiven byu = (tu1, tu2, tu3), u1 ∈ Sp(a), u2 ∈ Hq(b), u3 ∈ Rn−p−q−1, t ∈ R+.In this paper, we consider the space-like hypersurfaces Mn with conformalform C = 0, which also have harmonic curvature. Here is our main theoremTheorem 2. Let x :Mn→Qn+11 be a space-like hypersurface in Qn+11 with-out umbilics. If its conformal form C = 0 and its curvature tensor is harmonic,then its conformal second fundamental form is parallel.Every manifold with parallel Ricci tensor has harmonic curvature. This ap-plies, for instance, to Einstein manifolds. Consequently, we have the followingcorollaryCorollary 1. Let x :Mn→Qn+11 be a space-like hypersurface in Qn+11 with-out umbilics. If its conformal form C = 0 and Mn is Einstein hypersurface withrespect to conformal metric g, then Mn is conformally equivalent to an openpart of one of the following hypersurfaces in Qn+11 :1) Hk(a)×Hn−k(b) ⊂ Hn+11 ;2) H1(a)×Rn−1 ⊂ Rn+11 .1 Conformal invariants for space-like hypersurfacesin Qn+11Let x :Mn→Qn+11 be a space-like hypersurface in the conformal space Qn+11 .The cone of light in Rn+3 is given byCn+2 = {ξ ∈ Rn+3|〈ξ, ξ〉2 = 0, ξ 6= 0}.Then there exists a unique lift Y :Mn → Cn+2 of x such that g = 〈dY, dY 〉up to a sign, Y is called the canonical lift of x. Then we haveOvals with Invariant Maximal Quadrilaterals 65Theorem 3. [6]Two space-like hypersurfaces x, x˜ : Mn→Qn+11 are confor-mally equivalent if and only if there exists a pseudo-orthogonal transformationT∈O(n, 2) in Rn+32 such that Y = Y˜ T .It follows immediately from Theorem 3 that g = 〈dY, dY 〉 = e2τdx·dx, e2τ =nn− 1(∑ij(hij)2 − nH2) is a conformal invariant, which is called the conformalmetric of x :Mn→Qn+11 .Let ∆ be the Laplacian operator with respect to g. We defineN = − 1n∆Y − 12n2〈∆Y,∆Y 〉Y. (1)It is easy to see that〈∆Y, Y 〉2 = −n, 〈Y, dY 〉2 = 0, (2)〈Y, Y 〉2 = 〈N,N〉2 = 0, 〈Y,N〉2 = 1. (3)Let {Ei := e−τei} be a local orthonormal basis for the conformal metric gwith dual basis {ωi = eτθi}. Writing {Yi = Ei(Y )}, we have〈Yi, Yj〉2 = δij , 〈Yi, Y 〉2 = 〈Yi, N〉2 = 0, 1 ≤ i, j ≤ n. (4)If we denote by V is the orthogonal complement space of the subspacespan{Y,N, Y1, · · · , Yn} in Rn+32 , then we haveRn+32 = span{Y,N}⊕span{Y1, · · · , Yn} ⊕ V,here V is called the conformal normal bundle of x : Mn→Sn+1. We definethe local orthonormal basis of V byE = En+1 := (H,Hx+ en+1), (5)then {Y,N, Y1, · · · , Yn, E} is a moving frame of Rn+32 along Mn, we can writethe structure equations asdY =∑iYiωi, (6)dN =∑iψiYi + CE, (7)dYi = −ψiY − ωiN +∑jωijYj + ωi,n+1E, (8)dE = CY +∑iωi,n+1Yi. (9)The tensors A =∑i,jAijωiωj , B =∑i,jBijωiωj , C =∑iCiωi are called theBlaschke tensor, the conformal second fundamental form and the conformal66 Anghelform respectively. All of them are conformal invariants. The relations betweenconformal invariants and Euclidean invariants of x are given byBij = e−τ (hij −Hδij), (10)Ci = e−2τ (Hτi −∑jhijτj −Hi), (11)Aij = e−2τ [τiτj − τi,j −Hhij + 12(H2 −∑k(τk)2 + ǫ)δij ]. (12)Here τi = ei(τ), Hi = ei(H). τi,j and ∇ are called the Hessian-matrix and thegradient with respect to I = dx · dx respectively.We define the covariant derivatives of Ci, Aij , Bij as follows∑jCi,jωj = dCi −∑jCjωji, (13)∑kAij,kωk = dAij −∑kAikωkj −∑kAkjωki, (14)∑kBij,kωk = dBij −∑kBikωkj −∑kBkjωki. (15)Then the structure equations (6)–(9) are equivalent toAij,k −Aik,j = BikCi −BijCk, (16)Ci,j − Cj,i =∑k(BikAkj −BkjAki), (17)Bij,k −Bik,j = δijCk − δikCj , (18)Rijkl = δikAjl − δilAjk + δjlAik − δjkAil − (BikBjl −BilBjk), (19)Rij :=∑kRikjk =∑kBikBjk + (trA)δij + (n− 2)Aij , (20)∑iBii = 0,∑i,j(Bij)2 =n− 1n, trA =∑iAii =12n(n2κ− 1). (21)Here Rijkl is the curvature tensor of g, Q =∑i,jRijωi ⊗ ωj is the conformalRicci curvature and κ =1n(n− 1)∑iRii is the normalized conformal scalarcurvature of x :Mn→Sn+1.Theorem 4. [6]Two space-like hypersurfaces x :Mn→Qn+11 and x˜ : M˜n →Qn+11 (n ≥ 3) are conformally equivalent if and only if there exists a diffeomor-phism σ : Mn → M˜nwhich preserves the conformal metric g and conformalsecond fundamental form B.Ovals with Invariant Maximal Quadrilaterals 672 The proof of main resultsProof of Theorem 2. Let x : Mn→Qn+11 be a space-like hypersurface inQn+11 with conformal form C = 0. From (17), we choose a local orthonormalframe {ei} with respect to g such that A,B are diagonalizable at the same time,i.e.,Bij = biδij , Aij = aiδij , 1 ≤ i, j ≤ n. (22)We are assuming that x has harmonic conformal curvature, i.e.,∑iRijkl,i =0. This happens if and only if the Ricci tensor is Codazzi tensor, i.e., Rij,k =Rik,j . Thus the scalar curvature of x with respect to g is constant and tr(A) isconstant too. From (20), we haveRij,k =∑lBil,kBlj +∑lBilBlj,k + (n− 2)Aij,k, (23)Rik,j =∑lBil,jBlk +∑lBilBlk,j + (n− 2)Aik,j , (24)Since C = 0, form(17), (18), we have Bij,k = Bik,j , Aij,k = Aik,j . Thus from(22) and (23), we get ∑lBil,kBlj =∑lBil,jBlk.By using (22), for any indices of i, j, k, we getBij,kbj = Bik,jbk, (25)If bj 6= bk, then we haveBij,k = 0. (26)If bj = bk, since∑lBjk,lωl = dBjk +∑lBjlωlk +∑lBljωli= dBjk + (bj − bk)ωjk(27)It is easy to see from (22) thatBij,k = 0. (28)From (26), (28) and∑jBij,j = 0, for any indices of i, j, k, we getBij,k = 0. (29)Therefore we obtain our main Theorem 2.68 AnghelNow let t be the number of the distinct eigenvalues of A, and a1, a2, · · · , atbe all of distinct eigenvalues. Taking a suitably local orthonormal frame field{E1, E2, · · · , En} such that the matrix (Aij) can be written as(Aij) = Diag(a1, · · · , a1︸ ︷︷ ︸k1, a2, · · · , a2︸ ︷︷ ︸k2, · · · , at, · · · , at︸ ︷︷ ︸kt),that isA1 = · · · = Ak1 = a1, · · · , An−kt+1 = · · · = An = at,here a1, · · · , at are not necessarily different from each other.Similarly, under the same orthonormal frame field, the matrix (Bij) can bewritten as(Bij) = Diag(b1, · · · , b1︸ ︷︷ ︸k1, b2, · · · , b2︸ ︷︷ ︸k2, · · · , bt, · · · , bt︸ ︷︷ ︸kt),or equivalentlyB1 = · · · = Bk1 = b1, · · · , Bn−kt+1 = · · · = Bn = bt,and b1, · · · , bt are not necessarily different from each other.Proposition 1. If the number of the distinct eigenvalues is t ≥ 3, thent = 3.Proof. If t > 3, then there exist at least four indices i1, i2, i3, i4, such thatAi1 , Ai2 , Ai3 , Ai4 are distinct from each other.Making the convention on the ranges of indices as follows1 ≤ i1, j1 ≤ k1, k1 + 1 ≤ i2, j2 ≤ k1 + k2, · · · , k1 + k2 + 1 ≤ it, jt ≤ n.From (15) and Bij,k = 0, we have ωimin = 0(m 6= n, 1 ≤ m,n ≤ t). UsingGauss equation and from (19), we obtainBi1Bi2 +Ai1 +Ai2 = 0, Bi3Bi4 +Ai3 +Ai4 = 0,Bi1Bi3 +Ai1 +Ai3 = 0, Bi2Bi4 +Ai2 +Ai4 = 0.Consequently, (Ai1−Ai4)(Ai2−Ai3) = 0, it contradicts with the assumptionthat Ai1 , Ai2 , Ai3 , Ai4 are distinct from each other.Proposition 2. The Einstein space-like hypersurfaces with vanishing con-formal form in conformal space have at most two different conformal principalcurvatures.Ovals with Invariant Maximal Quadrilaterals 69Proof. If that doesn’t happen, t = 3. Taking a local orthonormal frame field{E1, E2, · · · , En} such that(Bij) = Diag(b1, · · · , b1︸ ︷︷ ︸k1, b2, · · · , b2︸ ︷︷ ︸k2, b3, · · · , b3︸ ︷︷ ︸k3)with the multiplicity are k1, k2, k3 respectively, and k1 + k2 + k3 = n.(Aij) = Diag(a1, · · · , a1︸ ︷︷ ︸k1, a2, · · · , a2︸ ︷︷ ︸k2, a3, · · · , a3︸ ︷︷ ︸k3).Since B is parallel, By Using Gauss equation and from (15), (19), we obtainb1b2 + a1 + a2 = 0,b1b3 + a1 + a3 = 0,b2b3 + a2 + a3 = 0.(30)Obviously, we haveb3(b1 − b2) = −(a1 − a2). (31)Since M is Einstein manifold, i.e., the Ricci curvature Rij =rnδij = (n −1)κδij(n ≥ 3), so its conformal scalar curvature κ is constant. From (20) wehave (n− 1)κ = tr(A) + (n− 2)a1 + b21,(n− 1)κ = tr(A) + (n− 2)a2 + b22,(n− 1)κ = tr(A) + (n− 2)a3 + b23.(32)Subtracting the second formula from the the first formula, we get(b1 + b2)(b1 − b2) = −(n− 2)(a1 − a2). (33)Substituting (31) into (33), we obtainb1 + b2 = −(n− 2)b3.Similarly, we haveb1 + b3 = −(n− 2)b2.Making subtraction in above two formulas and obtain b2 = b3, it contradictswith the assumption, so we complete the proof of Proposition 2.Next, we give the proof of Corollary 1.Proof of Corollary 1. Suppose that the number of different principle cur-vatures of the Einstein space-like hypersurfaces is t = 2, by using the first twoformulas of (21) to calculate b1, b2, we obtainb1 =1n (n− k)(n− 1)k, b2 = − 1n (n− 1)kn− k .70 Anghela1 + a2 = b1 · b2 = −n− 1n2< 0.Since(M, g) = (M1, g1)×(M2, g2), dimM1 = k, dimM2 = n−k. From (19),(M1, g1) and (M2, g2) have constant curvature R1 and R2. By direct calculation,we getR1 = 2a1 − b21, R2 = 2a2 − b22.R1 +R2 = −b21 − b22 + 2(a1 + a2) = −(b1 − b2)2 < 0.Then at least one of R1, R2 is negative, without generality, we let R1 < 0, i.e.,2a1 − b21 < 0.Since M is Einstein manifold, from (20), we have(n− 1)κδij = Rij = tr(A)δij + (n− 2)Aij +∑kBikBjk.Furthermore, we have®(n− 1)κ = tr(A) + (n− 2)a1 + b21,(n− 1)κ = tr(A) + (n− 2)a2 + b22.(34)Adding the above two formulas, we get2(n− 1)κ = 2tr(A) + (n− 2)(a1 + a2) + b21 + b22.From (21), we obtainκ =(1− k)(n− k − 1)k(n− k)(n− 2) .a2 =1n− 2[(n− 1)κ− tr(A)− b22]=1n− 2[(1− k)(n− k − 1)2k(n− k) +12n− (n− 1)kn2(n− k) ].By direct calculation, we haveR2 = 2a2 − b22=2n− 2[(1− k)(n− k − 1)2k(n− k) +12n− (n− 1)kn2(n− k) ]−(n− 1)kn2(n− k)=(1− k)(n− 1)k(n− 2)(n− k) ≤ 0.Here we get ” = ” if and only if k = 1. So we complete the proof of theCorollary 1.Ovals with Invariant Maximal Quadrilaterals 71References[1] Derdzinski,A.: On compact Riemannian manifolds with harmonic curvature, Math.Ann., 1982,259:145-152.[2] Hu Z.J.,Li H.: Classification of hypersurfaces with parallel Mo¨bius second fundamentalform in Sn+1, Science in China Ser.A, Mathematics, 2004,47:417-430.[3] Li H.,Wang C.: Surfaces with vanishing Mo¨bius form in Sn,Acta Math. Sinica, Englishseries, 2003, 19:617-678.[4] Li T.Z., Sun H.: Hypersurfaces with harmonic Mo¨bius curvature in Sn+1, Advancesin Mathematics(China), 2008, 37:57-66.[5] Nie C., Wu C : Space-like hypersurfaces with parallel second fundamental forms in theconformal space, Acta Math. Sinica, 2008, 51:685-692.[6] Nie C., Li T. : Conformal isoparametric hypersurfaces with two distinct conformal prin-cipal curvatures in conformal space. Sci China Math, 2010,53(4):953-963.

Space-like hypersurfaces with vanishing conformal forms in the conformal space

ESE - Salento University Publishing

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 32 (2012) no. 2, 63–71. doi:10.1285/i15900932v32n2p63Space-like hypersurfaces with vanishingconformal forms in the conformal spaceYang Huizhang iDepartment of Mathematics, University of Honghe,Mengzi, Yunnan, 661100 P.R.Chinayanghuizhangyn@163.comSun XiaDepartment of Arts and Science, University of Yunnan Radio & TV ,Kunming, Yunnan, 650223 P.R.Chinasunx@sina.comReceived: 14.4.2010; accepted: 7.3.2012.Abstract. We study the space-like hypersurfaces with vanishing conformal form in the con-formal geometry, and classify the the Einstein space-like hypersurfaces in the conformal space.Keywords: space-like hypersurfaces; conformal space; conformal form.MSC 2000 classification: 53A30; 53C21; 53C40IntroductionWe define the pseudo-Euclidean inner product 〈·, ·〉s in Rn+p as〈X,Y 〉s = −s∑i=1xiyi +n+p∑i=s+1xiyi, X = (xi), Y = (yi) ∈ Rn+p,Let RPn+2 be a real projective space. < ·, · >2 is the pseudo-Euclidean innerproduct in Rn+3. The quadratic surfaceQn+11 = {[ξ] ∈ RPn+2|〈ξ, ξ〉2 = 0}in RPn+2 is called the conformal space.Suppose that x : Mn→Qn+11 is a space-like hypersurface in the conformalspace Qn+11 , {ei} is a local orthonormal frame of Mn for the standard metricI = dx · dx with dual basis {θi}. Then we define the first fundamental form I,the second fundamental form II and the mean curvature of x asI = 〈dx, dx〉1 =∑iθi ⊗ θi; II =∑ijhijθi ⊗ θj ; H = 1n∑ihii.iThis work is partially supported by the Science Foundation of Honghe Univer-sity(10XJY121) of Chinahttp://siba-ese.unisalento.it/ c© 2012 Universita` del Salento64 AnghelIn [5], Nie and Wu classified the hypersurfaces with parallel conformal secondfundamental form.They obtainedTheorem 1. [5] Let x :Mn→Qn+11 be a space-like hypersurface with parallelconformal second fundamental form, then Mn is conformally equivalent to anopen part of one of the following hypersurfaces in Qn+11 :1) Sk(a)×Hn−k(b) ⊂ Sn+112) Hk(a)×Hn−k(b) ⊂ Hn+11 ;3) Hk(a)×Rn−k ⊂ Rn+11 ;4) WP (p, q, a) ⊂ Rn+11 , WP (p, q, a) is the warped product embedding u :Sp(a)×Hq(b)×R+ ×Rn−p−q−1 ⊂ Rn+21 → Rn+11 , a > 1, b =√a2 − 1, which isgiven byu = (tu1, tu2, tu3), u1 ∈ Sp(a), u2 ∈ Hq(b), u3 ∈ Rn−p−q−1, t ∈ R+.In this paper, we consider the space-like hypersurfaces Mn with conformalform C = 0, which also have harmonic curvature. Here is our main theoremTheorem 2. Let x :Mn→Qn+11 be a space-like hypersurface in Qn+11 with-out umbilics. If its conformal form C = 0 and its curvature tensor is harmonic,then its conformal second fundamental form is parallel.Every manifold with parallel Ricci tensor has harmonic curvature. This ap-plies, for instance, to Einstein manifolds. Consequently, we have the followingcorollaryCorollary 1. Let x :Mn→Qn+11 be a space-like hypersurface in Qn+11 with-out umbilics. If its conformal form C = 0 and Mn is Einstein hypersurface withrespect to conformal metric g, then Mn is conformally equivalent to an openpart of one of the following hypersurfaces in Qn+11 :1) Hk(a)×Hn−k(b) ⊂ Hn+11 ;2) H1(a)×Rn−1 ⊂ Rn+11 .1 Conformal invariants for space-like hypersurfacesin Qn+11Let x :Mn→Qn+11 be a space-like hypersurface in the conformal space Qn+11 .The cone of light in Rn+3 is given byCn+2 = {ξ ∈ Rn+3|〈ξ, ξ〉2 = 0, ξ 6= 0}.Then there exists a unique lift Y :Mn → Cn+2 of x such that g = 〈dY, dY 〉up to a sign, Y is called the canonical lift of x. Then we haveOvals with Invariant Maximal Quadrilaterals 65Theorem 3. [6]Two space-like hypersurfaces x, x˜ : Mn→Qn+11 are confor-mally equivalent if and only if there exists a pseudo-orthogonal transformationT∈O(n, 2) in Rn+32 such that Y = Y˜ T .It follows immediately from Theorem 3 that g = 〈dY, dY 〉 = e2τdx·dx, e2τ =nn− 1(∑ij(hij)2 − nH2) is a conformal invariant, which is called the conformalmetric of x :Mn→Qn+11 .Let ∆ be the Laplacian operator with respect to g. We defineN = − 1n∆Y − 12n2〈∆Y,∆Y 〉Y. (1)It is easy to see that〈∆Y, Y 〉2 = −n, 〈Y, dY 〉2 = 0, (2)〈Y, Y 〉2 = 〈N,N〉2 = 0, 〈Y,N〉2 = 1. (3)Let {Ei := e−τei} be a local orthonormal basis for the conformal metric gwith dual basis {ωi = eτθi}. Writing {Yi = Ei(Y )}, we have〈Yi, Yj〉2 = δij , 〈Yi, Y 〉2 = 〈Yi, N〉2 = 0, 1 ≤ i, j ≤ n. (4)If we denote by V is the orthogonal complement space of the subspacespan{Y,N, Y1, · · · , Yn} in Rn+32 , then we haveRn+32 = span{Y,N}⊕span{Y1, · · · , Yn} ⊕ V,here V is called the conformal normal bundle of x : Mn→Sn+1. We definethe local orthonormal basis of V byE = En+1 := (H,Hx+ en+1), (5)then {Y,N, Y1, · · · , Yn, E} is a moving frame of Rn+32 along Mn, we can writethe structure equations asdY =∑iYiωi, (6)dN =∑iψiYi + CE, (7)dYi = −ψiY − ωiN +∑jωijYj + ωi,n+1E, (8)dE = CY +∑iωi,n+1Yi. (9)The tensors A =∑i,jAijωiωj , B =∑i,jBijωiωj , C =∑iCiωi are called theBlaschke tensor, the conformal second fundamental form and the conformal66 Anghelform respectively. All of them are conformal invariants. The relations betweenconformal invariants and Euclidean invariants of x are given byBij = e−τ (hij −Hδij), (10)Ci = e−2τ (Hτi −∑jhijτj −Hi), (11)Aij = e−2τ [τiτj − τi,j −Hhij + 12(H2 −∑k(τk)2 + ǫ)δij ]. (12)Here τi = ei(τ), Hi = ei(H). τi,j and ∇ are called the Hessian-matrix and thegradient with respect to I = dx · dx respectively.We define the covariant derivatives of Ci, Aij , Bij as follows∑jCi,jωj = dCi −∑jCjωji, (13)∑kAij,kωk = dAij −∑kAikωkj −∑kAkjωki, (14)∑kBij,kωk = dBij −∑kBikωkj −∑kBkjωki. (15)Then the structure equations (6)–(9) are equivalent toAij,k −Aik,j = BikCi −BijCk, (16)Ci,j − Cj,i =∑k(BikAkj −BkjAki), (17)Bij,k −Bik,j = δijCk − δikCj , (18)Rijkl = δikAjl − δilAjk + δjlAik − δjkAil − (BikBjl −BilBjk), (19)Rij :=∑kRikjk =∑kBikBjk + (trA)δij + (n− 2)Aij , (20)∑iBii = 0,∑i,j(Bij)2 =n− 1n, trA =∑iAii =12n(n2κ− 1). (21)Here Rijkl is the curvature tensor of g, Q =∑i,jRijωi ⊗ ωj is the conformalRicci curvature and κ =1n(n− 1)∑iRii is the normalized conformal scalarcurvature of x :Mn→Sn+1.Theorem 4. [6]Two space-like hypersurfaces x :Mn→Qn+11 and x˜ : M˜n →Qn+11 (n ≥ 3) are conformally equivalent if and only if there exists a diffeomor-phism σ : Mn → M˜nwhich preserves the conformal metric g and conformalsecond fundamental form B.Ovals with Invariant Maximal Quadrilaterals 672 The proof of main resultsProof of Theorem 2. Let x : Mn→Qn+11 be a space-like hypersurface inQn+11 with conformal form C = 0. From (17), we choose a local orthonormalframe {ei} with respect to g such that A,B are diagonalizable at the same time,i.e.,Bij = biδij , Aij = aiδij , 1 ≤ i, j ≤ n. (22)We are assuming that x has harmonic conformal curvature, i.e.,∑iRijkl,i =0. This happens if and only if the Ricci tensor is Codazzi tensor, i.e., Rij,k =Rik,j . Thus the scalar curvature of x with respect to g is constant and tr(A) isconstant too. From (20), we haveRij,k =∑lBil,kBlj +∑lBilBlj,k + (n− 2)Aij,k, (23)Rik,j =∑lBil,jBlk +∑lBilBlk,j + (n− 2)Aik,j , (24)Since C = 0, form(17), (18), we have Bij,k = Bik,j , Aij,k = Aik,j . Thus from(22) and (23), we get ∑lBil,kBlj =∑lBil,jBlk.By using (22), for any indices of i, j, k, we getBij,kbj = Bik,jbk, (25)If bj 6= bk, then we haveBij,k = 0. (26)If bj = bk, since∑lBjk,lωl = dBjk +∑lBjlωlk +∑lBljωli= dBjk + (bj − bk)ωjk(27)It is easy to see from (22) thatBij,k = 0. (28)From (26), (28) and∑jBij,j = 0, for any indices of i, j, k, we getBij,k = 0. (29)Therefore we obtain our main Theorem 2.68 AnghelNow let t be the number of the distinct eigenvalues of A, and a1, a2, · · · , atbe all of distinct eigenvalues. Taking a suitably local orthonormal frame field{E1, E2, · · · , En} such that the matrix (Aij) can be written as(Aij) = Diag(a1, · · · , a1︸ ︷︷ ︸k1, a2, · · · , a2︸ ︷︷ ︸k2, · · · , at, · · · , at︸ ︷︷ ︸kt),that isA1 = · · · = Ak1 = a1, · · · , An−kt+1 = · · · = An = at,here a1, · · · , at are not necessarily different from each other.Similarly, under the same orthonormal frame field, the matrix (Bij) can bewritten as(Bij) = Diag(b1, · · · , b1︸ ︷︷ ︸k1, b2, · · · , b2︸ ︷︷ ︸k2, · · · , bt, · · · , bt︸ ︷︷ ︸kt),or equivalentlyB1 = · · · = Bk1 = b1, · · · , Bn−kt+1 = · · · = Bn = bt,and b1, · · · , bt are not necessarily different from each other.Proposition 1. If the number of the distinct eigenvalues is t ≥ 3, thent = 3.Proof. If t > 3, then there exist at least four indices i1, i2, i3, i4, such thatAi1 , Ai2 , Ai3 , Ai4 are distinct from each other.Making the convention on the ranges of indices as follows1 ≤ i1, j1 ≤ k1, k1 + 1 ≤ i2, j2 ≤ k1 + k2, · · · , k1 + k2 + 1 ≤ it, jt ≤ n.From (15) and Bij,k = 0, we have ωimin = 0(m 6= n, 1 ≤ m,n ≤ t). UsingGauss equation and from (19), we obtainBi1Bi2 +Ai1 +Ai2 = 0, Bi3Bi4 +Ai3 +Ai4 = 0,Bi1Bi3 +Ai1 +Ai3 = 0, Bi2Bi4 +Ai2 +Ai4 = 0.Consequently, (Ai1−Ai4)(Ai2−Ai3) = 0, it contradicts with the assumptionthat Ai1 , Ai2 , Ai3 , Ai4 are distinct from each other.Proposition 2. The Einstein space-like hypersurfaces with vanishing con-formal form in conformal space have at most two different conformal principalcurvatures.Ovals with Invariant Maximal Quadrilaterals 69Proof. If that doesn’t happen, t = 3. Taking a local orthonormal frame field{E1, E2, · · · , En} such that(Bij) = Diag(b1, · · · , b1︸ ︷︷ ︸k1, b2, · · · , b2︸ ︷︷ ︸k2, b3, · · · , b3︸ ︷︷ ︸k3)with the multiplicity are k1, k2, k3 respectively, and k1 + k2 + k3 = n.(Aij) = Diag(a1, · · · , a1︸ ︷︷ ︸k1, a2, · · · , a2︸ ︷︷ ︸k2, a3, · · · , a3︸ ︷︷ ︸k3).Since B is parallel, By Using Gauss equation and from (15), (19), we obtainb1b2 + a1 + a2 = 0,b1b3 + a1 + a3 = 0,b2b3 + a2 + a3 = 0.(30)Obviously, we haveb3(b1 − b2) = −(a1 − a2). (31)Since M is Einstein manifold, i.e., the Ricci curvature Rij =rnδij = (n −1)κδij(n ≥ 3), so its conformal scalar curvature κ is constant. From (20) wehave (n− 1)κ = tr(A) + (n− 2)a1 + b21,(n− 1)κ = tr(A) + (n− 2)a2 + b22,(n− 1)κ = tr(A) + (n− 2)a3 + b23.(32)Subtracting the second formula from the the first formula, we get(b1 + b2)(b1 − b2) = −(n− 2)(a1 − a2). (33)Substituting (31) into (33), we obtainb1 + b2 = −(n− 2)b3.Similarly, we haveb1 + b3 = −(n− 2)b2.Making subtraction in above two formulas and obtain b2 = b3, it contradictswith the assumption, so we complete the proof of Proposition 2.Next, we give the proof of Corollary 1.Proof of Corollary 1. Suppose that the number of different principle cur-vatures of the Einstein space-like hypersurfaces is t = 2, by using the first twoformulas of (21) to calculate b1, b2, we obtainb1 =1n (n− k)(n− 1)k, b2 = − 1n (n− 1)kn− k .70 Anghela1 + a2 = b1 · b2 = −n− 1n2< 0.Since(M, g) = (M1, g1)×(M2, g2), dimM1 = k, dimM2 = n−k. From (19),(M1, g1) and (M2, g2) have constant curvature R1 and R2. By direct calculation,we getR1 = 2a1 − b21, R2 = 2a2 − b22.R1 +R2 = −b21 − b22 + 2(a1 + a2) = −(b1 − b2)2 < 0.Then at least one of R1, R2 is negative, without generality, we let R1 < 0, i.e.,2a1 − b21 < 0.Since M is Einstein manifold, from (20), we have(n− 1)κδij = Rij = tr(A)δij + (n− 2)Aij +∑kBikBjk.Furthermore, we have®(n− 1)κ = tr(A) + (n− 2)a1 + b21,(n− 1)κ = tr(A) + (n− 2)a2 + b22.(34)Adding the above two formulas, we get2(n− 1)κ = 2tr(A) + (n− 2)(a1 + a2) + b21 + b22.From (21), we obtainκ =(1− k)(n− k − 1)k(n− k)(n− 2) .a2 =1n− 2[(n− 1)κ− tr(A)− b22]=1n− 2[(1− k)(n− k − 1)2k(n− k) +12n− (n− 1)kn2(n− k) ].By direct calculation, we haveR2 = 2a2 − b22=2n− 2[(1− k)(n− k − 1)2k(n− k) +12n− (n− 1)kn2(n− k) ]−(n− 1)kn2(n− k)=(1− k)(n− 1)k(n− 2)(n− k) ≤ 0.Here we get ” = ” if and only if k = 1. So we complete the proof of theCorollary 1.Ovals with Invariant Maximal Quadrilaterals 71References[1] Derdzinski,A.: On compact Riemannian manifolds with harmonic curvature, Math.Ann., 1982,259:145-152.[2] Hu Z.J.,Li H.: Classiﬁcation of hypersurfaces with parallel Mo¨bius second fundamentalform in Sn+1, Science in China Ser.A, Mathematics, 2004,47:417-430.[3] Li H.,Wang C.: Surfaces with vanishing Mo¨bius form in Sn,Acta Math. Sinica, Englishseries, 2003, 19:617-678.[4] Li T.Z., Sun H.: Hypersurfaces with harmonic Mo¨bius curvature in Sn+1, Advancesin Mathematics(China), 2008, 37:57-66.[5] Nie C., Wu C : Space-like hypersurfaces with parallel second fundamental forms in theconformal space, Acta Math. Sinica, 2008, 51:685-692.[6] Nie C., Li T. : Conformal isoparametric hypersurfaces with two distinct conformal prin-cipal curvatures in conformal space. Sci China Math, 2010,53(4):953-963.

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Space-like hypersurfaces with vanishing conformal forms in the conformal space

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