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

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

Yang Huizhang ⁱ

Department of Mathematics, University of Honghe, Mengzi, Yunnan, 661100 P.R.China

yanghuizhangyn@163.com

Sun Xia

Department of Arts and Science, University of Yunnan Radio & TV , Kunming, Yunnan, 650223 P.R.China

sunx@sina.com

Received: 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; 53C40

Introduction

We define the pseudo-Euclidean inner product h·, ·i s in R ^n+p as hX, Y i s = −

X s i=1

x _i y _i +

n+p X

i=s+1

x _i y _i , X = (x _i ), Y = (y _i ) ∈ R ^n+p ,

Let RP ⁿ⁺² be a real projective space. < ·, · > 2 is the pseudo-Euclidean inner product in R ⁿ⁺³ . The quadratic surface

Q ⁿ⁺¹ ₁ = {[ξ] ∈ RP ⁿ⁺² |hξ, ξi 2 = 0 } in RP ⁿ⁺² is called the conformal space.

Suppose that x : M ⁿ →Q ⁿ⁺¹ 1 is a space-like hypersurface in the conformal space Q ⁿ⁺¹ ₁ , {e i } is a local orthonormal frame of M ⁿ for the standard metric I = dx · dx with dual basis {θ i }. Then we define the first fundamental form I, the second fundamental form II and the mean curvature of x as

I = hdx, dxi 1 = ^X

i

θ _i ⊗ θ i ; II = ^X

ij

h _ij θ _i ⊗ θ j ; H = 1 n

X

i

h _ii .

i

This work is partially supported by the Science Foundation of Honghe Univer- sity(10XJY121) of China

http://siba-ese.unisalento.it/ c 2012 Universit`a del Salento

In [5], Nie and Wu classified the hypersurfaces with parallel conformal second fundamental form.They obtained

Theorem 1. [5] Let x : M ⁿ →Q ⁿ⁺¹ 1 be a space-like hypersurface with parallel conformal second fundamental form, then M ⁿ is conformally equivalent to an open part of one of the following hypersurfaces in Q ⁿ⁺¹ ₁ :

1) S ^k (a) ×H ^n−k (b) ⊂ S 1 ⁿ⁺¹

2) H ^k (a) ×H ^n−k (b) ⊂ H 1 ⁿ⁺¹ ; 3) H ^k (a) ×R ^n−k ⊂ R ⁿ⁺¹ 1 ;

4) W P (p, q, a) ⊂ R ⁿ⁺¹ ₁ , W P (p, q, a) is the warped product embedding u : S ^p (a) × H ^q (b) × R ⁺ × R ^{n−p−q−1} ⊂ R ⁿ⁺² 1 → R ⁿ⁺¹ 1 , a > 1, b = √

a ² − 1, which is given by

u = (tu ₁ , tu ₂ , tu ₃ ), u ₁ ∈ S ^p (a), u ₂ ∈ H ^q (b), u ₃ ∈ R ^{n−p−q−1} , t ∈ R ⁺ . In this paper, we consider the space-like hypersurfaces M ⁿ with conformal form C = 0, which also have harmonic curvature. Here is our main theorem

Theorem 2. Let x : M ⁿ →Q ⁿ⁺¹ 1 be a space-like hypersurface in Q ⁿ⁺¹ ₁ 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 following corollary

Corollary 1. Let x : M ⁿ →Q ⁿ⁺¹ 1 be a space-like hypersurface in Q ⁿ⁺¹ ₁ with- out umbilics. If its conformal form C = 0 and M ⁿ is Einstein hypersurface with respect to conformal metric g, then M ⁿ is conformally equivalent to an open part of one of the following hypersurfaces in Q ⁿ⁺¹ ₁ :

1) H ^k (a) ×H ^n−k (b) ⊂ H 1 ⁿ⁺¹ ; 2) H ¹ (a) ×R ⁿ⁻¹ ⊂ R ⁿ⁺¹ 1 .

1 Conformal invariants for space-like hypersurfaces in Q ⁿ⁺¹ ₁

Let x : M ⁿ →Q ⁿ⁺¹ 1 be a space-like hypersurface in the conformal space Q ⁿ⁺¹ ₁ . The cone of light in R ⁿ⁺³ is given by

C ⁿ⁺² = {ξ ∈ R ⁿ⁺³ |hξ, ξi 2 = 0, ξ 6= 0}.

Then there exists a unique lift Y : M ⁿ → C ⁿ⁺² of x such that g = hdY, dY i

up to a sign, Y is called the canonical lift of x. Then we have

Theorem 3. [6]Two space-like hypersurfaces x, ˜ x : M ⁿ →Q ⁿ⁺¹ 1 are confor- mally equivalent if and only if there exists a pseudo-orthogonal transformation T ∈O(n, 2) in R ⁿ⁺³ ₂ such that Y = ˜ Y T .

It follows immediately from Theorem 3 that g = hdY, dY i = e ^2τ dx ·dx, e ^2τ = n

n − 1 ( ^X

ij

(h _ij ) ² − nH ² ) is a conformal invariant, which is called the conformal metric of x : M ⁿ →Q ⁿ⁺¹ 1 .

Let ∆ be the Laplacian operator with respect to g. We define N = − 1

n ∆Y − 1

2n ² h∆Y, ∆Y iY. (1)

It is easy to see that

h∆Y, Y i 2 = −n, hY, dY i 2 = 0, (2) hY, Y i 2 = hN, Ni 2 = 0, hY, Ni 2 = 1. (3) Let {E i := e ^−τ e _i } be a local orthonormal basis for the conformal metric g with dual basis {ω i = e ^τ θ _i }. Writing {Y i = E _i (Y ) }, we have

hY i , Y _j i 2 = δ _ij , hY i , Y i 2 = hY i , N i 2 = 0, 1 ≤ i, j ≤ n. (4) If we denote by V is the orthogonal complement space of the subspace span {Y, N, Y 1 , · · · , Y n } in R ⁿ⁺³ 2 , then we have

R ⁿ⁺³ ₂ = span {Y, N}⊕span{Y 1 , · · · , Y n } ⊕ V,

here V is called the conformal normal bundle of x : M ⁿ →S ⁿ⁺¹ . We define the local orthonormal basis of V by

E = E _n+1 := (H, Hx + e _n+1 ), (5) then {Y, N, Y 1 , · · · , Y n , E } is a moving frame of R ⁿ⁺³ 2 along M ⁿ , we can write the structure equations as

dY = ^X

i

Y _i ω _i , (6)

dN = ^X

i

ψ _i Y _i + CE, (7)

dY _i = −ψ i Y − ω i N + ^X

j

ω _ij Y _j + ω _i,n+1 E, (8)

dE = CY + ^X

i

ω _i,n+1 Y _i . (9)

The tensors A = ^P

i,j

A _ij ω _i ω _j , B = ^P

i,j

B _ij ω _i ω _j , C = ^P

i

C _i ω _i are called the

Blaschke tensor, the conformal second fundamental form and the conformal

form respectively. All of them are conformal invariants. The relations between conformal invariants and Euclidean invariants of x are given by

B _ij = e ^−τ (h _ij − Hδ ij ), (10) C _i = e ^−2τ (Hτ _i − ^X

j

h _ij τ _j − H i ), (11)

A _ij = e ^−2τ [τ _i τ _j − τ i,j − Hh ij + 1

2 (H ² − ^X

k

(τ _k ) ² + ǫ)δ _ij ]. (12) Here τ _i = e _i (τ ), H _i = e _i (H). τ _i,j and ∇ are called the Hessian-matrix and the gradient with respect to I = dx · dx respectively.

We define the covariant derivatives of C _i , A _ij , B _ij as follows X

j

C _i,j ω _j = dC _i − ^X

j

C _j ω _ji , (13)

X

k

A _ij,k ω _k = dA ij − ^X

k

A _ik ω _kj − ^X

k

A _kj ω _ki , (14) X

k

B _ij,k ω _k = dB _ij − ^X

k

B _ik ω _kj − ^X

k

B _kj ω _ki . (15) Then the structure equations (6)–(9) are equivalent to

A _ij,k − A ik,j = B _ik C _i − B ij C _k , (16) C _i,j − C j,i = ^X

k

(B _ik A _kj − B kj A _ki ), (17) B _ij,k − B ik,j = δ _ij C _k − δ ik C _j , (18) R _ijkl = δ _ik A _jl − δ il A _jk + δ _jl A _ik − δ jk A _il − (B ik B _jl − B il B _jk ), (19)

R ij := ^X

k

R _ikjk = ^X

k

B _ik B _jk + (trA)δ ij + (n − 2)A ij , (20) X

i

B _ii = 0, ^X

i,j

(B _ij ) ² = n − 1

n , trA = ^X

i

A _ii = 1

2n (n ² κ − 1). (21) Here R _ijkl is the curvature tensor of g, Q = ^P

i,j

R _ij ω _i ⊗ ω j is the conformal Ricci curvature and κ = 1

n(n − 1) X

i

R _ii is the normalized conformal scalar curvature of x : M ⁿ →S ⁿ⁺¹ .

Theorem 4. [6]Two space-like hypersurfaces x : M ⁿ →Q ⁿ⁺¹ 1 and ˜ x : ˜ M ⁿ →

Q ⁿ⁺¹ ₁ (n ≥ 3) are conformally equivalent if and only if there exists a diffeomor-

phism σ : M ⁿ → ˜ M ⁿ which preserves the conformal metric g and conformal

second fundamental form B.

2 The proof of main results

Proof of Theorem 2. Let x : M ⁿ →Q ⁿ⁺¹ 1 be a space-like hypersurface in Q ⁿ⁺¹ ₁ with conformal form C = 0. From (17), we choose a local orthonormal frame {e i } with respect to g such that A, B are diagonalizable at the same time, i.e.,

B _ij = b _i δ _ij , A _ij = a _i δ _ij , 1 ≤ i, j ≤ n. (22) We are assuming that x has harmonic conformal curvature, i.e., ^P

i R _ijkl,i = 0. This happens if and only if the Ricci tensor is Codazzi tensor, i.e., R _ij,k = R _ik,j . Thus the scalar curvature of x with respect to g is constant and tr(A) is constant too. From (20), we have

R _ij,k = ^X

l

B _il,k B _lj + ^X

l

B _il B _lj,k + (n − 2)A ij,k , (23)

R _ik,j = ^X

l

B _il,j B _lk + ^X

l

B _il B _lk,j + (n − 2)A ik,j , (24) Since C = 0, form(17), (18), we have B _ij,k = B _ik,j , A _ij,k = A _ik,j . Thus from (22) and (23), we get _X

l

B _il,k B _lj = ^X

l

B _il,j B _lk . By using (22), for any indices of i, j, k, we get

B _ij,k b _j = B _ik,j b _k , (25) If b _j 6= b k , then we have

B _ij,k = 0. (26)

If b _j = b _k , since P

l

B _jk,l ω _l = dB _jk + ^P

l

B _jl ω _lk + ^P

l

B _lj ω _li

= dB _jk + (b _j − b k )ω _jk (27) It is easy to see from (22) that

B _ij,k = 0. (28)

From (26), (28) and ^P

j B _ij,j = 0, for any indices of i, j, k, we get

B _ij,k = 0. (29)

Therefore we obtain our main Theorem 2.

Now let t be the number of the distinct eigenvalues of A, and a ₁ , a ₂ , · · · , a t

be all of distinct eigenvalues. Taking a suitably local orthonormal frame field {E 1 , E ₂ , · · · , E n } such that the matrix (A ij ) can be written as

(A _ij ) = Diag(a ₁ , · · · , a 1

| {z }

k

, a ₂ , · · · , a 2

| {z }

k

, · · · , a t , · · · , a t

| {z }

k

),

that is

A ₁ = · · · = A k

= a ₁ , · · · , A n−k

+1 = · · · = A n = a _t , here a 1 , · · · , a t are not necessarily different from each other.

Similarly, under the same orthonormal frame field, the matrix (B _ij ) can be written as

(B _ij ) = Diag(b ₁ , · · · , b 1

| {z }

k

, b ₂ , · · · , b 2

| {z }

k

, · · · , b t , · · · , b t

| {z }

k

),

or equivalently

B ₁ = · · · = B k

= b ₁ , · · · , B n−k

+1 = · · · = B n = b _t , and b ₁ , · · · , b t are not necessarily different from each other.

Proposition 1. If the number of the distinct eigenvalues is t ≥ 3, then t = 3.

Proof. If t > 3, then there exist at least four indices i ₁ , i ₂ , i ₃ , i ₄ , such that A _i

, A _i

, A _i

, A _i

are distinct from each other.

Making the convention on the ranges of indices as follows

1 ≤ i 1 , j ₁ ≤ k 1 , k ₁ + 1 ≤ i 2 , j ₂ ≤ k 1 + k ₂ , · · · , k 1 + k ₂ + 1 ≤ i t , j _t ≤ n.

From (15) and B _ij,k = 0, we have ω _i

_i

= 0(m 6= n, 1 ≤ m, n ≤ t). Using Gauss equation and from (19), we obtain

B _i

B _i

+ A _i

+ A _i

= 0, B _i

B _i

+ A _i

+ A _i

= 0, B _i

B _i

+ A _i

+ A _i

= 0, B _i

B _i

+ A _i

+ A _i

= 0.

Consequently, (A _i

−A i

)(A _i

−A i

) = 0, it contradicts with the assumption that A _i

, A _i

, A _i

, A _i

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 principal

curvatures.

Proof. If that doesn’t happen, t = 3. Taking a local orthonormal frame field {E 1 , E 2 , · · · , E n } such that

(B ij ) = Diag(b 1 , · · · , b 1

| {z }

k

, b 2 , · · · , b 2

| {z }

k

, b 3 , · · · , b 3

| {z }

k

)

with the multiplicity are k ₁ , k ₂ , k ₃ respectively, and k ₁ + k ₂ + k ₃ = n.

(A _ij ) = Diag(a ₁ , · · · , a 1

| {z }

k

, a ₂ , · · · , a 2

| {z }

k

, a ₃ , · · · , a 3

| {z }

k

).

Since B is parallel, By Using Gauss equation and from (15), (19), we obtain

 



 

b ₁ b ₂ + a ₁ + a ₂ = 0, b 1 b 3 + a 1 + a 3 = 0, b ₂ b ₃ + a ₂ + a ₃ = 0.

(30)

Obviously, we have

b ₃ (b ₁ − b 2 ) = −(a 1 − a 2 ). (31) Since M is Einstein manifold, i.e., the Ricci curvature R _ij = r

n δ _ij = (n − 1)κδ _ij (n ≥ 3), so its conformal scalar curvature κ is constant. From (20) we

have ^ _



 

(n − 1)κ = tr(A) + (n − 2)a 1 + b ² ₁ , (n − 1)κ = tr(A) + (n − 2)a 2 + b ² ₂ , (n − 1)κ = tr(A) + (n − 2)a 3 + b ² ₃ .

(32) Subtracting the second formula from the the first formula, we get

(b ₁ + b ₂ )(b ₁ − b 2 ) = −(n − 2)(a 1 − a 2 ). (33) Substituting (31) into (33), we obtain

b ₁ + b ₂ = −(n − 2)b 3 . Similarly, we have

b 1 + b 3 = −(n − 2)b 2 .

Making subtraction in above two formulas and obtain b 2 = b 3 , it contradicts with 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 two formulas of (21) to calculate b ₁ , b ₂ , we obtain

b ₁ = 1 n

  (n − k)(n − 1)

k , b ₂ = − 1 n

  (n − 1)k

n − k .

a ₁ + a ₂ = b ₁ · b 2 = − n − 1 n ² < 0.

Since(M, g) = (M ₁ , g ₁ ) ×(M 2 , g ₂ ), dimM ₁ = k, dimM ₂ = n −k. From (19), (M ₁ , g ₁ ) and (M ₂ , g ₂ ) have constant curvature R ₁ and R ₂ . By direct calculation, we get

R ₁ = 2a ₁ − b ² 1 , R ₂ = 2a ₂ − b ² 2 .

R ₁ + R ₂ = −b ² 1 − b ² 2 + 2(a ₁ + a ₂ ) = −(b 1 − b 2 ) ² < 0.

Then at least one of R ₁ , R ₂ is negative, without generality, we let R ₁ < 0, i.e., 2a ₁ − b ² 1 < 0.

Since M is Einstein manifold, from (20), we have (n − 1)κδ ij = R _ij = tr(A)δ _ij + (n − 2)A ij + ^X

k

B _ik B _jk .

Furthermore, we have

® (n − 1)κ = tr(A) + (n − 2)a 1 + b ² ₁ ,

(n − 1)κ = tr(A) + (n − 2)a 2 + b ² ₂ . (34) Adding the above two formulas, we get

2(n − 1)κ = 2tr(A) + (n − 2)(a 1 + a ₂ ) + b ² ₁ + b ² ₂ . From (21), we obtain

κ = (1 − k)(n − k − 1) k(n − k)(n − 2) .

a ₂ = 1

n − 2 [(n − 1)κ − tr(A) − b ² 2 ]

= 1

n − 2 [ (1 − k)(n − k − 1) 2k(n − k) + 1

2n − (n − 1)k n ² (n − k) ].

By direct calculation, we have R ₂ = 2a ₂ − b ² 2

= 2

n − 2 [ (1 − k)(n − k − 1) 2k(n − k) + 1

2n − (n − 1)k

n ² (n − k) ] − (n − 1)k n ² (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 the

Corollary 1.

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