Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$
We study the conformal geometry of conformal spacelike hypersurfaces in the conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$. We obtain a complete classification of conformal isoparametric spacelike hypersurfaces in $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$.
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Institute of Mathematics, NAS of Ukraine
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| author | Bianping, Su Shu, Shichang Біанпинг, Су Шу, Шичанґ |
| author_facet | Bianping, Su Shu, Shichang Біанпинг, Су Шу, Шичанґ |
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| description | We study the conformal geometry of conformal spacelike hypersurfaces in the conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$.
We obtain a complete classification of conformal isoparametric spacelike hypersurfaces in $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$. |
| first_indexed | 2026-03-24T02:26:29Z |
| format | Article |
| fulltext |
UDC 517.91
Shichang Shu (School Math. and Inform. Sci., Xianyang Normal Univ., China),
Bianping Su (Xi’an Univ. Architecture and Technology, China)
CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES
IN CONFORMAL SPACES Q4
1 and Q5
1
*
КОНФОРМНI IЗОПАРАМЕТРИЧНI ПРОСТОРОПОДIБНI ГIПЕРПОВЕРХНI
У КОНФОРМНИХ ПРОСТОРАХ Q4
1 I Q5
1
We study the conformal geometry of conformal spacelike hypersurfaces in the conformal spaces Q4
1 and Q5
1. We obtain a
complete classification of conformal isoparametric spacelike hypersurfaces in Q4
1 and Q5
1.
Вивчено конформну геометрiю конформних простороподiбних гiперповерхонь у конформних просторах Q4
1 i Q5
1.
Отримано повну класифiкацiю конформних iзопараметричних простороподiбних гiперповерхонь у Q4
1 та Q5
1.
1. Introduction. Let 〈 , 〉s be the Lorentzian inner product with s negative index of the (n + s)-di-
mensional Euclidean space Rn+s. Denoted by
〈X,Y 〉s =
n∑
i=1
xiyi −
n+s∑
i=n+1
xiyi, X = (xi), Y = (yi) ∈ Rn+s.
Let RPn+2 be (n+ 2)-dimensional real projective space. The quadric surface
Qn+1
1 = {[ξ] ∈ RPn+2|〈ξ, ξ〉2 = 0},
is called conformal space. We define the Lorentzian space Rn+1
1 , de Sitter sphere Sn+1
1 and anti-de
Sitter sphere Hn+1
1 by
Rn+1
1 = (Rn+1, 〈 , 〉1), Sn+1
1 = {u ∈ Rn+2|〈u, u〉1 = 1},
Hn+1
1 = {u ∈ Rn+2|〈u, u〉2 = −1}.
We call Lorentzian space Rn+1
1 , de Sitter sphere Sn+1
1 and anti-de Sitter sphere Hn+1
1 Lorentzian
space forms.
Denote π = {[x] ∈ Qn+1
1 |x1 = xn+3}, π+ = {[x] ∈ Qn+1
1 |xn+3 = 0} and π− = {[x] ∈
∈ Qn+1
1 |x1 = 0}. Observe the conformal diffeomorphisms
σ0 : Rn1 → Qn+1
1 \π, u 7→
[(
〈u, u〉 − 1
2
, u,
〈u, u〉+ 1
2
)]
,
σ1 : Sn+1
1 → Qn+1
1 \π+, u 7→ [(u, 1)],
σ−1 : Hn+1
1 → Qn+1
1 \π−, u 7→ [(1, u)].
From [13], we may regard Qn+1
1 as the common compactified space of Rn+1
1 , Sn+1
1 and Hn+1
1 ,
while Rn+1
1 , Sn+1
1 and Hn+1
1 are regarded as the subsets of Qn+1
1 .
*Project supported by NSF of Shaanxi Province (SJ08A31) and NSF of Shaanxi Educational Committee (11JK0479,
2010JK642) and Talent Fund of Xi’an University of Architecture and Technology.
c© SHICHANG SHU, BIANPING SU, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4 557
558 SHICHANG SHU, BIANPING SU
Suppose that x : M → Qn+1
1 is a nondegenerated hypersurface, that is, x∗(TM) is nondegen-
erated subbundle of TQn+1
1 . Let y : U → Rn+3
2 be a lift of x : M → Qn+1
1 defined in an open
subset U of M. We denote by ∆ and κ Laplacian and the normalized scalar curvature of the local
nondegerated metric 〈dy, dy〉. Then know that on M the 2-form g = ε(〈∆y,∆y〉 − n2κ)〈dy, dy〉 is
a globally defined invariant of x : M → Qn+1
1 under the conformal group transformations of Qn+1
1 .
When the 2-form g = ε(〈∆y,∆y〉 − n2κ)〈dy, dy〉 is nondegenerated, we call x : M → Qn+1
1 a
conformal regular hypersurface and g = ε(〈∆y,∆y〉 − n2κ)〈dy, dy〉 the conformal metric of x,
where ε = −1 (spacelike) or ε = 1 (timelike). From [13], we know that there exists a unique lift
Y : U → Rn+3
2 such that g = 〈dY, dY 〉 up to a signature and we call Y the canonical lift of x. It is
obvious that g ≡ 0 if and only if x : M → Qn+1
1 is a umbilical hypersurface.
Let x : M → Qn+1
1 be an n-dimensional immersed conformal regular spacelike hypersurface
in conformal space Qn+1
1 . We choose a local orthonormal basis {ei} for the induced metric I =
= 〈dx, dx〉 with dual basis {θi}. Let II =
∑
i,j
hijθi ⊗ θj be the second fundamental form and
H =
1
n
∑
i
hii the mean curvature of the immersion x. From [7], we know that the conformal metric
of the immersion x can be defined by g =
n
n− 1
{∑
i,j
h2ij − nH2
}
〈dx, dx〉 := e2τ 〈dx, dx〉, which
is a conformal invariant. Denote
Φ =
n∑
i=1
eτCiθi, A =
n∑
i,j=1
e2τAijθi ⊗ θj , B =
n∑
i,j=1
e2τBijθi ⊗ θj , (1.1)
where Ci, Aij and Bij are defined by formulas (2.1) – (2.3) in Section 2. We call Φ, A and B
conformal form, conformal Blaschke tensor and conformal second fundamental form of the immersion
x, respectively. It is easy to prove that Φ, A and B are conformal invariants.
The conformal geometry of regular hypersurfaces in the conformal space is determined by confor-
mal metric. The negative index of conformal space Qn+1
1 is 1. If the negative index is degenerate, we
obtain the Möbius geometry in the unit sphere which had been studied by many authors (see [1 – 7,
9, 10, 16 – 18]). We call the eigenvalues of B the conformal principal curvatures of the immersion x,
while the eigenvalues of A are called the conformal Blaschke eigenvalues of x. A regular spacelike
hypersurface x : M → Qn+1
1 is called a conformal isoparametric spacelike hypersurface, if Φ ≡ 0
and the conformal principal curvatures of the immersion x are constant.
Let Sk(a) and Hk(a) denote k-dimensional sphere and k-dimensional hyperbolic surface with
radius
1
a
, Sk1(a) and Hk
1(a) denote k-dimensional de Sitter sphere and k-dimensional anti-de Sitter
sphere with radius
1
a
, where a is a constant parametric. Recently, C. X. Nie et al. [11 – 14] studied the
conformal geometry of regular spacelike hypersurfaces in the conformal space Qn+1
1 and obtained
the following results:
Theorem 1.1 [12]. If x : M → Qn+1
1 is a conformal regular spacelike hypersurface in Qn+1
1
with parallel conformal second fundamental form, then M is conformal equivalent to an open part
of these standard embeddings:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES . . . 559
(i) the Riemannian product Sm(a)×Hn−m −
(√
a2 − n− 1
m(n−m)
)
in Sn+1
1
(√
n− 1
m(n−m)
)
,
a >
√
n− 1
m(n−m)
;
(ii) the Riemannian product Rm ×Hn−m
(√
n− 1
m(n−m)
)
in Rn+1
1 ;
(iii) the Riemannian product Hm(a) × Hn−m
(√
n− 1
m(n−m)
− a2
)
in Hn+1
1
(√
n− 1
m(n−m)
)
,
0 < a <
√
n− 1
m(n−m)
;
(iv) the spacelike hypersurface x = σ0 ◦ u : Sp(a) × R+ × Rn−p−q−1 × Hq(b) → Qn+1
1 with
b =
√
a2 − 1, p ≥ 1, q ≥ 1, p+q < n, where u : Sp(a)×R+×Rn−p−q−1×Hq(b)→ Rn+2
1 ⊂ Rn+1
1 :
u(u′, t, u′′, u′′′) = (tu′, u′′, tu′′′), u′ ∈ Sp(a), t ∈ R+, u′′ ∈ Rn−p−q−1, u′′′ ∈ Hq(b).
Theorem 1.2 [13]. If x : M → Qn+1
1 is a conformal isoparametric spacelike hypersurface with
two distinct principle curvatures, then M is conformal equivalent to an open part of these standard
embeddings:
(i) the Riemannian product Sm(a) × Hn−m
(√
a2 − n− 1
m(n−m)
)
in Sn+1
1
(√
n− 1
m(n−m)
)
,
a >
√
n− 1
m(n−m)
;
(ii) the Riemannian product Rm ×Hn−m
(√
n− 1
m(n−m)
)
in Rn+1
1 ;
(iii) the Riemannian product Hm(a) × Hn−m
(√
n− 1
m(n−m)
− a2
)
in Hn+1
1
(√
n− 1
m(n−m)
)
,
0 < a <
√
n− 1
m(n−m)
.
We notice that in [4] and [5], the authors classified the Möbius isoparametric hypersurfaces in
the unit spheres S4 and S5. In this paper, we obtain the complete classification of conformal isopara-
metric spacelike hypersurfaces in Q4
1 and Q5
1.
Theorem 1.3. Let x : M → Q4
1 be a conformal isoparametric spacelike hypersurface in Q4
1.
Then M is conformal equivalent to an open part of these standard embeddings:
(i) the Riemannian product Sm(a) × H3−m
(√
a2 − 2
m(3−m)
)
in S41
(√
2
m(3−m)
)
, a >
>
√
2
m(3−m)
, m = 1, 2;
(ii) the Riemannian product Rm ×H3−m
(√
2
m(3−m)
)
in R4
1, m = 1, 2;
(iii) the Riemannian product Hm(a)×H3−m
(√
2
m(3−m)
− a2
)
in H4
1
(√
2
m(3−m)
)
, 0 <
< a <
√
2
m(3−m)
, m = 1, 2;
(iv) the spacelike hypersurface x = σ0 ◦u : S1(a)×R+×H1(b)→ Q4
1 with b =
√
a2 − 1, where
u : S1(a)× R+ ×H1(b)→ R5
1 ⊂ R4
1 :
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
560 SHICHANG SHU, BIANPING SU
u(u′, t, u′′′) = (tu′, tu′′′), u′ ∈ S1(a), t ∈ R+, u′′′ ∈ H1(b).
Theorem 1.4. Let x : M → Q5
1 be a conformal isoparametric spacelike hypersurface in Q5
1.
Then M is conformal equivalent to an open part of these standard embeddings:
(i) the Riemannian product Sm(a) × H4−m
(√
a2 − 3
m(4−m)
)
in S51
(√
3
m(4−m)
)
, a >
>
√
3
m(4−m)
, m = 1, 2, 3;
(ii) the Riemannian product Rm ×H4−m
(√
3
m(4−m)
)
in R5
1, m = 1, 2, 3;
(iii) the Riemannian product Hm(a)×H4−m
(√
3
m(4−m)
− a2
)
in H5
1
(√
3
m(4−m)
)
, 0 <
< a <
√
3
m(4−m)
, m = 1, 2, 3;
(iv) the spacelike hypersurface x = σ0 ◦ u : Sp(a) × R+ × R4−p−q−1 × Hq(b) → Q5
1 with
b =
√
a2 − 1, p ≥ 1, q ≥ 1, p+ q < 4, where u : Sp(a)× R+ × R4−p−q−1 ×Hq(b)→ R6
1 ⊂ R5
1 :
u(u′, t, u′′, u′′′) = (tu′, u′′, tu′′′), u′ ∈ Sp(a), t ∈ R+, u′′ ∈ R4−p−q−1, u′′′ ∈ Hq(b).
2. Fundamental formulas on conformal geometry. In this section, we review the conformal
invariants and fundamental formulas on conformal geometry of spacelike hypersurfaces in Qn+1
1 , for
more details (see [14]).
Let x : M → Qn+1
1 be an n-dimensional conformal regular spacelike hypersurface with Φ ≡ 0
in Qn+1
1 . We have (see [13])
〈∆Y,∆Y 〉 = (n2κ− 1),
where Y is the canonical lift of x defined in Section 1 and n(n−1)κ is the conformal scalar curvature
of x. Let {E1, . . . , En} denote a local orthonormal frame on (M, g) with dual frame {ω1, . . . , ωn}.
Putting Yi = Ei(Y ), then we have
N = − 1
n
∆Y − 1
2n2
〈∆Y,∆Y 〉Y,
〈N,Y 〉 = 1, 〈N,N〉 = 0, 〈Yi, N〉 = 0, 〈Yi, Yj〉 = δij , 1 ≤ i, j ≤ n.
Let V be the orthogonal complement to the subspace Span{Y,N, Y1, . . . , Yn} in Rn+2
1 . Along M,
we have the following orthogonal decomposition:
Rn+2
1 = Span{Y,N} ⊕ Span{Y1, . . . , Yn} ⊕ V,
where V is called conformal normal bundle of the immersion x. Let ξ be a unit basis of V and
〈ξ, ξ〉 = −1. Then {Y,N, Y1, . . . , Yn, ξ, } forms a moving frame in Rn+2
1 along M. We use the
following range of indices throughout this paper:
1 ≤ i, j, k, l,m ≤ n.
The structure equations on M with respect to the conformal metric g can be written as
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES . . . 561
dY =
∑
i
ωiYi,
dN =
∑
i
ψiYi + φξ,
dYi = −ψiY − ωiN +
∑
j
ωijYj + ωin+1ξ,
dξ = φY +
∑
i
ωin+1Yi,
where {ψi, ωij , ωin+1, φ} are 1-forms on M with
ωij + ωji = 0.
By exterior differentiation of these equations, we get∑
i
ωi ∧ ψi = 0,
∑
i
ωin+1 ∧ ωi = 0,
dωi =
∑
j
ωij ∧ ωj ,
dψi =
∑
j
ωij ∧ ψj + ωin+1 ∧ φ,
dφ =
∑
i
ωin+1 ∧ ψi,
dωij =
∑
k
ωik ∧ ωkj + ωin+1 ∧ ωjn+1 − ωi ∧ ψj − ψi ∧ ωj ,
dωin+1 =
∑
j
ωij ∧ ωjn+1 + ωi ∧ φ,
where
ψi =
∑
j
Aijωj , Aij = Aji, ωin+1 =
∑
j
Bijωj , Bij = Bji, φ =
∑
i
Ciωi.
Let the conformal metric g = e2τI. Then the local orthonormal frame {E1, . . . , En} on (M, g) and
the dual frame {ω1, . . . , ωn} satisfy Ei = e−τei and ωi = eτθi. Aij , Bij and Ci are locally defined
functions and satisfy
e2τCi = Hτi −Hi −
∑
j
hijτj , (2.1)
e2τAij = τiτj − τi,j −Hhij −
1
2
(∑
k
τkτk −H2 − ε
)
Iij , (2.2)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
562 SHICHANG SHU, BIANPING SU
eτBij = hij −HIij , (2.3)
where τi,j is Hessian of τ with respect to the first fundamental form I, τ i =
∑
j
Iijτj , (Iij) =
= (Iij)
−1, Hi = ei(H) and ε = 0 for Rn+1
1 , ε = 1 for Sn+1
1 and ε = −1 for Hn+1
1 (see [14])∑
i
Bii = 0,
∑
i,j
B2
ij =
n− 1
n
, trA =
1
2n
(n2κ− 1). (2.4)
Defining the covariant derivative of Ci, Aij , Bij by∑
j
Ci,jωj = dCi +
∑
j
Cjωji, (2.5)
∑
k
Aij,kωk = dAij +
∑
k
Aikωkj +
∑
k
Akjωki, (2.6)
∑
k
Bij,kωk = dBij +
∑
k
Bikωkj +
∑
k
Bkjωki, (2.7)
dωij =
∑
k
ωik ∧ ωkj −
1
2
∑
k,l
Rijklωk ∧ ωl, Rijkl = −Rjikl, (2.8)
we have
Aij,k −Aik,j = BijCk −BikCj , (2.9)
Ci,j − Cj,i =
∑
k
(BikAkj −BkjAki), (2.10)
Bij,k −Bik,j = δijCk − δikCj , (2.11)
Rijkl = −(BikBjl −BilBjk) + δikAjl + δjlAik − δilAjk − δjkAil, (2.12)
where Rijkl denotes the curvature tensor with respect to the conformal metric g on M. Since the
conformal form Φ ≡ 0, we have for all indices i, j, k
Aij,k = Aik,j , Bij,k = Bik,j ,
∑
k
BikAkj =
∑
k
BkjAki. (2.13)
Defining the second covariant derivative of Bij by∑
l
Bij,klωl = dBij,k +
∑
l
Blj,kωli +
∑
l
Bil,kωlj +
∑
l
Bij,lωlk, (2.14)
we have the following Ricci identities:
Bij,kl −Bij,lk =
∑
m
BmjRmikl +
∑
m
BimRmjkl. (2.15)
3. Some examples and propositions. We cite some examples of conformal regular spacelike
hypersurfaces in Qn+1
1 :
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES . . . 563
Example 3.1. Spacelike hypersurface x : Sm(a)×Hn−m(
√
a2 − r2)→ Sn+1
1 (r), r < a. Let
x = (x1, x2) ∈ Sm(a)×Hn−m
(√
a2 − r2
)
⊂ Rm+1
1 × Rn−m+1
1 ,
〈x1, x1〉 = a2, 〈x2, x2〉 = −
(
a2 − r2
)
,
and
en+1 =
(
−
√
a2 − r2
a
x1
r
,− a√
a2 − r2
x2
r
)
be the unit normal vector of x such that 〈en+1, en+1〉 = −1. By a direct calculation, we know
that x has two distinct conformal principal curvatures
c
r
and
1
rc
with multiplicities m and n −m,
respectively, where c =
√
a2 − r2
a
. The conformal second fundamental form of x is parallel.
Example 3.2. Spacelike hypersurface x : Rm ×Hn−m(r)→ Rn+1
1 .
Let x = (x1, x2), x1 ∈ Rm, x2 ∈ Hn−m(r) ⊂ Rn−m+1
1 , 〈x2, x2〉 = −r2 and en+1 =
(
0,
x2
r
)
be the unit normal vector of x such that 〈en+1, en+1〉 = −1. By a direct calculation, we know
that x has two distinct conformal principal curvatures 0 and −1
r
with multiplicities m and n −m,
respectively. The conformal second fundamental form of x is parallel.
Example 3.3. Spacelike hypersurface x : Hm(a) × Hn−m(
√
r2 − a2) → Hn+1
1 (r), 0 < a < r.
Let
x = (x1, x2) ∈ Hm(a)×Hn−m
(√
r2 − a2
)
⊂ Rm+1
1 × Rn−m+1
1 ,
〈x1, x1〉 = −a2, 〈x2, x2〉 = −
(
r2 − a2
)
,
and
en+1 =
(
−
√
r2 − a2
a
x1
r
,
a√
r2 − a2
x2
r
)
be the unit normal vector of x such that 〈en+1, en+1〉 = −1. By a direct calculation, we know that
x has two distinct conformal principal curvatures
c
r
and − 1
rc
with multiplicities m and n − m,
respectively, where c =
√
r2 − a2
a
. The conformal second fundamental form of x is parallel.
Example 3.4 [12]. For any natural number p, q, p + q < n and real number a ∈ (1,+∞) and
b =
√
a2 − 1, consider the immersed hypersurface u : Sp(a)×R+ ×Rn−p−q−1 ×Hq(b)→ Rn+2
1 ⊂
⊂ Rn+1
1 :
u(u′, t, u′′, u′′′) = (tu′, u′′, tu′′′), u′ ∈ Sp(a), t ∈ R+, u′′ ∈ Rn−p−q−1, u′′′ ∈ Hq(b).
Then x = σ0 ◦ u : Sp(a) × R+ × Rn−p−q−1 × Hq(b) → Qn+1
1 is a conformal regular spacelike
hypersurface in Qn+1
1 , it is denoted byWP (p, q, a) = x(Sp(a)×R+×Rn−p−q−1×Hq(b)). From [12],
by a direct calculation, we know that WP (p, q, a) has three distinct conformal principal curvatures
and the conformal second fundamental form is parallel.
From Nomizu [15], Li and Xie [8], we know that the following:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
564 SHICHANG SHU, BIANPING SU
Proposition 3.1 [15, 8]. Let x be Euclidean isoparametric spacelike hypersurfaces in Lorentzian
space forms. Then x can have at most two distinct Euclidean principal curvatures.
Proposition 3.2. Let x : M → Qn+1
1 be an n-dimensional conformal isoparametric spacelike
hypersurfaces in Qn+1
1 with constant normalized conformal scalar curvature κ and κ 6= 1. Then x is
an n-dimensional Euclidean isoparametric spacelike hypersurfaces.
Proof. Let κ and R be the normalized conformal scalar curvature and the normalized Euclidean
scalar curvature. From [14], we know that κ = R. Let Bi and λi be the conformal principal curvatures
and the Euclidean principal curvatures of x. Since (2.3) implies that the matrix (Bij) and (hij) are
commutative, we can choose a local orthonormal basis such that Bij = Biδij and hij = λiδij . From
(2.3), we have
eτBi = λi −H. (3.1)
From (2.1), we have
0 = Hτi −Hi − λiτi = (H − λi)τi −Hi. (3.2)
From the Gaussian equation of x, we have n(n− 1)(R− 1) =
∑
i,j
h2ij − n2H2. Thus
e2τ =
n
n− 1
∑
i,j
h2ij − nH2
= n2(R− 1 +H2). (3.3)
Since κ is constant, we know that R is constant. From (3.3), τi =
HHi
R− 1 +H2
. From (3.2),
0 =
R− 1 + λiH
R− 1 +H2
Hi. (3.4)
If H is not constant, then there is some i such that H,i 6= 0. Thus R− 1 +λiH = 0 for such i. From
(3.1), we have that R − 1 + (eτBi + H)H = 0 for such i. Combining with (3.3), we see that for
such i
R− 1 + (n
√
R− 1 +H2Bi +H)H = 0.
Thus, we see that for such i
(n2B2
i − 1)H4 + (n2B2
i − 2)(R− 1)H2 − (R− 1)2 = 0. (3.5)
Since Bi is constant, if n2B2
i − 1 = 0, from (3.5) and R 6= 1, we infer that H2 = 1 − R is
constant, this is a contradiction. If n2B2
i − 1 6= 0, by (3.5) and R 6= 1, we see that H2 = 1 − R or
H2 =
R− 1
n2B2
i − 1
, also a contradiction. We conclude that H must be constant. From (3.1), we know
that λi are constant for all i .
Proposition 3.2 is proved.
4. Proofs of theorems. Proof of Theorem 1.3. From (2.4), we know that the number γ of distinct
conformal principal curvatures can only take the values γ = 2, 3. From (2.13), we know that we can
choose the local orthonormal basis Ei to diagonalize the matrix (Bij) and (Aij), that is, Bij = Biδij
and Aij = Aiδij .
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CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES . . . 565
Let B1, B2, B3 be the constant conformal principal curvatures of x. From (2.7), we have∑
k
Bij,kωk = (Bi −Bj)ωij . (4.1)
We consider the following cases:
(1) If γ = 2, from Theorem 1.2, we know that Theorem 1.3 is true.
(2) If γ = 3 and the conformal second fundamental form is parallel. From Theorem 1.1, we know
that Theorem 1.3 is true. If γ = 3 and the conformal second fundamental form is not parallel. We
can prove that this case does not occur. In fact, since B1 6= B2 6= B3, from (4.1), we have
Bii,k = 0, for all i, k, (4.2)
and
ωij =
∑
k
Bij,k
Bi −Bj
ωk, for i 6= j. (4.3)
Since the conformal second fundamental form is not parallel, combining with (4.2), we know that
B12,3 6= 0. We may prove that B12,3 is constant. In fact, from (2.14), (4.2) and (4.3), we have∑
k
B12,3kωk = dB12,3, (4.4)
∑
k
Bii,jkωk = 2
∑
l 6=i,j
Bli,jωli = 2
∑
k
∑
l 6=i,j
Bli,jBli,k
Bl −Bi
ωk. (4.5)
Thus,
Bii,jk = 2
∑
l 6=i,j
Bli,jBli,k
Bl −Bi
. (4.6)
From (4.2) and (4.6), we know that
Bii,ji = Bii,jl = 0, for distinct i, j, l. (4.7)
From (2.15), we have
Bij,kl −Bij,lk = (Bi −Bj)Rijkl.
From (2.12), we know that if three of {i, j, k, l} are either the same or distinct, then Rijkl = 0. Thus,
if three of {i, j, k, l} are either the same or distinct, then
Bij,kl = Bij,lk. (4.8)
From (4.7), (4.8) and (2.13), we have B12,31 = B11,23 = 0, B12,32 = B22,13 = 0, B12,33 = B33,12 =
= 0. Thus, (4.4) implies that dB12,3 = 0. Therefore, we know that B12,3 is constant. From (4.3)
and (2.8),
−1
2
∑
k,l
R12klωk ∧ ωl = dω12 − ω13 ∧ ω32 = −
2B2
12,3
(B1 −B3)(B2 −B3)
ω1 ∧ ω2,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
566 SHICHANG SHU, BIANPING SU
−1
2
∑
k,l
R13klωk ∧ ωl = dω13 − ω12 ∧ ω23 = −
2B2
12,3
(B1 −B2)(B3 −B2)
ω1 ∧ ω3,
−1
2
∑
k,l
R23klωk ∧ ωl = dω23 − ω21 ∧ ω13 = −
2B2
12,3
(B2 −B1)(B3 −B1)
ω2 ∧ ω3.
Thus,
R1212 =
2B2
12,3
(B1 −B3)(B2 −B3)
,
R1313 =
2B2
12,3
(B1 −B2)(B3 −B2)
,
R2323 =
2B2
12,3
(B2 −B1)(B3 −B1)
.
We have
κ =
1
6
∑
i 6=j
Rijij = R1212 +R1313 +R2323 = 0.
From (3.1) and Proposition 3.2, we know that x is a 3-dimensional Euclidean isoparametric
spacelike hypersurfaces with three distinct Euclidean principal curvatures. This is in contradiction
with Proposition 3.1.
Theorem 1.3 is proved.
Proof of Theorem 1.4. From (2.4), we know that the number γ of distinct conformal principal
curvatures can only take the values γ = 2, 3, 4. From (2.13), we know that we can choose the local
orthonormal basis Ei to diagonalize the matrix (Bij) and (Aij), that is, Bij = Biδij and Aij = Aiδij .
Let B1, B2, B3, B4 be the constant conformal principal curvatures of x. We consider the follow-
ing cases:
(1) If γ = 2, from Theorem 1.2, we know that Theorem 1.4 is true.
(2) If γ = 3 and the conformal second fundamental form is parallel. From Theorem 1.1, we
know that Theorem 1.4 is true. If γ = 3 and the conformal second fundamental form is not parallel.
We can prove that this case does not occur. In fact, without loss of generality, we may assume that
B1 6= B2 6= B3 = B4. From (4.1), we have
Bii,k = 0, B34,k = 0, for all i, k, (4.9)
and
ωij =
∑
k
Bij,k
Bi −Bj
ωk, for Bi 6= Bj . (4.10)
From (4.9), (4.10) and (2.14), we have∑
l
B13,4lωl = B12,4ω23 +B12,3ω24 =
2B12,3B12,4
B2 −B3
ω1, (4.11)
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CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES . . . 567
∑
l
B11,3lωl = 2B12,3ω21 =
2B2
12,3
B2 −B1
ω3 +
2B12,3B12,4
B2 −B1
ω4. (4.12)
Comparing two side of (4.11) and (4.12), we have
B13,41 =
2B12,3B12,4
B2 −B3
, B13,42 = B13,43 = B13,44 = 0, (4.13)
B11,33 =
2B2
12,3
B2 −B1
, B11,34 =
2B12,3B12,4
B2 −B1
, B11,32 = 0. (4.14)
From (4.8), (2.13), (4.13) and (4.14), we have B12,3B12,4 = 0. Since the conformal second funda-
mental form is not parallel, without loss of generality, we may assume that B12,3 6= 0 and B12,4 = 0.
We may also prove that B12,3 is constant. In fact, from (2.14), (4.9) and (4.10), we have∑
k
B12,3kωk = dB12,3, (4.15)
∑
k
Bii,jkωk = 2
∑
l 6=i,j
Bli,jωli = 2
∑
k
∑
l 6=i,j
Bli,jBli,k
Bl −Bi
ωk, for Bl 6= Bi. (4.16)
Thus,
Bii,jk = 2
∑
l 6=i,j
Bli,jBli,k
Bl −Bi
, for Bl 6= Bi. (4.17)
From (4.9) and (4.17), we know that
Bii,ji = Bii,jl = 0, for distinct i, j, l. (4.18)
From (4.18), (4.8) and (2.13), we have
B12,31 = B11,23 = 0, B12,32 = B22,13 = 0, B12,33 = B33,12 = 0. (4.19)
On the other hand, from (4.9), (4.10) and B12,4 = 0, we have∑
k
B34,1kωk = B12,3ω24 =
∑
k
B12,3B24,k
B2 −B4
ωk.
Thus,
B34,1k =
B12,3B24,k
B2 −B4
,
and we have B34,12 = 0. From (4.8) and (2.13), we have
B12,34 = B34,12 = 0. (4.20)
(4.19) and (4.20) imply that dB12,3 = 0. Therefore, we know that B12,3 is constant.
From (4.9) and (4.10), we have
ω12 =
B12,3
B1 −B2
ω3, ω13 =
B12,3
B1 −B3
ω2, ω23 =
B12,3
B2 −B3
ω1, ω14 = ω24 = 0. (4.21)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
568 SHICHANG SHU, BIANPING SU
From (4.21) and (2.8), by a simple calculation, we have
−1
2
∑
k,l
R12klωk ∧ ωl = dω12 − ω13 ∧ ω32 =
= −
2B2
12,3
(B1 −B3)(B2 −B3)
ω1 ∧ ω2 −
B12,3
B1 −B2
ω4 ∧ ω34, (4.22)
−1
2
∑
k,l
R13klωk ∧ ωl = dω13 − ω12 ∧ ω23 = −
2B2
12,3
(B1 −B2)(B3 −B2)
ω1 ∧ ω3, (4.23)
−1
2
∑
k,l
R14klωk ∧ ωl = −ω13 ∧ ω34 = − B12,3
B1 −B3
ω2 ∧ ω34, (4.24)
−1
2
∑
k,l
R23klωk ∧ ωl = dω23 − ω21 ∧ ω13 = −
2B2
12,3
(B2 −B1)(B3 −B1)
ω2 ∧ ω3, (4.25)
−1
2
∑
k,l
R24klωk ∧ ωl = dω24 − ω23 ∧ ω34 = − B12,3
B2 −B3
ω1 ∧ ω34. (4.26)
Let ω34 =
∑
k
Γ3
k4ωk, Γ3
k4 = −Γ4
k3. Comparing two side of (4.22) – (4.26), we have
R1212 =
2B2
12,3
(B1 −B3)(B2 −B3)
,
R1313 =
2B2
12,3
(B1 −B2)(B3 −B2)
,
R2323 =
2B2
12,3
(B2 −B1)(B3 −B1)
, R1414 = R2424 = 0.
From (4.22), (4.24) and (4.26), we know that
1
2
R12k4 =
B12,3
B2 −B1
Γ3
k4,
1
2
R142k =
B12,3
B1 −B3
Γ3
k4,
1
2
R24k1 =
B12,3
B3 −B2
Γ3
k4. (4.27)
Since we know that the Bianchi identities of curvature tensors Rijkl are Rijkl + Riklj + Riljk = 0
and Rijkl = Rklij , Rijlk = Rjikl, we have R142k +R12k4 +R24k1 = 0. Thus, from (4.27), we have
Γ3
k4 = 0 for all k. Thus ω34 = 0. From (4.21) and (2.8)
−1
2
∑
k,l
R34klωk ∧ ωl = dω34 −
∑
k
ω3k ∧ ωk4 = 0.
This implies that R3434 = 0. We have
κ =
1
12
∑
i 6=j
Rijij = R1212 +R1313 +R1414 +R2323 +R2424 +R3434 = 0.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
CONFORMAL ISOPARAMETRIC SPACELIKE HYPERSURFACES . . . 569
From (3.1) and Proposition 3.2, we know that x is a 4-dimensional Euclidean isoparametric
spacelike hypersurfaces with three distinct Euclidean principal curvatures. This is in contradiction
with Proposition 3.1. Thus, we know that Theorem 1.4 is true.
(3) If γ = 4, from [12], we know that the conformal second fundamental form is not parallel. We
can prove that this case does not occur. In fact, we may assume that B1 6= B2 6= B3 6= B4. Denote
by i, j, k, l the four distinct elements of {1, 2, 3, 4} with order arbitrarily given, then from (2.7), we
have
ωij =
Bij,kωk +Bij,lωl
Bi −Bj
, for i 6= j. (4.28)
From (4.28) and (2.8), by a simple calculation (see [5]), we have
−1
2
∑
s,t
Rijstωs ∧ ωt = dωij − ωik ∧ ωkj − ωil ∧ ωlj ≡
≡ −
(
2B2
ij,k
(Bi −Bk)(Bj −Bk)
+
2B2
ij,l
(Bi −Bl)(Bj −Bl)
)
ωi ∧ ωj
mod (ωs ∧ ωt, (s, t) 6= (i, j), (j, i)) .
Comparing two side of the above equation, we have
Rijij =
2B2
ij,k
(Bi −Bk)(Bj −Bk)
+
2B2
ij,l
(Bi −Bl)(Bj −Bl)
.
Thus,
R1212 =
2B2
12,3
(B1 −B3)(B2 −B3)
+
2B2
12,4
(B1 −B4)(B2 −B4)
,
R1313 =
2B2
12,3
(B1 −B2)(B3 −B2)
+
2B2
13,4
(B1 −B4)(B3 −B4)
,
R1414 =
2B2
13,4
(B1 −B3)(B4 −B3)
+
2B2
12,4
(B1 −B2)(B4 −B2)
,
R2323 =
2B2
12,3
(B2 −B1)(B3 −B1)
+
2B2
23,4
(B2 −B4)(B3 −B4)
,
R2424 =
2B2
12,4
(B2 −B1)(B4 −B1)
+
2B2
23,4
(B2 −B3)(B4 −B3)
,
R3434 =
2B2
13,4
(B3 −B1)(B4 −B1)
+
2B2
23,4
(B3 −B2)(B4 −B2)
.
We have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
570 SHICHANG SHU, BIANPING SU
κ =
1
12
∑
i 6=j
Rijij = R1212 +R1313 +R1414 +R2323 +R2424 +R3434 = 0.
From (3.1) and Proposition 3.2, we know that x is a 4-dimensional Euclidean isoparametric
spacelike hypersurfaces with four distinct Euclidean principal curvatures. This is in contradiction
with Proposition 3.1.
Theorem 1.4 is proved.
1. Akivis M. A., Goldberg V. V. Conformal differential geometry and its generalizations. – New York: Wiley, 1996.
2. Akivis M. A., Goldberg V. V. A conformal differential invariant and the conformal rigidity of hypersurfaces // Proc.
Amer. Math. Soc. – 1997. – 125. – P. 2415 – 2424.
3. Cheng Q.-M., Shu S. C. A Möbius characterization of submanifolds // J. Math. Soc. Jap. – 2006 . – 58. – P. 903 – 925.
4. Hu Z. J., Li H. Z. Classification of Moebius isoparametric hypersurfaces in S4 // Nagoya Math. J. – 2005. – 179. –
P. 147 – 162.
5. Hu Z. J., Li H. Z., Wang C. P. Classification of Moebius isoparametric hypersurfaces in S5 // Monatsh. Math. – 2007.
– 151. – S. 201 – 222.
6. Li H., Liu H. L., Wang C. P., Zhao G. S. Möbius isoparametric hypersurface in Sn+1 with two distinct principal
curvatures // Acta Math. Sinica. English Ser. – 2002. – 18. – P. 437 – 446.
7. Li H., Wang C. P., Wu F. Möbius characterization of Veronese surfaces in Sn // Math. Ann. – 2001. – 319. –
P. 707 – 714.
8. Li Z. Q., Xie Z. H. Spacelike isoparametric hypersurfaces in Lorentzian space forms // Front. Math. China. – 2006. –
1. – P. 130 – 137.
9. Li X. X., Zhang F. Y. Immersed hypersurfaces in the unit sphere Sm+1 with constant Blaschke eigenvalues // Acta
Math. Sinica. English Ser. – 2007. – 23. – P. 533 – 548.
10. Liu H. L., Wang C. P., Zhao G. S. Möbius isotropic submanifolds in Sn // Tohoku Math. J. – 2001. – 53. – P. 553 – 569.
11. Nie C. X., Wu C. X. Regular submanifolds in conformal spaces (in Chinese) // Chinese Ann. Math. Ser. A. – 2008. –
29. – P. 315 – 324.
12. Nie C. X., Wu C. X. Space-like hyperspaces with parallel conformal second fundamental forms in the conformal space
// Acta Math. Sinica. Chinese Ser. – 2008. – 51. – P. 685 – 692.
13. Nie C. X., Li T. Z., He Y., Wu C. X. Conformal isoparametric hypersurfaces with two distinct conformal principal
curvatures in conformal space // Sci. China (Math.). – 2010. – 53. – P. 953 – 965.
14. Nie C. X. Conformal geometry of hypersurfaces and surfaces in Lorentzian space forms (in Chinese): Diss. Doctoral
Degree. – Beijing, Peking Univ., 2006.
15. Nomizu K. On isoparametric hypersurfaces in the Lorentzian space forms // Jap. J. Math. – 1981. – 7. – P. 217 – 226.
16. Shu S. C., Liu S. Y. Submanifolds with Möbius flat normal bundle in Sn // Acta Math. Sinica. Chinese Ser. – 2005. –
48. – P. 1221 – 1232.
17. Wang C. P. Möbius geometry of submanifolds in Sn // Manuscr. Math. – 1998. – 96. – P. 517 – 534.
18. Zhong D. X., Sun H. A., Zhang T. F. The hypersurfaces in S5 with constant para-Blaschke eigenvalues // Acta Math.
Sinica. Chinese Ser. – 2010. – 53. – P. 263 – 278.
Received 21.09.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
|
| id | umjimathkievua-article-2595 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:29Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/40/37a70c8e0da1bf33642bdc495bcb8540.pdf |
| spelling | umjimathkievua-article-25952020-03-18T19:30:15Z Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ Конформнi iзопараметричнi простороподiбнi гiперповерхнi у конформних просторах $\mathbb{Q}^4_1$ і $\mathbb{Q}^5_1$ Bianping, Su Shu, Shichang Біанпинг, Су Шу, Шичанґ We study the conformal geometry of conformal spacelike hypersurfaces in the conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$. We obtain a complete classification of conformal isoparametric spacelike hypersurfaces in $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$. Вивчено конформну геометрiю конформних простороподiбних гiперповерхонь у конформних просторах $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1.$ Отримано повну класифiкацiю конформних iзопараметричних простороподiбних гiперповерхонь у $\mathbb{Q}^4_1$ та $\mathbb{Q}^5_1$. Institute of Mathematics, NAS of Ukraine 2012-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2595 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 4 (2012); 557-570 Український математичний журнал; Том 64 № 4 (2012); 557-570 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2595/1949 https://umj.imath.kiev.ua/index.php/umj/article/view/2595/1950 Copyright (c) 2012 Bianping Su; Shu Shichang |
| spellingShingle | Bianping, Su Shu, Shichang Біанпинг, Су Шу, Шичанґ Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ |
| title | Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ |
| title_alt | Конформнi iзопараметричнi простороподiбнi гiперповерхнi у конформних просторах $\mathbb{Q}^4_1$ і $\mathbb{Q}^5_1$ |
| title_full | Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ |
| title_fullStr | Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ |
| title_full_unstemmed | Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ |
| title_short | Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$ |
| title_sort | conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{q}^4_1$ and $\mathbb{q}^5_1$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2595 |
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