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Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms
Journal of Geometry and Physics 92 (2015) 69–77
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Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms Jiancheng Liu a,∗ , Li Du a,b , Juan Zhang b a
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
b
Department of Mathematics, Dingxi Teachers College, Dingxi 743000, China
article
abstract
info
Article history: Received 16 November 2013 Accepted 10 February 2015 Available online 18 February 2015
In this paper, we investigate the minimality or the constraint of the mean curvature of three kinds of biharmonic space-like submanifolds in pseudo-Riemannian space forms: (1) pseudo-umbilical ones; (2) the ones with parallel mean curvature vector; (3) with constant mean curvature. © 2015 Elsevier B.V. All rights reserved.
MSC: 58E20 Keywords: Biharmonic space-like submanifolds Pseudo-Riemannian space forms Mean curvature Parallel mean curvature vector
1. Introduction The study of biharmonic maps between Riemannian manifolds, as a generalization of harmonic maps, was suggested by J. Eells and J. H. Sampson in [1]. Thus, whilst a harmonic map φ : (M , g ) → (N , h) between two Riemannian manifolds is defined as a critical point of the energy functional E (φ, M ) = 21 M |dφ|2 vg , a biharmonic map is a critical point of the bienergy functional E2 (φ, M ) = 12 M |τ (φ)|2 vg , where τ (φ) = tr∇ dφ is the tension field that vanishes for harmonic maps. The Euler–Lagrange equation for the bienergy functional was derived firstly by G.Y. Jiang [2] in 1986
τ2 (φ) := −∆τ (φ) − tr RN (dφ, τ (φ))dφ = 0, where ∆ is the rough Laplacian defined on sections of φ −1 (TN ) and RN is the curvature tensor of N. Clearly, harmonic maps are biharmonic maps. We call a non-harmonic biharmonic map a proper biharmonic map. The study of biharmonic maps has attracted more and more attention of mathematicians and has become a fascinating area of research. One of the central topics in this area is the study of biharmonic submanifolds, i.e., those submanifolds whose inclusion maps are biharmonic. Independently, B.Y. Chen [3] defined the biharmonic submanifolds in a Euclidean space as those with harmonic mean curvature vector field. Using the characterization formula of biharmonic maps for Riemannian immersions into Euclidean spaces (cf. [4,5]), Chen’s notion of biharmonic submanifold can be recovered. During the last decade, focused on the two fundamental problems: existence problem and classification problem, important progress has been made in the study of biharmonic submanifolds. We refer to [4,6] for a review, and [5,7,8] with references therein for recent progress.
∗
Corresponding author. E-mail address: [email protected] (J. Liu).
http://dx.doi.org/10.1016/j.geomphys.2015.02.008 0393-0440/© 2015 Elsevier B.V. All rights reserved.
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
Biharmonic submanifolds in pseudo-Euclidean spaces Em s were originally studied by Chen in [3,9] and by several other authors thereafter (cf. [10–14]). For submanifolds in pseudo-Riemannian space forms, C.Z. Ouyang [15] obtained biharmonic equations using moving frame (similar to that of in [2] for Riemannian case, see also in [16]). Very recently, T. Sasahara [17] classified proper biharmonic curves and surfaces in de Sitter 3-space and anti-de Sitter 3-space. It turns out that the existence problem for proper biharmonic submanifolds in pseudo-Riemannian manifolds often appears considerably different from that of in Riemannian manifolds. And the classification problems are more complicated. As we know, even in the non flat pseudo-Riemannian space forms, there is no much more progress. The difficulties, on our opinion, show up in the following two aspects: (i) Submanifolds in pseudo-Riemannian manifolds may be space-like type or Lorentz type, the shape operator of a Riemannian submanifold is always diagonalizable, but this is not the case for that of a Lorentzian submanifold (e.g. [18,19]). (ii) The mean curvature vector may be space-like type, time-like type or light-like type according to the index of ambient spaces. In general, each of them will imply different properties of submanifolds. Motivated by above consideration, in this paper, we concentrate our attention on studying the minimality or the constraint of the mean curvature of space-like type biharmonic submanifolds in pseudo-Riemannian space forms. More pre-
− →
cisely, in Section 3, we deal with pseudo-umbilical biharmonic space-like submanifolds with mean curvature vector H , and prove that n +p
Theorem 1.1. Let M n be a pseudo-umbilical biharmonic space-like submanifold in pseudo-Riemannian space forms Nq
(c ).
− →
• When c = 0 and H is not light-like, then M n is minimal. • When c > 0. − → (i) It is impossible for H to be time-like. − → (ii) If H is space-like, then either M n is minimal, or H 2 = c. • When c < 0. − → (i) If H is space-like, then M n is minimal. − → (ii) If H is time-like, then H 2 = |c |. As an application of Theorem 1.1, we also obtain several bounds for the scalar curvature of pseudo-umbilical biharmonic n +p space-like submanifolds in pseudo-Riemannian space forms Nq (c ) (cf. Proposition 3.5). In Section 4, we investigate biharmonic space-like submanifolds with parallel mean curvature vector in pseudon +p Riemannian space forms Nq (c ), and prove that Theorem 1.2. Let M n be a biharmonic space-like submanifold with parallel mean curvature vector in pseudo-Riemannian space n +p forms Nq (c ).
• When c = 0, then H = 0. • When c > 0. − → (i) It is impossible for H to be time-like. − → − → (ii) If H is space-like, then H 2 ≤ c. In particular, if H ̸= 0, then H 2 = c holds if and only if M n is pseudo-umbilical. • When c < 0. − → (i) If H is space-like, then M n is minimal. − → (ii) If H is time-like, then H 2 ≤ |c |. The equality holds if and only if M n is pseudo-umbilical. In last section, we derive firstly a general inequality concerning the mean curvature H and the squared norm S of the n +p second fundamental form of a space-like submanifold M n in pseudo-Riemannian space forms Np (c ). As its application to n anti-de Sitter space, we obtain a sufficient condition for such submanifold M being minimal (see Corollary 5.2) and a gap property (see Corollary 5.3). 2. Preliminaries n +p
Let Eq denote an (n + p)-dimensional real vector space endowed with an inner product of index q given by ⟨x, y⟩ = − i=1 xi yi + nj=+qp+1 xj yj , where x = (x1 , x2 , . . . , xn+p ) is the natural coordinate of Eqn+p . The manifold Enq+p is called pseudo-
q
n +p
Euclidean space and it has constant curvature c = 0. We also define the pseudo-Riemannian manifolds Sq with c > 0 and c < 0 respectively, called de Sitter space, anti-de Sitter space as follows:
Sqn+p
(c ) = (x1 , x2 , . . . , xn+p+1 ) ∈
Enq+p+1
:−
q
n+p+1
x2i
+
i=1
n+p+1
Hqn+p (c ) = (x1 , x2 , . . . , xn+p+1 ) ∈ Eq+1
:−
q +1 i =1
x2j
1
=
c
j =q +1 n+p+1
x2i +
x2j =
j=q+2 n+p
These spaces are called pseudo-Riemannian space forms, denoted by Nq
1
,
c
(c ).
.
(c ) and Hqn+p (c ),
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
71
n+p
A smooth immersion φ : M n → Nq (c ) of an n-dimensional connected manifold M n is said to be a space-like submanifold if the induced metric via φ is a Riemannian metric on M n . We choose a local field of pseudo-Riemannian orthonormal frame {e1 , . . . , en+p } on Nqn+p (c ) such that, restricted to M n , e1 , . . . , en are tangent to M n . Let {ω1 , . . . , ωn+p } be its dual frame field n+p n +p 2 so that the pseudo-Riemannian metric of Nq (c ) is given by ds¯2 = A=1 εA ωA , where εi = ⟨ei , ei ⟩ = 1, εα = ⟨eα , eα ⟩ = −1 (here and in the sequel, we use ⟨ , ⟩ as an alternative notation of metric, and make the following convention on the range of indices: 1 ≤ i, j, k, · · · ≤ n; n + 1 ≤ α, β, γ , · · · ≤ n + p). It is well known that (cf. [18,19])
ωα = 0,
ωα i =
hαij ωj ,
j
Rijkl = c (δik δjl − δil δjk ) +
α
hαij = hαji ,
εα (hαik hαjl − hαil hαjk ), − →
α where Rijkl are the components of the curvature tensor of M n . The quadratic form h = α,i,j hij ωi ⊗ ωj ⊗ eα and H = n are the second fundamental form and the mean curvature vector of M , respectively. The square length of h and the mean curvature of M n can be expressed as, respectively,
α 2 S= εα (hij ) , α i ,j
α 2 hii ) . εα ( i
1
H =
n
α
1 tr h n
(1)
The covariant derivatives hαijk of hαij satisfy
hαijk ωk = dhαij +
k
hαik ωkj +
k
hαjk ωki −
β
hij ωβα .
β
k
Similarly, we have the second covariant derivatives hαijkl of hαij so that
hαijkl ωl = dhαijk +
l
hαijk ωli +
l
α
hαilk ωlj +
l
α
hαijl ωlk −
The Laplacian △hij of hij is defined by △hij =
− →
k
β
hijk ωβα .
β
l
α
α
hijkk .
− → α
If H is parallel in the normal bundle of M , i.e., ∇ ⊥ H = 0, then M n is called a submanifold with parallel mean curvature n
vector. It is well known that ∇
→ ⊥−
H = 0 if and only if
k
hkkj = 0, ∀ α, j.
n
− →
− →
The submanifold M is called minimal if the mean curvature vector vanishes identically, that is, H = 0. Obviously, if H is not light-like, then M n is minimal if and only if its mean curvature H = 0. A space-like submanifold is said pseudo-umbilical, if it is umbilical with respect to the direction of the mean curvature
− →
− →
− → − →
vector H , i.e., ⟨h(ei , ej ), H ⟩ = ⟨ H , H ⟩⟨ei , ej ⟩, ∀ i, j. n +p As it is known [18,19], a nonzero vector v tangent to Nq (c ) is called space-like, light-like, or time-like (or isotropic) according to whether ⟨v, v⟩ > 0, ⟨v, v⟩ = 0 or ⟨v, v⟩ < 0, respectively. The zero vector is considered to be space-like. n +p 2 2 Denote byα g2˜ the metric on tensor bundle T1 (M ) induced fromn ds¯ (the metric of Nq (c )). Then g˜ (h, h) = α εα i,j (hij ) . According to [20], the second fundamental form of M is called locally space-like, light-like, or time-like, when g˜ (h, h) ≥ 0, g˜ (h, h) = 0 and h ̸= 0, or g˜ (h, h) < 0, respectively. We need the following biharmonic equations due to C.Z. Ouyang, for space-like submanifolds in pseudo-Riemannian n +p space forms Nq (c ). It can be seen as the dual version of [21, Corollary 3.1] for that of submanifolds in Riemannian space forms. n +p
Lemma 2.1 (C.Z. Ouyang, [15]). Let M n be a Riemannian submanifold of pseudo-Riemannian space forms Nq Then it is biharmonic if and only if
εα (2hαiik hαkj + hαkkj hαii ) = 0, ∀ j, α,i,k β β α εβ hii hkj hαjk ) + nc hαii = 0, (hiikk − i ,k
β,j
∀ α.
(c ), 0 ≤ q ≤ p.
(2)
i
3. Pseudo-umbilical biharmonic space-like submanifolds 3.1. When mean curvature vector is light-like We start with the following existence example of biharmonic space-like submanifold with light-like mean curvature vector. Let M 2 be an open portion of the Euclidean plane E2 with Euclidean coordinates (u, v) and φ(u, v) be a smooth
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
function on M 2 such that ∆φ(u, v) ̸= 0 and ∆2 φ(u, v) = 0. Let x : M 2 → E41 be defined by x(u, v) = (φ(u, v), φ(u, v), u, v). Obviously, it is a biharmonic space-like surface with light-like mean curvature vector in E41 . Moreover, it is pseudo-umbilical (see the proof of Theorem 6.1 in [3]). Indeed, we can prove that biharmonic space-like submanifolds with light-like mean curvature vector in pseudon+p Riemannian space forms Nq (c ) are all pseudo-umbilical. Theorem 3.1. Let M n be a biharmonic space-like submanifold with light-like mean curvature vector in pseudo-Riemannian space n+p forms Nq (c ). Then M n is pseudo-umbilical.
− →
− → − →
− →
Proof. Because H is light-like, i.e., ⟨ H , H ⟩ = 0, and H ̸= 0. So we may choose local pseudo-Riemannian orthonormal n +p frame field {ei , eα } on Nq (c ) such that
− →
H = f (en+1 + en+2 ),
(3)
where en+1 is time-like and en+2 is space-like, f is a nonzero function on M n . Thus (3) implies
β hii = nf , β = n + 1, n + 2, i β hii = 0, ∀β > n + 2.
(4)
i
Substituting (4) into the second equation of (2), we obtain, for any α ≥ n + 1,
hαiikk − (nf )
i,k
εn+1 hnkj+1 hαjk +
j,k
εn+2 hnkj+2 hαjk + nc hαii = 0.
j,k
(5)
i
Evaluating (5) for α = n + 1, n + 2, respectively, and subtracting each other, we have
n +1 hkj − hkjn+2
2
= 0.
j ,k
Therefore, n+2 . hnkj+1 = hkj
(6)
Using (3) and (6), it is easy to check that
− →
⟨h(ei , ej ), H ⟩ =
α
hαij eα , f (en+1 + en+2 ) = f (hijn+1 εn+1 + hnij+2 εn+2 ) = 0,
− →
− →
− → − →
together with H being light-like, which implies ⟨h(ei , ej ), H ⟩ = ⟨ H , H ⟩δij . Equivalently, M n is pseudo-umbilical.
3.2. When mean curvature vector is not light-like Now, we will concentrate on the study of pseudo-umbilical biharmonic space-like submanifolds in pseudo-Riemannian n +p
spaces Nq
− → (c ), under the assumption that the mean curvature vector H is either space-like or time-like.
n +p
Proposition 3.2. Let M n be a pseudo-umbilical biharmonic space-like submanifold in pseudo-Euclidean space Eq light-like, then M n is minimal.
− →
. If H is not
Remark 3.1. Indeed, the pseudo-umbilical condition in Proposition 3.2 is essential. For instance, let M 2 be a surface in pseudo-Euclidean space E41 defined by x(u, v) =
a2 6
u , 3
a 2
u ,− 2
a2 6
u + u, v 3
,
2 for some nonzero constant a. It is easy to show that M is a biharmonic space-like surface, and its mean curvature
− → − →
− →
H = |⟨ H , H ⟩| = 12 |a| ̸= 0, which also implies H is not light-like. In fact, M 2 is not a pseudo-umbilical surface (see the proof of Theorem 5.1 in [3]).
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
73
− →
Proof of Proposition 3.2. According to the assumption that H is not light-like, we can choose pseudo-Riemannian ortho-
− →
normal frame field {eA } (A = 1, 2, . . . , n + p) such that H = Hen+1 . Then
tr Hα = 0, α > n + 1, tr Hn+1 = nH ,
(7)
where Hα denotes the matrix (hαij )n× n . Using (7) together with c = 0, the biharmonic equation (2) can be simplified as
n+1 n+1 +1 n+1 + hnkkj hii = 0. ε n+1 2hiik hkj i ,k +1 hniikk − εn+1 hnkj+1 hαjk hnii+1 = 0, ∀α. i ,k
(8)
j
Now, suppose on the contrary that H ̸= 0. Since M n is pseudo-umbilical, we know that hijn+1 = H δij .
(9)
Putting (9) into the first equation of (8) yields H (nH )j = 0,
∀j.
(10)
Meanwhile, taking α = n + 1 in the second equation of (8) and using (9), we have
(nH )kk = εn+1 (n2 H 3 ).
(11)
k n +p
Combining (10) and (11), we obtain that a pseudo-umbilical space-like submanifold M n in Eq
is biharmonic if and only if
(nH )kk = εn+1 (n H ), 2
k
H (nH )j = 0,
3
∀j ,
which implies that H = 0. That is a contradiction.
n +p
Proposition 3.3. Let M n be a pseudo-umbilical biharmonic space-like submanifold in de Sitter space Sq
− →
(c ).
(i) If H is space-like, then either M is minimal, or H = c. − → (ii) It is impossible for H to be time-like. n
2
n+p
Remark 3.2. In fact, let M n be a pseudo-umbilical space-like submanifold with space-like mean curvature vector in Sq If H 2 = c, then in view of (12) below, we know that M n is a proper biharmonic submanifold.
(c ).
− →
Proof. We choose pseudo-Riemannian orthonormal frame field {ei , eα } such that H = Hen+1 . Then (7) holds.
− →
(i) Assume that H ̸= 0 (otherwise, M n is minimal). Similar to the proof of Proposition 3.2, we can obtain that a pseudon +p umbilical space-like submanifold M n in Sq (c ) is biharmonic if and only if
(nH )kk = εn+1 (n2 H ) H 2 − εn+1 c ,
k
H (nH )j = 0,
(12)
∀j,
which implies that H 2 = εn + 1 c .
(13)
− →
Because of H being space-like, i.e., εn+1 = 1, then we have H 2 = c from (13). (ii) Suppose that there exists a pseudo-umbilical biharmonic space-like submanifold M n with time-like mean curvature n+p vector in Sq (c ), then its mean curvature H ̸= 0, and εn+1 = −1. In this case, (13) becomes
(H 2 + c ) = 0, which is a contradiction.
When c < 0, using the similar method as in the proof of Proposition 3.3, we can easily prove the following result. n +p
Proposition 3.4. Let M n be a pseudo-umbilical biharmonic space-like submanifold in anti-de Sitter space Hq
− → (i) If H is space-like, then M n is minimal. − → (ii) If H is time-like, then H 2 = |c |.
(c ).
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
Remark 3.3. Similar to Remark 3.2, we know that a√ pseudo-umbilical space-like submanifold with time-like mean curvature n +p vector in Hq (c ) whose mean curvature equal to |c | is proper biharmonic. Finally, combining Propositions 3.2–3.4, we obtain our main Theorem 1.1 stated in the introduction. 3.3. Applications As an application of Theorem 1.1, we also get several bounds for the scalar curvature of pseudo-umbilical biharmonic n +p space-like submanifold in pseudo-Riemannian space forms Nq (c ). n+p
Proposition 3.5. Let M n be a pseudo-umbilical biharmonic space-like submanifold of Nq of M n .
(c ). Denote by R the scalar curvature
• When c = 0 and h is locally space-like, then R ≤ 0. The equality holds if and only if M n is totally geodesic. • When c > 0 and h is locally space-like, then R ≤ n(n − 1)c. • When c < 0. (i) If h is locally time-like, then R ≥ 2n(n − 1)c. The equality holds if and only if M n is totally umbilical. (ii) If h is locally space-like, then R ≤ n(n − 1)c. The equality holds if and only if M n is totally geodesic. Proof. From Gauss equation, we have n+p
R = n(n − 1)c +
εα
α=n+1
hαii
2
n +p
−
εα
α=n+1
i
(hαij )2 .
(14)
i ,j
Choosing a frame field {ei , eα } such that one of the following holds (cf. [20]): (i) For h being locally time-like,
n+p
h=
hαij ωi ⊗ ωj ⊗ eα ,
εα = −1,
(15)
α=n+q+1 i,j
α
hij = 0,
α = n + 1, . . . , n + q.
(ii) For h being locally space-like,
h= α
n +q
hαij ωi ⊗ ωj ⊗ eα ,
εα = +1,
(16)
α=n+1 i,j
hij = 0,
α = n + q + 1, . . . , n + p.
Making use of (15) and (16) separately, we have
− →
H =
1
n+p
n α=n+q+1
hαii eα ,
for h being locally time-like
(17)
i
and
− →
H =
n+q 1 α hii eα , n α=n+1 i
for h being locally space-like.
− →
(18)
− →
Clearly, (17) implies that H is time-like, while (18) implies that H is space-like. When h is locally time-like, εα = −1 (n + q + 1 ≤ α ≤ n + p). Applying Cauchy inequality, we have from (1) and (17) that n2 H 2 =
n +p
α=n+q+1
hαii
2
i
≤n
n +p
α=n+q+1
(hαii )2 ≤ n i
n +p
(hαij )2 ,
α=n+q+1 i,j
that is, nH 2 ≤
n +p
α=n+q+1 i,j
(hαij )2 .
(19)
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
75
Similarly, when h is locally space-like, (1) and (18) lead to nH 2 ≤
n+q (hαij )2 .
(20)
α=n+1 i,j
− →
Case 1: c = 0. If h is locally space-like, then H is space-like as we have proved before. Using Proposition 3.2 together with (15), (14) becomes R=−
n +q (hαij )2 .
α=n+1 i,j
Obviously, R ≤ 0 and the equality holds if and only if M n is totally geodesic. − → Case 2: c > 0. If h is locally space-like, then H is space-like (cf. Eq. (18)), together with (14), (16) and (18), we have R = n(n − 1)c +
n+q
α=n+1
hαii
2
−
n +q
(hαij )2 .
α=n+1 i,j
i
Using this equation, Proposition 3.3(i) and (20), we have R ≤ n(n − 1)c, or R ≤ 2n(n − 1)c. Moreover R ≤ n(n − 1)c. Case 3: c < 0. − → (i) If h is locally time-like, then H is time-like (cf. Eq. (17)), together with (14), (15) and (17), we can easily obtain n +p
R = n(n − 1)c −
α=n+q+1
hαii
2
+
i
n +p
(hαij )2 ,
α=n+q+1 i,j
which combining with (19) and Proposition 3.4(ii) implies that R ≥ n(n − 1)c − n2 H 2 + nH 2 = 2n(n − 1)c . Obviously, the equality holds if and only if M n is totally umbilical. (ii) If h is locally space-like, by taking the similar methods as (i), and using (20) and Proposition 3.4(i), we have R ≤ n(n − 1)c. Clearly, the equality holds if and only if M n is totally geodesic. 4. Biharmonic space-like submanifolds with parallel mean curvature vector In [15], C.Z. Ouyang proved the following non-existence results for proper biharmonic submanifolds with parallel mean curvature vector in pseudo-Riemannian space forms. Proposition 4.1 (C.Z. Ouyang, [15]). Let M n be a biharmonic space-like submanifold with parallel mean curvature vector in n+p pseudo-Riemannian space forms Nq (c ). (1) If c = 0, then H = 0. (2) If c > 0 and the mean curvature vector of M n is not space-like, then H = 0. (3) If c < 0 and the mean curvature vector of M n is not time-like, then H = 0. As a supplement of Proposition 4.1, in this section, we will study continuously the same problem in the following two cases: (i) when c > 0 and the mean curvature vector is space-like; (ii) when c < 0 and the mean curvature vector is time-like. More precisely, we prove the following proposition. Proposition 4.2. Let M n be a biharmonic space-like submanifold with parallel mean curvature vector in pseudo-Riemannian n +p space forms Nq (c ).
− → (i) If c ̸= 0, then H is not light-like. − → − → (ii) If c > 0 and H is space-like, then H 2 ≤ c. In particular, if H ̸= 0, then H 2 = c holds if and only if M n is pseudoumbilical. − → (iii) If c < 0 and H is time-like, then H 2 ≤ |c |. The equality holds if and only if M n is pseudo-umbilical. − →
Proof. (i) Suppose on the contrary that there exists such a submanifold with light-like mean curvature vector H , making use of the similar methods as Theorem 3.1, we know that (4)–(6) also hold. Moreover, (5) becomes
(nf )kk + nc (nf ) = 0. k
− →
Because H is parallel, so
(nf )j = 0,
∀ j.
Those two facts lead to c = 0, contradiction.
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
− →
− →
− →
(ii) When H is space-like, choose a local pseudo-orthonormal frame {ei , eα } such that H = Hen+1 , then (7) holds. Since
H is parallel, H is a constant. Using (7) and c > 0, we can simplify the second equation of (2) as
(nH )(tr Hn2+1 − nc ) = 0.
(21)
It follows that H = 0,
tr Hn2+1 = nc .
or
If H = 0, then H 2 ≤ c. If tr Hn2+1 = nc, making use of Cauchy inequality and noticing that (7), we get n2 H 2 =
hnii+1
2
≤ n
(hnii+1 )2 ≤ n (hnij+1 )2 = n2 c ,
i
(22)
i ,j
i
which also implies H 2 ≤ c. When H 2 = c, all equalities in (22) hold. Therefore
hn11+1 = hn22+1 = · · · = hnnn+1 , hijn+1 = 0,
i ̸= j.
This fact together with (7) yields hnij+1 = H δij . In other words, M n is pseudo-umbilical.
− →
Conversely, when H ̸= 0 and M n is pseudo-umbilical, by (21) we obtain that H 2 = c.
− →
(iii) When c < 0 and H is time-like, we can also choose a local pseudo-Riemannian orthonormal frame {ei , eα } such
− →
that H = Hen+1 . The remaining proof is similar to that of (ii). We complete the proof of Proposition 4.2.
Finally, combining Propositions 4.1 and 4.2, we obtain our main Theorem 1.2 stated in introduction. 5. Biharmonic space-like submanifolds with constant mean curvature In this section, we prove firstly a general inequality concerning the mean curvature H and the squared norm S of the n+p second fundamental form of a submanifold M n in pseudo-Riemannian space forms Np (c ) of index p ≥ 1. n +p
Lemma 5.1. Let M n be a biharmonic space-like submanifold in pseudo-Riemannian space forms Np inequality holds 1 2
∆(n2 H 2 ) ≥ (−n2 H 2 )(S + nc ).
(23)
Proof. Multiplying the second equation in (2) by
α
hαii
hαiikk
=
i ,k
i
β
εβ hkj
i
β
j ,k
(c ). Then the following
hαii and taking sum for α yields β
hii
α
i
hαjk
hαii
− nc
α
i
hαii
2
.
(24)
i
On one hand, a direct calculation gives 1 2
∆(n2 H 2 ) =
≥
1 2
∆
α
2
=
α
i
α
hαii
hαii
hαiik
2
+
α
i
hαii
hαiikk
i ,k
i
hαiikk .
(25)
i ,k
i
On the other hand, using Cauchy inequality, notice that εβ = −1, we have
j,k
β
β
εβ hkj
i
β
hii
α
hαjk
i
hαii
=−
j,k
β
β
hkj
β
hii
2
i
β β 2 ≥− (hkj )2 hii = −n2 H 2 S . β
j ,k
β
i
Substituting (24) and (26) into (25), we complete the proof of Lemma 5.1.
As a result of Lemma 5.1, we can easily get the following two conclusions.
(26)
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
77 n +p
Corollary 5.2. Let M n be a biharmonic space-like submanifold with constant mean curvature in anti-de Sitter space Hp S < n|c |, then M n is minimal.
(c ). If
Proof. Since H is constant, it follows from (23) that n2 H 2 (S − n|c |) ≥ 0, which leads to H = 0 provided that S < n|c |.
Corollary 5.3. Let M n be a proper biharmonic space-like submanifold with constant mean curvature in anti-de Sitter space n+p Hp (c ). If S ≤ n|c |, then S = n|c |. Remark 5.1. Corollary 5.2 generalizes Ouyang’s result [15, Corollary 4.2] to higher codimension: let M n be a biharmonic space-like hypersurface with constant mean curvature in anti-de Sitter space H1n+1 (c ). If S ̸= n|c |, then M n is minimal. Corollary 5.3 can be seen as a gap phenomenon. Remark 5.2. It is easy to see that, when c ≥ 0 and H is constant, (23) holds automatically. We cannot obtain the similar results as Corollaries 5.2 and 5.3 for de Sitter space. Acknowledgments The first author was supported by the National Natural Science Foundation of China (Grant No. 11261051, 11171246), and Fundamental Research Funds of the Gansu Universities (2012). The second and the third authors were supported by the Young Talents Project and the Youth Project of Dingxi Teacher’s College (No. 2012-2017, 1333). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J. Eells, J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109–160. G.Y. Jiang, 2-Harmonic maps and their first and second variational formulas, Chin. Ann. Math. A7 (4) (1986) 389–402. B.Y. Chen, S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Kyushu Univ. A45 (2) (1991) 323–347. A. Balmuş, Biharmonic maps and submanifolds, in: Balkan Society of Geometers Differential Geometry–Dynamical Systems, Monographs, Geometry Balkan Press, Bucharest, 2009. Y.L. Ou, Biharmonic hypersurfaces in general Riemannian manifolds, Pacific J. Math. 248 (1) (2010) 217–232. S. Montaldo, C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2) (2006) 1–22. L.J. Alías, S. Carolina García-Martínez, M. Rigoli, Biharmonic hypersurfaces in complete Riemannian manifolds, Pacific J. Math. 263 (1) (2013) 1–12. Y.L. Ou, L. Tang, On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J. 61 (3) (2012) 531–542. B.Y. Chen, S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math. 52 (1998) 167–185. A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, Hypersurfaces of Es4 with proper mean curvature vector, J. Math. Soc. Japan 59 (3) (2007) 797–809. A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, B.J. Papantoniou, Biharmonic Lorentz hypersurfaces in E14 , Pacific J. Math. 229 (3) (2007) 293–306. A. Arvanitoyeorgos, G. Kaimakamis, Hypersurfaces of type M23 in E42 with proper mean curvature vector, J. Geom. Phys. 63 (2013) 99–106. A. Arvanitoyeorgos, G. Kaimakamis, M. Magid, Lorentz hypersurfaces in E41 satisfying △H = α H, Illinois J. Math. 53 (2) (2009) 581–590. F. Defever, G. Kaimakamis, B.J. Papantoniou, Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space Es4 , J. Math. Anal. Appl. 315 (2006) 276–286. C.Z. Ouyang, 2-Harmonic space-like submanifolds in pseudo-Riemannian space forms, Chin. Ann. Math. A21 (6) (2000) 649–654. X.F. Wang, L. Wu, Proper biharmonic submanifolds in a sphere, Acta Math. Sin. (Engl. Ser.) 28 (1) (2012) 205–218. T. Sasahara, Biharmonic submaifolds in nonflat Lorentz 3-space forms, Bull. Aust. Math. Soc. 85 (2012) 422–432. B.Y. Chen, Pseudo-Riemannian Geometry, δ -Invariants and Applications, World Scientific, Hackensack, NJ, 2011. B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983. M. Mariano, On complete spacelike submanifolds with second fundamental form locally timelike in a semi-Riemannian space form, J. Geom. Phys. 60 (5) (2010) 720–728. G.Y. Jiang, 2-Harmonic isometric immersions between Riemannian manifolds, Chin. Ann. Math. A7 (2) (1986) 130–144.
Contents lists available at ScienceDirect
Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms Jiancheng Liu a,∗ , Li Du a,b , Juan Zhang b a
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
b
Department of Mathematics, Dingxi Teachers College, Dingxi 743000, China
article
abstract
info
Article history: Received 16 November 2013 Accepted 10 February 2015 Available online 18 February 2015
In this paper, we investigate the minimality or the constraint of the mean curvature of three kinds of biharmonic space-like submanifolds in pseudo-Riemannian space forms: (1) pseudo-umbilical ones; (2) the ones with parallel mean curvature vector; (3) with constant mean curvature. © 2015 Elsevier B.V. All rights reserved.
MSC: 58E20 Keywords: Biharmonic space-like submanifolds Pseudo-Riemannian space forms Mean curvature Parallel mean curvature vector
1. Introduction The study of biharmonic maps between Riemannian manifolds, as a generalization of harmonic maps, was suggested by J. Eells and J. H. Sampson in [1]. Thus, whilst a harmonic map φ : (M , g ) → (N , h) between two Riemannian manifolds is defined as a critical point of the energy functional E (φ, M ) = 21 M |dφ|2 vg , a biharmonic map is a critical point of the bienergy functional E2 (φ, M ) = 12 M |τ (φ)|2 vg , where τ (φ) = tr∇ dφ is the tension field that vanishes for harmonic maps. The Euler–Lagrange equation for the bienergy functional was derived firstly by G.Y. Jiang [2] in 1986
τ2 (φ) := −∆τ (φ) − tr RN (dφ, τ (φ))dφ = 0, where ∆ is the rough Laplacian defined on sections of φ −1 (TN ) and RN is the curvature tensor of N. Clearly, harmonic maps are biharmonic maps. We call a non-harmonic biharmonic map a proper biharmonic map. The study of biharmonic maps has attracted more and more attention of mathematicians and has become a fascinating area of research. One of the central topics in this area is the study of biharmonic submanifolds, i.e., those submanifolds whose inclusion maps are biharmonic. Independently, B.Y. Chen [3] defined the biharmonic submanifolds in a Euclidean space as those with harmonic mean curvature vector field. Using the characterization formula of biharmonic maps for Riemannian immersions into Euclidean spaces (cf. [4,5]), Chen’s notion of biharmonic submanifold can be recovered. During the last decade, focused on the two fundamental problems: existence problem and classification problem, important progress has been made in the study of biharmonic submanifolds. We refer to [4,6] for a review, and [5,7,8] with references therein for recent progress.
∗
Corresponding author. E-mail address: [email protected] (J. Liu).
http://dx.doi.org/10.1016/j.geomphys.2015.02.008 0393-0440/© 2015 Elsevier B.V. All rights reserved.
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
Biharmonic submanifolds in pseudo-Euclidean spaces Em s were originally studied by Chen in [3,9] and by several other authors thereafter (cf. [10–14]). For submanifolds in pseudo-Riemannian space forms, C.Z. Ouyang [15] obtained biharmonic equations using moving frame (similar to that of in [2] for Riemannian case, see also in [16]). Very recently, T. Sasahara [17] classified proper biharmonic curves and surfaces in de Sitter 3-space and anti-de Sitter 3-space. It turns out that the existence problem for proper biharmonic submanifolds in pseudo-Riemannian manifolds often appears considerably different from that of in Riemannian manifolds. And the classification problems are more complicated. As we know, even in the non flat pseudo-Riemannian space forms, there is no much more progress. The difficulties, on our opinion, show up in the following two aspects: (i) Submanifolds in pseudo-Riemannian manifolds may be space-like type or Lorentz type, the shape operator of a Riemannian submanifold is always diagonalizable, but this is not the case for that of a Lorentzian submanifold (e.g. [18,19]). (ii) The mean curvature vector may be space-like type, time-like type or light-like type according to the index of ambient spaces. In general, each of them will imply different properties of submanifolds. Motivated by above consideration, in this paper, we concentrate our attention on studying the minimality or the constraint of the mean curvature of space-like type biharmonic submanifolds in pseudo-Riemannian space forms. More pre-
− →
cisely, in Section 3, we deal with pseudo-umbilical biharmonic space-like submanifolds with mean curvature vector H , and prove that n +p
Theorem 1.1. Let M n be a pseudo-umbilical biharmonic space-like submanifold in pseudo-Riemannian space forms Nq
(c ).
− →
• When c = 0 and H is not light-like, then M n is minimal. • When c > 0. − → (i) It is impossible for H to be time-like. − → (ii) If H is space-like, then either M n is minimal, or H 2 = c. • When c < 0. − → (i) If H is space-like, then M n is minimal. − → (ii) If H is time-like, then H 2 = |c |. As an application of Theorem 1.1, we also obtain several bounds for the scalar curvature of pseudo-umbilical biharmonic n +p space-like submanifolds in pseudo-Riemannian space forms Nq (c ) (cf. Proposition 3.5). In Section 4, we investigate biharmonic space-like submanifolds with parallel mean curvature vector in pseudon +p Riemannian space forms Nq (c ), and prove that Theorem 1.2. Let M n be a biharmonic space-like submanifold with parallel mean curvature vector in pseudo-Riemannian space n +p forms Nq (c ).
• When c = 0, then H = 0. • When c > 0. − → (i) It is impossible for H to be time-like. − → − → (ii) If H is space-like, then H 2 ≤ c. In particular, if H ̸= 0, then H 2 = c holds if and only if M n is pseudo-umbilical. • When c < 0. − → (i) If H is space-like, then M n is minimal. − → (ii) If H is time-like, then H 2 ≤ |c |. The equality holds if and only if M n is pseudo-umbilical. In last section, we derive firstly a general inequality concerning the mean curvature H and the squared norm S of the n +p second fundamental form of a space-like submanifold M n in pseudo-Riemannian space forms Np (c ). As its application to n anti-de Sitter space, we obtain a sufficient condition for such submanifold M being minimal (see Corollary 5.2) and a gap property (see Corollary 5.3). 2. Preliminaries n +p
Let Eq denote an (n + p)-dimensional real vector space endowed with an inner product of index q given by ⟨x, y⟩ = − i=1 xi yi + nj=+qp+1 xj yj , where x = (x1 , x2 , . . . , xn+p ) is the natural coordinate of Eqn+p . The manifold Enq+p is called pseudo-
q
n +p
Euclidean space and it has constant curvature c = 0. We also define the pseudo-Riemannian manifolds Sq with c > 0 and c < 0 respectively, called de Sitter space, anti-de Sitter space as follows:
Sqn+p
(c ) = (x1 , x2 , . . . , xn+p+1 ) ∈
Enq+p+1
:−
q
n+p+1
x2i
+
i=1
n+p+1
Hqn+p (c ) = (x1 , x2 , . . . , xn+p+1 ) ∈ Eq+1
:−
q +1 i =1
x2j
1
=
c
j =q +1 n+p+1
x2i +
x2j =
j=q+2 n+p
These spaces are called pseudo-Riemannian space forms, denoted by Nq
1
,
c
(c ).
.
(c ) and Hqn+p (c ),
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
71
n+p
A smooth immersion φ : M n → Nq (c ) of an n-dimensional connected manifold M n is said to be a space-like submanifold if the induced metric via φ is a Riemannian metric on M n . We choose a local field of pseudo-Riemannian orthonormal frame {e1 , . . . , en+p } on Nqn+p (c ) such that, restricted to M n , e1 , . . . , en are tangent to M n . Let {ω1 , . . . , ωn+p } be its dual frame field n+p n +p 2 so that the pseudo-Riemannian metric of Nq (c ) is given by ds¯2 = A=1 εA ωA , where εi = ⟨ei , ei ⟩ = 1, εα = ⟨eα , eα ⟩ = −1 (here and in the sequel, we use ⟨ , ⟩ as an alternative notation of metric, and make the following convention on the range of indices: 1 ≤ i, j, k, · · · ≤ n; n + 1 ≤ α, β, γ , · · · ≤ n + p). It is well known that (cf. [18,19])
ωα = 0,
ωα i =
hαij ωj ,
j
Rijkl = c (δik δjl − δil δjk ) +
α
hαij = hαji ,
εα (hαik hαjl − hαil hαjk ), − →
α where Rijkl are the components of the curvature tensor of M n . The quadratic form h = α,i,j hij ωi ⊗ ωj ⊗ eα and H = n are the second fundamental form and the mean curvature vector of M , respectively. The square length of h and the mean curvature of M n can be expressed as, respectively,
α 2 S= εα (hij ) , α i ,j
α 2 hii ) . εα ( i
1
H =
n
α
1 tr h n
(1)
The covariant derivatives hαijk of hαij satisfy
hαijk ωk = dhαij +
k
hαik ωkj +
k
hαjk ωki −
β
hij ωβα .
β
k
Similarly, we have the second covariant derivatives hαijkl of hαij so that
hαijkl ωl = dhαijk +
l
hαijk ωli +
l
α
hαilk ωlj +
l
α
hαijl ωlk −
The Laplacian △hij of hij is defined by △hij =
− →
k
β
hijk ωβα .
β
l
α
α
hijkk .
− → α
If H is parallel in the normal bundle of M , i.e., ∇ ⊥ H = 0, then M n is called a submanifold with parallel mean curvature n
vector. It is well known that ∇
→ ⊥−
H = 0 if and only if
k
hkkj = 0, ∀ α, j.
n
− →
− →
The submanifold M is called minimal if the mean curvature vector vanishes identically, that is, H = 0. Obviously, if H is not light-like, then M n is minimal if and only if its mean curvature H = 0. A space-like submanifold is said pseudo-umbilical, if it is umbilical with respect to the direction of the mean curvature
− →
− →
− → − →
vector H , i.e., ⟨h(ei , ej ), H ⟩ = ⟨ H , H ⟩⟨ei , ej ⟩, ∀ i, j. n +p As it is known [18,19], a nonzero vector v tangent to Nq (c ) is called space-like, light-like, or time-like (or isotropic) according to whether ⟨v, v⟩ > 0, ⟨v, v⟩ = 0 or ⟨v, v⟩ < 0, respectively. The zero vector is considered to be space-like. n +p 2 2 Denote byα g2˜ the metric on tensor bundle T1 (M ) induced fromn ds¯ (the metric of Nq (c )). Then g˜ (h, h) = α εα i,j (hij ) . According to [20], the second fundamental form of M is called locally space-like, light-like, or time-like, when g˜ (h, h) ≥ 0, g˜ (h, h) = 0 and h ̸= 0, or g˜ (h, h) < 0, respectively. We need the following biharmonic equations due to C.Z. Ouyang, for space-like submanifolds in pseudo-Riemannian n +p space forms Nq (c ). It can be seen as the dual version of [21, Corollary 3.1] for that of submanifolds in Riemannian space forms. n +p
Lemma 2.1 (C.Z. Ouyang, [15]). Let M n be a Riemannian submanifold of pseudo-Riemannian space forms Nq Then it is biharmonic if and only if
εα (2hαiik hαkj + hαkkj hαii ) = 0, ∀ j, α,i,k β β α εβ hii hkj hαjk ) + nc hαii = 0, (hiikk − i ,k
β,j
∀ α.
(c ), 0 ≤ q ≤ p.
(2)
i
3. Pseudo-umbilical biharmonic space-like submanifolds 3.1. When mean curvature vector is light-like We start with the following existence example of biharmonic space-like submanifold with light-like mean curvature vector. Let M 2 be an open portion of the Euclidean plane E2 with Euclidean coordinates (u, v) and φ(u, v) be a smooth
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
function on M 2 such that ∆φ(u, v) ̸= 0 and ∆2 φ(u, v) = 0. Let x : M 2 → E41 be defined by x(u, v) = (φ(u, v), φ(u, v), u, v). Obviously, it is a biharmonic space-like surface with light-like mean curvature vector in E41 . Moreover, it is pseudo-umbilical (see the proof of Theorem 6.1 in [3]). Indeed, we can prove that biharmonic space-like submanifolds with light-like mean curvature vector in pseudon+p Riemannian space forms Nq (c ) are all pseudo-umbilical. Theorem 3.1. Let M n be a biharmonic space-like submanifold with light-like mean curvature vector in pseudo-Riemannian space n+p forms Nq (c ). Then M n is pseudo-umbilical.
− →
− → − →
− →
Proof. Because H is light-like, i.e., ⟨ H , H ⟩ = 0, and H ̸= 0. So we may choose local pseudo-Riemannian orthonormal n +p frame field {ei , eα } on Nq (c ) such that
− →
H = f (en+1 + en+2 ),
(3)
where en+1 is time-like and en+2 is space-like, f is a nonzero function on M n . Thus (3) implies
β hii = nf , β = n + 1, n + 2, i β hii = 0, ∀β > n + 2.
(4)
i
Substituting (4) into the second equation of (2), we obtain, for any α ≥ n + 1,
hαiikk − (nf )
i,k
εn+1 hnkj+1 hαjk +
j,k
εn+2 hnkj+2 hαjk + nc hαii = 0.
j,k
(5)
i
Evaluating (5) for α = n + 1, n + 2, respectively, and subtracting each other, we have
n +1 hkj − hkjn+2
2
= 0.
j ,k
Therefore, n+2 . hnkj+1 = hkj
(6)
Using (3) and (6), it is easy to check that
− →
⟨h(ei , ej ), H ⟩ =
α
hαij eα , f (en+1 + en+2 ) = f (hijn+1 εn+1 + hnij+2 εn+2 ) = 0,
− →
− →
− → − →
together with H being light-like, which implies ⟨h(ei , ej ), H ⟩ = ⟨ H , H ⟩δij . Equivalently, M n is pseudo-umbilical.
3.2. When mean curvature vector is not light-like Now, we will concentrate on the study of pseudo-umbilical biharmonic space-like submanifolds in pseudo-Riemannian n +p
spaces Nq
− → (c ), under the assumption that the mean curvature vector H is either space-like or time-like.
n +p
Proposition 3.2. Let M n be a pseudo-umbilical biharmonic space-like submanifold in pseudo-Euclidean space Eq light-like, then M n is minimal.
− →
. If H is not
Remark 3.1. Indeed, the pseudo-umbilical condition in Proposition 3.2 is essential. For instance, let M 2 be a surface in pseudo-Euclidean space E41 defined by x(u, v) =
a2 6
u , 3
a 2
u ,− 2
a2 6
u + u, v 3
,
2 for some nonzero constant a. It is easy to show that M is a biharmonic space-like surface, and its mean curvature
− → − →
− →
H = |⟨ H , H ⟩| = 12 |a| ̸= 0, which also implies H is not light-like. In fact, M 2 is not a pseudo-umbilical surface (see the proof of Theorem 5.1 in [3]).
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
73
− →
Proof of Proposition 3.2. According to the assumption that H is not light-like, we can choose pseudo-Riemannian ortho-
− →
normal frame field {eA } (A = 1, 2, . . . , n + p) such that H = Hen+1 . Then
tr Hα = 0, α > n + 1, tr Hn+1 = nH ,
(7)
where Hα denotes the matrix (hαij )n× n . Using (7) together with c = 0, the biharmonic equation (2) can be simplified as
n+1 n+1 +1 n+1 + hnkkj hii = 0. ε n+1 2hiik hkj i ,k +1 hniikk − εn+1 hnkj+1 hαjk hnii+1 = 0, ∀α. i ,k
(8)
j
Now, suppose on the contrary that H ̸= 0. Since M n is pseudo-umbilical, we know that hijn+1 = H δij .
(9)
Putting (9) into the first equation of (8) yields H (nH )j = 0,
∀j.
(10)
Meanwhile, taking α = n + 1 in the second equation of (8) and using (9), we have
(nH )kk = εn+1 (n2 H 3 ).
(11)
k n +p
Combining (10) and (11), we obtain that a pseudo-umbilical space-like submanifold M n in Eq
is biharmonic if and only if
(nH )kk = εn+1 (n H ), 2
k
H (nH )j = 0,
3
∀j ,
which implies that H = 0. That is a contradiction.
n +p
Proposition 3.3. Let M n be a pseudo-umbilical biharmonic space-like submanifold in de Sitter space Sq
− →
(c ).
(i) If H is space-like, then either M is minimal, or H = c. − → (ii) It is impossible for H to be time-like. n
2
n+p
Remark 3.2. In fact, let M n be a pseudo-umbilical space-like submanifold with space-like mean curvature vector in Sq If H 2 = c, then in view of (12) below, we know that M n is a proper biharmonic submanifold.
(c ).
− →
Proof. We choose pseudo-Riemannian orthonormal frame field {ei , eα } such that H = Hen+1 . Then (7) holds.
− →
(i) Assume that H ̸= 0 (otherwise, M n is minimal). Similar to the proof of Proposition 3.2, we can obtain that a pseudon +p umbilical space-like submanifold M n in Sq (c ) is biharmonic if and only if
(nH )kk = εn+1 (n2 H ) H 2 − εn+1 c ,
k
H (nH )j = 0,
(12)
∀j,
which implies that H 2 = εn + 1 c .
(13)
− →
Because of H being space-like, i.e., εn+1 = 1, then we have H 2 = c from (13). (ii) Suppose that there exists a pseudo-umbilical biharmonic space-like submanifold M n with time-like mean curvature n+p vector in Sq (c ), then its mean curvature H ̸= 0, and εn+1 = −1. In this case, (13) becomes
(H 2 + c ) = 0, which is a contradiction.
When c < 0, using the similar method as in the proof of Proposition 3.3, we can easily prove the following result. n +p
Proposition 3.4. Let M n be a pseudo-umbilical biharmonic space-like submanifold in anti-de Sitter space Hq
− → (i) If H is space-like, then M n is minimal. − → (ii) If H is time-like, then H 2 = |c |.
(c ).
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
Remark 3.3. Similar to Remark 3.2, we know that a√ pseudo-umbilical space-like submanifold with time-like mean curvature n +p vector in Hq (c ) whose mean curvature equal to |c | is proper biharmonic. Finally, combining Propositions 3.2–3.4, we obtain our main Theorem 1.1 stated in the introduction. 3.3. Applications As an application of Theorem 1.1, we also get several bounds for the scalar curvature of pseudo-umbilical biharmonic n +p space-like submanifold in pseudo-Riemannian space forms Nq (c ). n+p
Proposition 3.5. Let M n be a pseudo-umbilical biharmonic space-like submanifold of Nq of M n .
(c ). Denote by R the scalar curvature
• When c = 0 and h is locally space-like, then R ≤ 0. The equality holds if and only if M n is totally geodesic. • When c > 0 and h is locally space-like, then R ≤ n(n − 1)c. • When c < 0. (i) If h is locally time-like, then R ≥ 2n(n − 1)c. The equality holds if and only if M n is totally umbilical. (ii) If h is locally space-like, then R ≤ n(n − 1)c. The equality holds if and only if M n is totally geodesic. Proof. From Gauss equation, we have n+p
R = n(n − 1)c +
εα
α=n+1
hαii
2
n +p
−
εα
α=n+1
i
(hαij )2 .
(14)
i ,j
Choosing a frame field {ei , eα } such that one of the following holds (cf. [20]): (i) For h being locally time-like,
n+p
h=
hαij ωi ⊗ ωj ⊗ eα ,
εα = −1,
(15)
α=n+q+1 i,j
α
hij = 0,
α = n + 1, . . . , n + q.
(ii) For h being locally space-like,
h= α
n +q
hαij ωi ⊗ ωj ⊗ eα ,
εα = +1,
(16)
α=n+1 i,j
hij = 0,
α = n + q + 1, . . . , n + p.
Making use of (15) and (16) separately, we have
− →
H =
1
n+p
n α=n+q+1
hαii eα ,
for h being locally time-like
(17)
i
and
− →
H =
n+q 1 α hii eα , n α=n+1 i
for h being locally space-like.
− →
(18)
− →
Clearly, (17) implies that H is time-like, while (18) implies that H is space-like. When h is locally time-like, εα = −1 (n + q + 1 ≤ α ≤ n + p). Applying Cauchy inequality, we have from (1) and (17) that n2 H 2 =
n +p
α=n+q+1
hαii
2
i
≤n
n +p
α=n+q+1
(hαii )2 ≤ n i
n +p
(hαij )2 ,
α=n+q+1 i,j
that is, nH 2 ≤
n +p
α=n+q+1 i,j
(hαij )2 .
(19)
J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
75
Similarly, when h is locally space-like, (1) and (18) lead to nH 2 ≤
n+q (hαij )2 .
(20)
α=n+1 i,j
− →
Case 1: c = 0. If h is locally space-like, then H is space-like as we have proved before. Using Proposition 3.2 together with (15), (14) becomes R=−
n +q (hαij )2 .
α=n+1 i,j
Obviously, R ≤ 0 and the equality holds if and only if M n is totally geodesic. − → Case 2: c > 0. If h is locally space-like, then H is space-like (cf. Eq. (18)), together with (14), (16) and (18), we have R = n(n − 1)c +
n+q
α=n+1
hαii
2
−
n +q
(hαij )2 .
α=n+1 i,j
i
Using this equation, Proposition 3.3(i) and (20), we have R ≤ n(n − 1)c, or R ≤ 2n(n − 1)c. Moreover R ≤ n(n − 1)c. Case 3: c < 0. − → (i) If h is locally time-like, then H is time-like (cf. Eq. (17)), together with (14), (15) and (17), we can easily obtain n +p
R = n(n − 1)c −
α=n+q+1
hαii
2
+
i
n +p
(hαij )2 ,
α=n+q+1 i,j
which combining with (19) and Proposition 3.4(ii) implies that R ≥ n(n − 1)c − n2 H 2 + nH 2 = 2n(n − 1)c . Obviously, the equality holds if and only if M n is totally umbilical. (ii) If h is locally space-like, by taking the similar methods as (i), and using (20) and Proposition 3.4(i), we have R ≤ n(n − 1)c. Clearly, the equality holds if and only if M n is totally geodesic. 4. Biharmonic space-like submanifolds with parallel mean curvature vector In [15], C.Z. Ouyang proved the following non-existence results for proper biharmonic submanifolds with parallel mean curvature vector in pseudo-Riemannian space forms. Proposition 4.1 (C.Z. Ouyang, [15]). Let M n be a biharmonic space-like submanifold with parallel mean curvature vector in n+p pseudo-Riemannian space forms Nq (c ). (1) If c = 0, then H = 0. (2) If c > 0 and the mean curvature vector of M n is not space-like, then H = 0. (3) If c < 0 and the mean curvature vector of M n is not time-like, then H = 0. As a supplement of Proposition 4.1, in this section, we will study continuously the same problem in the following two cases: (i) when c > 0 and the mean curvature vector is space-like; (ii) when c < 0 and the mean curvature vector is time-like. More precisely, we prove the following proposition. Proposition 4.2. Let M n be a biharmonic space-like submanifold with parallel mean curvature vector in pseudo-Riemannian n +p space forms Nq (c ).
− → (i) If c ̸= 0, then H is not light-like. − → − → (ii) If c > 0 and H is space-like, then H 2 ≤ c. In particular, if H ̸= 0, then H 2 = c holds if and only if M n is pseudoumbilical. − → (iii) If c < 0 and H is time-like, then H 2 ≤ |c |. The equality holds if and only if M n is pseudo-umbilical. − →
Proof. (i) Suppose on the contrary that there exists such a submanifold with light-like mean curvature vector H , making use of the similar methods as Theorem 3.1, we know that (4)–(6) also hold. Moreover, (5) becomes
(nf )kk + nc (nf ) = 0. k
− →
Because H is parallel, so
(nf )j = 0,
∀ j.
Those two facts lead to c = 0, contradiction.
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
− →
− →
− →
(ii) When H is space-like, choose a local pseudo-orthonormal frame {ei , eα } such that H = Hen+1 , then (7) holds. Since
H is parallel, H is a constant. Using (7) and c > 0, we can simplify the second equation of (2) as
(nH )(tr Hn2+1 − nc ) = 0.
(21)
It follows that H = 0,
tr Hn2+1 = nc .
or
If H = 0, then H 2 ≤ c. If tr Hn2+1 = nc, making use of Cauchy inequality and noticing that (7), we get n2 H 2 =
hnii+1
2
≤ n
(hnii+1 )2 ≤ n (hnij+1 )2 = n2 c ,
i
(22)
i ,j
i
which also implies H 2 ≤ c. When H 2 = c, all equalities in (22) hold. Therefore
hn11+1 = hn22+1 = · · · = hnnn+1 , hijn+1 = 0,
i ̸= j.
This fact together with (7) yields hnij+1 = H δij . In other words, M n is pseudo-umbilical.
− →
Conversely, when H ̸= 0 and M n is pseudo-umbilical, by (21) we obtain that H 2 = c.
− →
(iii) When c < 0 and H is time-like, we can also choose a local pseudo-Riemannian orthonormal frame {ei , eα } such
− →
that H = Hen+1 . The remaining proof is similar to that of (ii). We complete the proof of Proposition 4.2.
Finally, combining Propositions 4.1 and 4.2, we obtain our main Theorem 1.2 stated in introduction. 5. Biharmonic space-like submanifolds with constant mean curvature In this section, we prove firstly a general inequality concerning the mean curvature H and the squared norm S of the n+p second fundamental form of a submanifold M n in pseudo-Riemannian space forms Np (c ) of index p ≥ 1. n +p
Lemma 5.1. Let M n be a biharmonic space-like submanifold in pseudo-Riemannian space forms Np inequality holds 1 2
∆(n2 H 2 ) ≥ (−n2 H 2 )(S + nc ).
(23)
Proof. Multiplying the second equation in (2) by
α
hαii
hαiikk
=
i ,k
i
β
εβ hkj
i
β
j ,k
(c ). Then the following
hαii and taking sum for α yields β
hii
α
i
hαjk
hαii
− nc
α
i
hαii
2
.
(24)
i
On one hand, a direct calculation gives 1 2
∆(n2 H 2 ) =
≥
1 2
∆
α
2
=
α
i
α
hαii
hαii
hαiik
2
+
α
i
hαii
hαiikk
i ,k
i
hαiikk .
(25)
i ,k
i
On the other hand, using Cauchy inequality, notice that εβ = −1, we have
j,k
β
β
εβ hkj
i
β
hii
α
hαjk
i
hαii
=−
j,k
β
β
hkj
β
hii
2
i
β β 2 ≥− (hkj )2 hii = −n2 H 2 S . β
j ,k
β
i
Substituting (24) and (26) into (25), we complete the proof of Lemma 5.1.
As a result of Lemma 5.1, we can easily get the following two conclusions.
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J. Liu et al. / Journal of Geometry and Physics 92 (2015) 69–77
77 n +p
Corollary 5.2. Let M n be a biharmonic space-like submanifold with constant mean curvature in anti-de Sitter space Hp S < n|c |, then M n is minimal.
(c ). If
Proof. Since H is constant, it follows from (23) that n2 H 2 (S − n|c |) ≥ 0, which leads to H = 0 provided that S < n|c |.
Corollary 5.3. Let M n be a proper biharmonic space-like submanifold with constant mean curvature in anti-de Sitter space n+p Hp (c ). If S ≤ n|c |, then S = n|c |. Remark 5.1. Corollary 5.2 generalizes Ouyang’s result [15, Corollary 4.2] to higher codimension: let M n be a biharmonic space-like hypersurface with constant mean curvature in anti-de Sitter space H1n+1 (c ). If S ̸= n|c |, then M n is minimal. Corollary 5.3 can be seen as a gap phenomenon. Remark 5.2. It is easy to see that, when c ≥ 0 and H is constant, (23) holds automatically. We cannot obtain the similar results as Corollaries 5.2 and 5.3 for de Sitter space. Acknowledgments The first author was supported by the National Natural Science Foundation of China (Grant No. 11261051, 11171246), and Fundamental Research Funds of the Gansu Universities (2012). The second and the third authors were supported by the Young Talents Project and the Youth Project of Dingxi Teacher’s College (No. 2012-2017, 1333). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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