Classification of f-biharmonic submanifolds in Lorentz space forms

Li Du

doi:10.1515/math-2021-0084

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Classification of f-biharmonic submanifolds in Lorentz space forms

Li Du

Published/Copyright: December 31, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 . When f is a function, we completely classify such pseudo-umbilical submanifolds.

Keywords: Lorentz space forms; f-biharmonic submanifolds; parallel normal mean curvature vector field; the shape operator; pseudo-umbilical

MSC 2010: 53C40

1 Introduction

The theory of harmonic maps between manifolds is a very active and rich subject as an extension of important concepts of geodesics, minimal surfaces, and has been studied extensively by many mathematicians and physicists, we refer to [1,2,3, 4,5] for details. As a generalization of harmonic maps, Eells and Lemaire in [1] suggested the ideas of k-harmonic maps. Later, Jiang in [6] defined the biharmonic maps or 2-harmonic maps (i.e., its bitension field vanishes identically) in the sense of Eells and Lemaire.

During the last decade, there have been tremendous developments in the study of biharmonic maps, which can be divided in two main research directions. On the one side is the analytic aspect from the point of view of partial differential equation (PDE): biharmonic maps are solutions of a fourth order strongly elliptic semilinear PDE, we refer to [7,8] for a review. On the other side, the differential geometric aspect has driven attention to biharmonic submanifolds as those submanifolds whose isometric immersions are biharmonic maps. The study of such submanifolds was initiated by Jiang as applications of biharmonic maps (cf. [6,9]), also independently by Chen (cf. [10]) in his study on finite type submanifolds, and had become a very dynamic research subject in modern differential geometry. We refer to [9,11,12, 13,14] for a review, [15,16] with references therein for recent progress.

As a generalization of biharmonic submanifolds, η-biharmonic submanifolds with η being a constant (i.e., its bitension field and tension field satisfy a linear relation, see Section 2.3 for details) are concerned by many geometers, and important progress has been made. For example, Ferrández and Lucas in [17] classified completely η-biharmonic surfaces of Lorentz spaces L 3 . In [18], Ferrández and Lucas classified η-biharmonic hypersurfaces in high-dimensional Lorentz spaces L n + 1 under the assumption that the minimal polynomial of the shape operator is at most of degree two. Most recently, Du (cf. [19]) obtained several full classification results of η-biharmonic surfaces in non-flat Lorentz 3-space forms and some examples of such nonminimal surfaces.

In [20], Arvanitoyeorgos et al. proved that η-biharmonic hypersurfaces of pseudo-Euclidean spaces E s 4 with diagonalizable shape operator have constant mean curvature, and the same conclusion holds for Lorentz hypersurfaces in L 4 (cf. [21]), hypersurfaces of index 2 in E 2 4 (cf. [22]), and hypersurfaces in E s n + 1 with certain geometric properties (cf. [23] with references therein), respectively. More general, it was shown in [24] that the conclusion remains true for η -biharmonic hypersurfaces with certain geometric conditions in pseudo-Riemannian space forms. We need to point out that the conclusion also holds for the corresponding surfaces or hypersurfaces in [17,18,19].

In this paper, we attempt to study f-biharmonic submanifolds (instead of hypersurfaces) in Lorentz space forms, where f is a function. The outline of the paper is as follows. In Section 2, we recall some basic notions and formulas, and different forms of the shape operator of submanifolds in Lorentz space forms. Meanwhile, we give a more precise statement of f-biharmonic submanifolds in pseudo-Riemannian manifolds and the relationship between biharmonic submanifolds and f-biharmonic submanifolds. In Section 3, we study f-biharmonic submanifolds with parallel normalized mean curvature vector field from two aspects: When f is a constant, under the assumption that the minimal polynomial of the shape operator is at most of degree two, we prove that submanifolds have parallel mean curvature vector field, which means that the conclusion in [11] holds for such η -biharmonic submanifolds, and obtain an upper bound of that curvature. When f is a function, we classify completely pseudo-umbilical submanifolds.

2 Preliminaries

2.1 Notions and formulas

Let N 1 n + p ( c ) be an ( n + p ) -dimensional pseudo-Riemannian manifold of constant curvature c ( c ∈ { − 1 , 0 , 1 } ) , of index 1, which we call a Lorentz space form. According to whether c = 1 , c = 0 or c = − 1 , it is called a de Sitter space S 1 n + p ( 1 ) , a Lorentz space L n + p or an anti-de Sitter space H 1 n + p ( − 1 ) . The curvature tensor R ˜ of N 1 n + p ( c ) is given by

(2.1) R ˜ ( X , Y ) Z = c ( ⟨ Y , Z ⟩ X − ⟨ X , Z ⟩ Y ) .

Let M r n be a submanifold with signature ( r , n − r ) in a Lorentz space form N 1 n + p ( c ) . Let ∇ and ∇ ˜ denote the Levi-Civita connections of M r n and N 1 n + p ( c ) , respectively. For any tangent vector fields X , Y , and a unit normal vector field ξ of M r n in N 1 n + p ( c ) , the Gauss and Weingarten formulas are given by (cf. [25] or [26])

∇ ˜ X Y = ∇ X Y + h ( X , Y ) , ∇ ˜ X ξ = − A ξ X + D X ξ ,

where h is the second fundamental form of M r n , A ξ is the shape operator with respect to ξ , and D is the normal connection of M r n . In general, the shape operator A ξ is not diagonalizable, and A ξ and h are related by

(2.2) ⟨ A ξ X , Y ⟩ = ⟨ h ( X , Y ) , ξ ⟩ .

The mean curvature vector field H → of M r n is defined by H → = 1 n trace h . The mean curvature H of M r n in N 1 n + p ( c ) is expressed as H = ∣ ⟨ H → , H → ⟩ ∣ .

For the second fundamental form h , the covariant derivative of h is defined by

(2.3) ( ∇ ˜ X h ) ( Y , Z ) = D X h ( Y , Z ) − h ( ∇ X Y , Z ) − h ( Y , ∇ X Z ) .

The Codazzi equation is given by

( ∇ ˜ X h ) ( Y , Z ) = ( ∇ ˜ Y h ) ( X , Z ) .

Denote the curvature tensor R of M r n by

(2.4) R ( X , Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X , Y ] Z .

The Gauss equation is given by

R ( X , Y , Z , W ) = c ( ⟨ Y , Z ⟩ ⟨ X , W ⟩ − ⟨ X , Z ⟩ ⟨ Y , W ⟩ ) − ⟨ h ( Y , Z ) , h ( X , W ) ⟩ + ⟨ h ( X , Z ) , h ( Y , W ) ⟩ .

If we denote the curvature tensor with the normal connection R D of M r n by

(2.5) R D ( X , Y ) ξ = D X D Y ξ − D Y D X ξ − D [ X , Y ] ξ ,

then, for any normal vector fields ξ and η of M r n , the Ricci equation is given by

R ( X , Y , ξ , η ) = ⟨ R D ( X , Y ) ξ , η ⟩ = ⟨ [ A ξ , A η ] X , Y ⟩ .

2.2 The shape operator of submanifolds

When a submanifold M r n is Riemannian, it is well known that the shape operator A α of M r n is always diagonalizable, then we can choose a suitable pseudo-Riemannian orthonormal frame basis { e 1 , e 2 , … , e n } , such that A α = diag { λ 1 α , λ 2 α , … , λ n α } , α = n + 1 , n + 2 , … , n + p .

When a submanifold M r n is Lorentzian, the shape operators A α ( α = n + 1 , n + 2 , … , n + p ) of M r n may not be diagonalizable. According to [26, p. 261] (or [25, pp. 33–34]), we know that A α ( α = n + 1 , n + 2 , … , n + p ) can be put into one of the following four forms, where D k is k × k diagonal,

(a) = λ α 0 1 λ α D n − 2 ; ( b ) = a α b α − b α a α D n − 2 ( b α ≠ 0 ) ; (c) = D n ; (d) = λ α 0 1 0 λ α 0 0 1 λ α D n − 3 .

Denote by, for k , l = 1 , 2 , … , n ,

δ k l = 1 , k = l ; 0 , k ≠ l .

The forms (a) and (b) are represented with respect to a pseudo-orthonormal frame basis { v 1 , v 2 , … , v n } satisfying

⟨ v 1 , v 1 ⟩ = ⟨ v 2 , v 2 ⟩ = 0 , ⟨ v 1 , v i ⟩ = ⟨ v 2 , v i ⟩ = 0 , ⟨ v 1 , v 2 ⟩ = − 1 , ⟨ v i , v j ⟩ = δ i j , for i = 3 , 4 , … , n ; j = 2 , 3 , … , n ,

while the forms (c) and (d) are represented with respect to a local pseudo-Riemannian orthonormal frame basis { e 1 , e 2 , … , e n } satisfying

⟨ e 1 , e 1 ⟩ = − 1 , ⟨ e 1 , e i ⟩ = 0 , ⟨ e i , e j ⟩ = δ i j , for i , j = 2 , 3 , … , n .

When the minimal polynomial of the shape operator A α is at most of degree two, combining with the four forms of A α : (a), (b), (c), and (d), a straightforward algebraic computation can prove that A α has one of the following four forms (cf. [18]):

( a ˜ ) = λ α 0 1 λ α λ α I n − 2 ; ( b ˜ ) = a α b α − b α a α ( b α ≠ 0 ) ; ( c ˜ ) = λ α I n ; ( d ˜ ) = λ α I n 1 μ α I n − n 1 ,

where I k is the unit matrix, and n 1 and n − n 1 are multiplicities of λ α and μ α in the form ( d ˜ ) , respectively.

2.3 The f-biharmonic submanifolds in pseudo-Riemannian manifolds

Let M r n and N q n + p be pseudo-Riemannian manifolds of dimensions n and ( n + p ) , respectively, and ϕ : M r n → N q n + p be an isometric immersion with the mean curvature vector field H → . Denote by ∇ ϕ the induced connection by ϕ on the bundle ϕ − 1 T N q n + p .

Harmonic isometric immersions ϕ : M r n → N q n + p between two pseudo-Riemannian manifolds are defined as critical points of the energy functional

E ( ϕ ) = 1 2 ∫ M r n ⟨ d ϕ , d ϕ ⟩ v g .

The corresponding Euler-Lagrange equation for E ( ϕ ) is given by the vanishing of the tension field

(2.6) τ ( ϕ ) = tr ∇ d ϕ = n H → .

We say that ϕ is biharmonic if it is a critical point of the bienergy functional

E 2 ( ϕ ) = 1 2 ∫ M r n ⟨ τ ( ϕ ) , τ ( ϕ ) ⟩ v g .

For the biharmonic isometric immersion ϕ , the bitension field τ 2 ( ϕ ) satisfies the associated Euler-Lagrange equation

τ 2 ( ϕ ) = − n ( Δ ϕ H → + trace R ˜ ( d ϕ , H → ) d ϕ ) = 0 ,

where Δ ϕ ≔ − trace ( ∇ ϕ ∇ ϕ − ∇ ∇ ϕ ) , which states the fact that M r n is a biharmonic submanifold if and only if its bitension field τ 2 ( ϕ ) vanishes identically (cf. [6,9,16] for details). This, together with the well-known fact that a submanifold is minimal if and only if the isometric immersion that defined the submanifold is harmonic, implies that a minimal submanifold is always a biharmonic one.

The energy functional E 21 η ( ϕ ) of ϕ : M r n → N q n + p between pseudo-Riemannian manifolds:

E 21 η ( ϕ ) = E 2 ( ϕ ) + η E ( ϕ ) ,

is called the ( 2 , 1 , η ) -energy functional of ϕ (cf. [27] or [28] for q = 0 , and [29] for q ≠ 0 ), and we say that ϕ is η -biharmonic or ( 2 , 1 , η ) -harmonic if it is a critical point of the ( 2 , 1 , η ) -energy functional. The Euler-Lagrange equation of E 21 η ( ϕ ) is

τ 2 ( ϕ ) = η τ ( ϕ ) .

In particular, when N q n + p is the pseudo-Euclidean space E q n + p , τ 2 ( ϕ ) = η τ ( ϕ ) is equivalent to (cf. [20])

Δ H → = η H → ,

with Δ be the Laplace operator of M r n .

If an isometric immersion ϕ : M r n → N q n + p is η -biharmonic, then M r n is called an η -biharmonic submanifold in N q n + p , which is named by Chen submanifolds with proper mean curvature vector when N q n + p = E q n + p .

More general, a submanifold is called a f-biharmonic submanifold if the isometric immersion ϕ satisfies τ 2 ( ϕ ) = f τ ( ϕ ) , where f is a function, which is different from Ou’s notion of f-biharmonic submanifolds (cf. [30]). It is obvious to see that minimal and biharmonic submanifolds (i.e., 0-biharmonic submanifolds) must be f-biharmonic.

Finally, we derive the equations for M r n of N q n + p to be f-biharmonic.

We choose a suitable local pseudo-Riemannian orthonormal frame field { e 1 , e 2 , … , e n + p } on N q n + p , and a long computation shows that the bitension field of M r n is given by (cf. [12,15,25])

(2.7) τ 2 ( ϕ ) = − n Δ D H → − n ∑ i = 1 n ε i h ( A H → ( e i ) , e i ) − n ∑ i = 1 n ε i A D e i H → ( e i ) − n ∑ i = 1 n ε i ( ∇ e i A H → ) ( e i ) − n ∑ i = 1 n ε i R ˜ ( e i , H → ) e i ⊥ − n ∑ i = 1 n ε i R ˜ ( e i , H → ) e i ⊤ ,

where

(2.8) Δ D = − ∑ i = 1 n ε i ( D e i D e i − D ∇ e i e i ) .

Combining with (2.6) and (2.7), and using the elementally argument, we obtain that M r n is f-biharmonic if and only if the following two equations hold:

(2.9) − Δ D H → − ∑ i = 1 n ε i h ( A H → ( e i ) , e i ) − ∑ i = 1 n ε i R ˜ ( e i , H → ) e i ⊥ = f H → , n ∇ ⟨ H → , H → ⟩ + 4 ∑ i = 1 n ε i A D e i H → ( e i ) + ∑ i = 1 n ε i R ˜ ( e i , H → ) e i ⊤ = 0 .

3 Main results and their proofs

A submanifold in a pseudo-Riemannian space form N q n + p ( c ) with constant sectional curvature c is said to have parallel normalized mean curvature vector field if it has nowhere zero mean curvature and the unit vector field in the direction of the mean curvature vector field is parallel in the normal bundle (cf. [10]), i.e., D ( H → / H ) = 0 .

In this paper, we investigate f-biharmonic submanifolds M r n with parallel normalized mean curvature vector field in Lorentz space forms N 1 n + p ( c ) .

3.1 Characterization of f-biharmonic submanifolds with f = constant

In this section, we consider such f-biharmonic submanifolds with f being a constant. Denote by η = f . We choose a local orthonormal frame field { e 1 , e 2 , … , e n + p } on N 1 n + p ( c ) such that e 1 , … , e n are tangent to M r n and e n + 1 = H → H , e n + 2 , … , e n + p are normal to M r n , then

(3.1) H → = H e n + 1 .

According to the definition of the mean curvature vector field and (2.2), we get

(3.2) H = ε n + 1 n trace A n + 1 ,

and

(3.3) trace A β = 0 , ∀ β > n + 1 .

Note that

(3.4) D e i e n + 1 = D e i ( H → / H ) = 0 , i = 1 , 2 , … , n ,

then (2.5) implies that

R D ( e i , e j ) e n + 1 = 0 , i , j = 1 , … , n ,

together with the Ricci equation, we have

(3.5) A n + 1 A a = A a A n + 1 , a = n + 2 , … , n + p .

While, it follows from (3.1) that, for any i = 1 , 2 , … , n ,

(3.6) D e i H → = D e i ( H e n + 1 ) = e i ( H ) e n + 1 .

In order to prove our main results, we give the following important lemma.

Lemma 3.1

Let M r n be a submanifold with parallel normalized mean curvature vector field in a Lorentz space form N 1 n + p ( c ) . Then M r n is η -biharmonic if and only if

(3.7) − Δ H − ε n + 1 H ∑ i n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ + n c H = η H , ∑ i n ε i 2 A n + 1 ( ∇ H ) + n ε n + 1 H ∇ H = 0 , ∑ i = 1 n ⟨ A n + 1 ( e i ) , A β ( e i ) ⟩ = 0 , ∀ β > n + 1 ,

where

∇ H = ∑ i n ε i e i ( H ) e i , Δ H = − ∑ i = 1 n ε i ( e i e i − ∇ e i e i ) H .

Proof

Using (2.8), (3.1), (3.4), and (3.6), a straightforward calculation gives

Δ D H → = Δ H e n + 1 ,

and

∑ i = 1 n ε i h ( A H → ( e i ) , e i ) = H ε n + 1 ∑ i n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ e n + 1 + H ∑ β = n + 2 n + p ε β ∑ i n ε i ⟨ A n + 1 ( e i ) , A β ( e i ) ⟩ e β .

Moreover, it follows from (2.1) that

∑ i = 1 n ε i R ˜ ( e i , H → ) e i ⊥ = − n c H e n + 1 , ∑ i = 1 n ε i R ˜ ( e i , H → ) e i ⊤ = 0 .

Putting those equations into the first equation of (2.9) yields

− Δ H − ε n + 1 H ∑ i n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ + n c H = η H ,

and

∑ i n ε i ⟨ A n + 1 ( e i ) , A β ( e i ) ⟩ = 0 , ∀ β > n + 1 .

Then, it follows from (3.1) and (3.6) that

∇ ⟨ H → , H → ⟩ = 2 ε n + 1 H ∇ H , ∑ i = 1 n ε i A D e i H → ( e i ) = A n + 1 ( ∇ H ) .

Taking into account the above two equations, the second equation of (2.9) becomes

2 A n + 1 ( ∇ H ) + n ε n + 1 H ∇ H = 0 ,

and we complete the proof of Lemma 3.1.□

As it is well known that non-minimal submanifolds with parallel mean curvature vector field (i.e., D H → = 0 ) must have parallel normalized mean curvature vector field, the converse is not true in general. The counterexamples are given in [10].

Using (3.6), we know that the submanifold with parallel normalized mean curvature vector field has parallel mean curvature vector field provided that its mean curvature is a constant. Next, we will prove that the mean curvature H of M r n is a constant when A α has the form ( a ˜ ) , ( b ˜ ) , ( c ˜ ) , or ( d ˜ ) , respectively.

Suppose on the contrary that H is not a constant, then ∇ H ≠ 0 on a certain open subset U . Using the second equation of (3.7), we obtain that one of the principal curvatures of A n + 1 is − n ε n + 1 H 2 .

Note that the forms ( a ˜ ) and ( b ˜ ) are not diagonalizable, and ( c ˜ ) and ( d ˜ ) are diagonalizable, then we show that the assumption runs into a contradiction from the following two sides.

Case 1

The shape operator of M r n is not diagonalizable.

Proposition 3.1

Let M r n be an η -biharmonic submanifold with parallel normalized mean curvature vector field in N 1 n + p ( c ) . Assume that the shape operator has the form ( a ˜ ) or ( b ˜ ) , then M r n has parallel mean curvature vector field.

Proof

When the shape operator has the form ( a ˜ ) , we choose a pseudo-orthonormal frame basis { u 1 , … , u n } with ⟨ u 1 , u 2 ⟩ = ⟨ u i , u i ⟩ = 1 for i = 2 , … , n , and the others be zero, such that (cf. [24]), the shape operator A n + 1 of M r n takes the form

λ n + 1 0 1 λ n + 1 λ n + 1 I n − 2 .

Now, we construct a pseudo-Riemannian orthonormal basis { e 1 , e 2 , … , e n } from the pseudo-orthonormal basis { u 1 , u 2 , … , u n } such that

e 1 = u 1 − u 2 2 , e 2 = u 1 + u 2 2 , e i = u i , for i = 3 , 4 , … , n ,

where e 1 is time-like, and the others are space-like. Then A n + 1 takes the following form with respect to this new basis,

λ n + 1 + 1 2 1 2 ⋯ 0 − 1 2 λ n + 1 − 1 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ λ n + 1 .

Note that

λ n + 1 = − 1 2 n ε n + 1 H .

Together with (3.2), we obtain that ( n + 2 ) H 2 = 0 , which shows that H = 0 . This contradicts the assumption that M r n has parallel normalized mean curvature vector field.

When the shape operator A 3 has the form ( b ˜ ) , it is easy to compute that the shape operator of M r 2 has two adjoint imaginary principal curvatures, which contradicts with the fact that one principal curvature of A 3 is ( − ε 3 H ) .□

Case 2

The shape operator of M r n is diagonalizable.

Proposition 3.2

Let M r n be an η -biharmonic submanifold with parallel normalized mean curvature vector field in N 1 n + p ( c ) . Assume that the shape operator has the form ( c ˜ ) or ( d ˜ ) , then M r n has parallel mean curvature vector field.

Proof

When the shape operator has the form ( c ˜ ) or ( d ˜ ) , then, correspondingly, A n + 1 takes the form

(III) = λ n + 1 I n or (IV) = λ n + 1 I n 1 μ n + 1 I n − n 1 .

Without loss of generality, we denote

(3.8) λ n + 1 = − 1 2 n ε n + 1 H .

When the shape operator A n + 1 has the form (III), H = 0 by (3.2) and (3.8), a contradiction.

When the shape operator A n + 1 has the form (IV), we choose a pseudo-Riemannian orthonormal frame field { e 1 , e 2 , … , e n } on U so that e 1 is parallel to ∇ H , e 1 , e 2 , … , e n 1 are principal directions corresponding to λ n + 1 ; e n 1 + 1 , e n 1 + 2 , … , e n are principal directions corresponding to μ n + 1 , i.e.,

(3.9) A n + 1 ( e i ) = λ i n + 1 e i , i = 1 , 2 , … , n ,

where λ 1 n + 1 = λ 2 n + 1 = ⋯ = λ n 1 n + 1 = λ n + 1 , λ n 1 + 1 n + 1 = λ n 1 + 2 n + 1 = ⋯ = λ n n + 1 = μ n + 1 . Then it follows from (3.2) that

(3.10) ε n + 1 n H = n 1 λ n + 1 + ( n − n 1 ) μ n + 1 .

Set

(3.11) ∇ e i e j = ∑ k = 1 n ε k Γ i j k e k , i , j = 1 , 2 , … , n .

Applying compatibility condition to calculate e i ⟨ e j , e k ⟩ , we conclude

(3.12) Γ i j k = − Γ i k j .

We consider an arbitrary integral curve γ of e 1 and denote by H ′ , H ″ the first and the second derivatives of H along this curve.

Upon the hypothesis that H is not a constant, in order to simplify the proving process of Proposition 3.2, we prove the following four lemmas which are the four key steps in the remaining proof of Proposition 3.2.□

Lemma 3.2

We have, for distinct i , j = 1 , 2 , … , n ,

(3.13) e i ( λ j n + 1 ) e j + ∑ l = 1 n ε l ( λ j n + 1 − λ l n + 1 ) Γ i j l e l = e j ( λ i n + 1 ) e i + ∑ l = 1 n ε l ( λ i n + 1 − λ l n + 1 ) Γ j i l e l .

Proof

Using (2.3) and the symmetry of the shape operator, we derive, for distinct i , j , k = 1 , 2 , … , n ,

(3.14) ⟨ ( ∇ ˜ e i h ) ( e j , e k ) , e n + 1 ⟩ = ⟨ D e i h ( e j , e k ) − h ( ∇ e i e j , e k ) − h ( e j , ∇ e i e k ) , e n + 1 ⟩ = ⟨ D e i h ( e j , e k ) , e n + 1 ⟩ − ⟨ A n + 1 ( ∇ e i e j ) , e k ⟩ − ⟨ A n + 1 ( ∇ e i e k ) , e j ⟩ ,

and

(3.15) ⟨ ( ∇ ˜ e j h ) ( e i , e k ) , e n + 1 ⟩ = ⟨ D e j h ( e i , e k ) , e n + 1 ⟩ − ⟨ A n + 1 ( ∇ e j e i ) , e k ⟩ − ⟨ A n + 1 ( ∇ e j e k ) , e i ⟩ .

Using (2.2) and (3.6), a direct computation gives

(3.16) ⟨ D e i h ( e j , e k ) , e n + 1 ⟩ = ∇ e i ⟨ h ( e j , e k ) , e n + 1 ⟩ = ∇ e i ⟨ A n + 1 ( e j ) , e k ⟩ , ⟨ A n + 1 ( ∇ e i e k ) , e j ⟩ = ∇ e i ⟨ A n + 1 ( e j ) , e k ⟩ − ⟨ ∇ e i ( A n + 1 ( e j ) ) , e k ⟩ ,

and

(3.17) ⟨ D e j h ( e i , e k ) , e n + 1 ⟩ = ∇ e j ⟨ h ( e i , e k ) , e n + 1 ⟩ = ∇ e j ⟨ A n + 1 ( e i ) , e k ⟩ , ⟨ A n + 1 ( ∇ e j e k ) , e i ⟩ = ∇ e j ⟨ A n + 1 ( e i ) , e k ⟩ − ⟨ ∇ e j ( A n + 1 ( e i ) ) , e k ⟩ .

Substituting (3.16) and (3.17) into (3.14) and (3.15), respectively, and according to the Codazzi equation ( ∇ ˜ e i h ) ( e j , e k ) = ( ∇ ˜ e j h ) ( e i , e k ) , we find

∇ e i ( A n + 1 ( e j ) ) − A n + 1 ( ∇ e i e j ) = ∇ e j ( A n + 1 ( e i ) ) − A n + 1 ( ∇ e j e i ) .

Together with (3.9), (3.11), and (3.12), we obtain (3.13) and complete the proof of Lemma 3.2.□

Lemma 3.3

Assume that H is not a constant, then we have

(3.18) 3 n ε n + 1 H = 2 ( n − 1 ) μ n + 1 .

Proof

We claim that λ α n + 1 ≠ λ 1 n + 1 , α = 2 , … , n 1 . Suppose on the contrary α > 1 , then there exists a fix index α ( 1 < α ≤ n 1 ) such that A n + 1 ( e α ) = λ n + 1 e α . Taking i = 1 and j = α in (3.13), we have

e 1 ( λ α n + 1 ) e α + ∑ l = 1 n ε l ( λ α n + 1 − λ l n + 1 ) Γ 1 α l e l = e α ( λ 1 n + 1 ) e 1 + ∑ l = 1 n ε l ( λ 1 n + 1 − λ l n + 1 ) Γ α 1 l e l .

Taking the scalar product with e α , we obtain that e 1 ( λ n + 1 ) = 0 , that is,

(3.19) H ′ = 0 .

Note that e 1 ‖ ∇ H , then

(3.20) e i ( H ) = 0 , i = 2 , 3 , … , n ,

thus

(3.21) ∇ H = ∑ i = 1 n ε i e i ( H ) e i = ε 1 H ′ e 1 ,

which together with (3.19) shows that ∇ H = 0 on U , which is a contradiction.

With n 1 = 1 , (3.8) and (3.10) lead to (3.18), which completes the proof of Lemma 3.3.□

Lemma 3.4

We have, for α , β = 2 , 3 , … , n ,

(3.22) ( n + 2 ) Γ α 1 α H = − 3 ε α H ′ .

(3.23) Γ β 1 α = 0 , β ≠ α .

(3.24) Γ 11 α = 0 .

Proof

Putting i = 1 and j = α in (3.13), we arrive at

(3.25) e 1 ( λ α n + 1 ) e α + ∑ l = 1 n ε l ( λ α n + 1 − λ l n + 1 ) Γ 1 α l e l = e α ( λ 1 n + 1 ) e 1 + ∑ l = 1 n ε l ( λ 1 n + 1 − λ l n + 1 ) Γ α 1 l e l ,

and taking the scalar product with e α , we obtain (3.22) from (3.18).

Putting i = 1 and j = α in (3.13) again, and taking the scalar product with e β ( β ≠ α ) and e 1 , respectively, we have from (3.20) that (3.23) and (3.24) hold.

We complete the proof of Lemma 3.4.□

Lemma 3.5

Assume that H is not a constant, then we obtain

(3.26) C 1 H H ″ + C 2 ( H ′ ) 2 = C 3 H 2 + C 4 H 4 ,

(3.27) C ˜ 1 H H ″ + C ˜ 2 ( H ′ ) 2 = C ˜ 3 H 2 + C ˜ 3 H 4 ,

where

C 1 = 4 ( n − 1 ) ( n + 2 ) , C 2 = − 12 ( n − 1 ) 2 , C 3 = − ε 1 ( n − 1 ) ( n + 2 ) ( n c − η ) , C 4 = ε 1 ε n + 1 ( n + 2 ) ( n + 8 ) n 2 ,

C ˜ 1 = 12 ( n − 1 ) ( n + 2 ) , C ˜ 2 = − 12 ( n − 1 ) ( n + 5 ) , C ˜ 3 = 4 ε 1 ( n − 1 ) ( n + 2 ) 2 c , C ˜ 4 = − 3 ε 1 ε n + 1 ( n + 2 ) 2 n 2 .

Proof

It follows from (3.8) and (3.18) that

(3.28) 4 ( n − 1 ) ∑ i = 1 n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ = ( n + 8 ) n 2 H 2 .

By using (3.20), (3.22), and (3.24), it is straightforward to verify

(3.29) Δ H = − ε 1 H ″ + 3 ( n − 1 ) ε 1 ( n + 2 ) H ( H ′ ) 2 .

Putting (3.28) and (3.29) into the first equation of (3.7) yields (3.26).

On the other hand, set A β ( e 1 ) = ∑ i = 1 n λ i β e i , β = n + 2 , … , n + p , where λ i β s are functions on U , then it follows from (3.9) that

A n + 1 A β ( e 1 ) = λ n + 1 e 1 + μ n + 1 ∑ j = 2 n e j , A β A n + 1 ( e 1 ) = λ n + 1 ∑ i = 1 n λ i β e i ,

together with (3.5), show that A β ( e 1 ) = λ 1 β e 1 , β = n + 2 , … , n + p . Also, the third equation of (3.7) and (3.9) lead to

λ n + 1 ε 1 ⟨ A β ( e 1 ) , ( e 1 ) ⟩ + μ n + 1 ∑ j = 2 n ε j ⟨ A β ( e j ) , ( e j ) ⟩ = 0 .

By (3.3), we find from the above two equations that λ 1 β = 0 . Those facts show that

(3.30) A β ( e 1 ) = 0 , β = n + 2 , … , n + p .

Note that

h ( e i , e j ) = ∑ β = n + 1 n + p ε β ⟨ A β ( e i ) , e j ⟩ e β , i , j = 1 , 2 , … , n .

Then, we infer from (3.30) that

(3.31) h ( e 1 , e α ) = 0 , h ( e 1 , e 1 ) = ε 1 ε n + 1 λ n + 1 e n + 1 , ⟨ h ( e 1 , e 1 ) , h ( e α , e α ) ⟩ = ε 1 ε α ε n + 1 λ n + 1 μ n + 1 , α = 2 , … , n .

Making use of (3.11), (3.12), (3.23), and (3.24), it follows from (2.4) that

⟨ R ( e 1 , e α ) e α , e 1 ⟩ = − e 1 ( Γ α 1 α ) − ε α ( Γ α 1 α ) 2 .

According to the Gauss equation, and combining with (3.22), (3.24), (3.31), we prove that (3.27) holds.

We complete the proof of Lemma 3.5.□

Coming back to the proof of Proposition 3.2. Multiplying (3.27) by 3 ε 1 ( n − 1 ) and (3.26) by ( n + 5 ) , subtracting the results, we get

(3.32) 8 ( n − 1 ) ( n − 4 ) H ″ = 4 ε 1 ( ( n − 1 ) ( n + 5 ) ( n c − η ) + ( n − 1 ) 2 ( n + 2 ) c ) H − ε 1 ε n + 1 ( 3 ( n − 1 ) ( n + 2 ) + ( n + 5 ) ( n + 8 ) ) n 2 H 3 .

When n = 4 (since M r n has two distinct principal curvatures, n > 1 ), (3.32) can be rewritten as

H 2 = ε n + 1 1 4 c − 1 24 η ,

which means that H is a constant along integral curve γ , then e 1 ( H ) = 0 on U , which together with (3.21) shows that ∇ H = 0 , a contradiction.

When n ≠ 4 , then if H ′ = 0 , then it follows from (3.32) that e 1 ( H ) = 0 . Together with (3.21) proves that Δ H = 0 on U , a contradiction occurs. Otherwise, multiplying (3.32) by H ′ , and integrating the result along γ gives

(3.33) 48 ( n − 1 ) ( n − 4 ) ( H ′ ) 2 = 24 ε 1 ( n − 1 ) ( ( n + 5 ) ( n c − η ) + ( n − 1 ) ( n + 2 ) c ) H 2 − 3 ε 1 ε n + 1 ( 3 ( n − 1 ) ( n + 2 ) + ( n + 5 ) ( n + 8 ) ) n 2 H 4 .

Also, multiplying (3.27) by ( − ε 1 ) , and adding it to (3.26) leads to

(3.34) 48 ( n − 1 ) ( n − 4 ) ( H ′ ) 2 = 24 ε 1 ( n − 1 ) ( n + 2 ) ( 2 ( 2 n + 1 ) c − 3 η ) H 2 − 12 ε 1 ε n + 1 ( n + 2 ) ( n + 5 ) n 2 H 4 .

It is easy to check that ε 1 ε n + 1 ( 3 ( n − 1 ) ( n + 2 ) + ( n + 5 ) ( n + 8 ) ) n 2 ≠ 4 ε 1 ε n + 1 ( n + 2 ) ( n + 5 ) n 2 , thus, (3.33) and (3.34) show that H is a constant. It is impossible.

We complete the proof of Proposition 3.2.□

Combining with Propositions 3.1 and 3.2, we have

Theorem 3.1

Let M r n be a submanifold with parallel normalized mean curvature vector field and the minimal polynomial of the shape operator being at most of degree two in N 1 n + p ( c ) . If M r n is η -biharmonic, then it has parallel mean curvature vector field.

Using Theorem 3.1, we obtain the following.

Theorem 3.2

Let M r n be an η -biharmonic submanifold with parallel normalized mean curvature vector field and the minimal polynomial of the shape operator being at most of degree two in N 1 n + p ( c ) , then H ≤ max ∣ c − η n ∣ , ( c − η 2 ) + b 2 , where b is a nonzero constant.

Proof

According to Theorem 3.1, we know that H is a nonzero constant. Then it follows from the first equation in (3.7) that

(3.35) ∑ i n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ = ε n + 1 ( n c − η ) .

Moreover, we note that A n + 1 has the following four forms:

(I) = λ n + 1 0 1 λ n + 1 λ n + 1 I n − 2 ; (II) = a n + 1 b n + 1 − b n + 1 a n + 1 ( b n + 1 ≠ 0 ) ; (III) = λ n + 1 I n ; (IV) = λ n + 1 I n 1 μ n + 1 I n − n 1 .

For the form (I), we can easily obtain
∑ i n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ = n ( λ n + 1 ) 2 ,
which together with (3.2) and (3.35), we find
(3.36) H 2 = ε n + 1 c − η n .

Since ε n + 1 = 1 , (3.36) shows that H 2 = c − η n and η < n c .

For the form (II), since ε 1 ε 2 = − 1 , a direct calculation then gives

∑ i 2 ε i ⟨ A 3 ( e i ) , A 3 ( e i ) ⟩ = 2 ( ( a n + 1 ) 2 − ( b n + 1 ) 2 ) .

Then we have from (3.35) that

(3.37) ( a n + 1 ) 2 − ( b n + 1 ) 2 = ε 3 ( c − η 2 ) .

Note that a n + 1 = H , we know that b n + 1 is a nonzero constant, denoted by b . Together with ε 3 = 1 , we have from (3.37) that H 2 = ( c − η 2 ) + b 2 .

For the form (III) or (IV), we have from (3.35) that

(3.38) ∑ i n ( λ i n + 1 ) 2 = ε n + 1 ( n c − η ) ,

where λ i n + 1 s are the principal curvatures of A n + 1 .

We claim that n c ≠ η . If this is not in the case, then (3.2) and (3.38) lead to H = 0 , a contradiction. So η > n c or η < n c .

When η < n c , then ε n + 1 = 1 , i.e., H → is spacelike, and

(3.39) ∑ i n ( λ i n + 1 ) 2 = ( n c − η ) .

If the shape operator of M r n is the form (III), then, according to the definition of mean curvature, (3.39) leads to
H 2 = c − η n .
If the shape operator of M r n is the form (IV), then, using Cauchy inequality, and combining with (3.2), (3.9), and (3.39), we obtain
( n H ) 2 = ∑ i n λ i n + 1 2 < n ∑ i n ( λ i n + 1 ) 2 = n ( n c − η ) ,
which shows that H 2 < c − η n .

Those two cases show that H ≤ c − η n .

When η > n c , then ε n + 1 = − 1 , i.e., H → is timelike, and

∑ i n ( λ n + 1 ) i 2 = − ( n c − η ) .

Using similar discussion of η < n c , we know that H ≤ − c − η n .

Summing up, we obtain that H ≤ max c − η n , c − η 2 + b 2 . Thus, the proof of Theorem 3.2 is complete.□

3.2 Classification of f-biharmonic submanifolds

Using similar methods to that in the proof of Lemma 3.1, we have from (2.9) that M r n is f-biharmonic with f being a function if and only if

(3.40) (i) − Δ H − ε n + 1 H ∑ i n ε i ⟨ A n + 1 ( e i ) , A n + 1 ( e i ) ⟩ + n c H = f H , (ii) ∑ i n ε i ⟨ A n + 1 ( e i ) , A β ( e i ) ⟩ = 0 , ∀ β > n + 1 , (iii) 2 A n + 1 ( ∇ H ) + n ε n + 1 H ∇ H = 0 .

A nonminimal submanifold is called pseudo-umbilical, if it is umbilical with respect to the direction of H → (cf. [25, p. 63]), i.e.,

(3.41) ⟨ A H → ( X ) , Y ⟩ = ⟨ H → , H → ⟩ ⟨ X , Y ⟩ .

In this section, we will classify completely pseudo-umbilical f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms. More precisely,

Theorem 3.3

Let M r n be a pseudo-umbilical submanifold with parallel normalized mean curvature vector field in Lorentz space L n + p . Assume that M r n is f-biharmonic, then f is a nonzero constant. Furthermore,

If f > 0 , then H → is timelike with H = f n , and M r n is a minimal submanifold in hyperbolic H n + p − 1 x → 0 , − f n ;
If f < 0 , then H → is spacelike with H = − f n , and M r n is a minimal submanifold in pseudo-hypersphere S 1 n + p − 1 x → 0 , − f n ,

where x → 0 is a constant vector in L n + p .

Proof

We choose a local orthonormal frame field { e 1 , e 2 , … , e n + p } on L n + p such that e 1 , … , e n are tangent to M r n , and e n + 1 = H → H , e n + 2 , … , e n + p are normal to M r n , then (3.1), (3.2), and (3.3) hold. Since M r n is pseudo-umbilical,

(3.42) A n + 1 = ε n + 1 H I .

We claim that H is a constant. Suppose on the contrary that H is not constant, then we have from (iii) in (3.40) that − n ε n + 1 H 2 is the principal curvature of M r n . This together with (3.42) which shows that H = 0 , a contradiction to the hypothesis that M r n is nonminimal. Then it follows from (i) in (3.40) that

(3.43) H 2 = − ε n + 1 f n ,

which implies that f is a nonzero constant. In the following, we study f > 0 or f < 0 .

If f > 0 , we have from (3.43) that ε n + 1 = − 1 , i.e., H → is timelike, and H 2 = f n , then (3.6) implies that D H → = 0 . Note that M r n is pseudo-umbilical, then we conclude from [25, Proposition 3.9] that M r n is a minimal submanifold in hyperbolic H n + p − 1 x → 0 , − f n with x → 0 being a constant vector.

Conversely, using Proposition 3.9 in [25, p. 63], we know that M r n has parallel mean curvature vector field, i.e., D H → = 0 , and

(3.44) A H → = ⟨ H → , H → ⟩ I .

Since H 2 = f n (a nonzero constant), it is not difficult to check that (i) and (iii) in (3.40) hold. Also, (3.3) and (3.44) lead to

∑ i = 1 n ε i ⟨ A β ( e i ) , A n + 1 ( e i ) ⟩ = ε n + 1 H ∑ i = 1 n ε i ⟨ A β ( e i ) , e i ⟩ = 0 , β > n + 1 ,

i.e., (ii) in (3.40) holds. Thus, M r n is f-biharmonic.

If f < 0 , it follows from (3.43) that ε n + 1 = 1 , i.e., H → is spacelike and H 2 = − f n , then (3.6) implies that D H → = 0 . Note that M r n is pseudo-umbilical, then we know from [25, Proposition 3.9] that M r n is a minimal submanifold in pseudo-hypersphere S 1 n + p − 1 x → 0 , − f n with x → 0 being a constant vector.

Conversely, using Proposition 3.9 in [25, p. 63], we know that D H → = 0 and A H → = ⟨ H → , H → ⟩ I . Note that H is a nonzero constant, then, be similar to the discussion of Case f > 0 , we prove that (3.40) holds, i.e., M r n is f -biharmonic.

We complete the proof of Theorem 3.3.□

When N 1 n + p ( c ) = S 1 n + p ( 1 ) , we can prove the following.

Theorem 3.4

Let M r n be a pseudo-umbilical submanifold with parallel normalized mean curvature vector field in de-Sitter space S 1 n + p ( 1 ) . Assume that M r n is f-biharmonic, then f is a constant which is not equal to n . Furthermore,

If f > n , then H → is spacelike with H = − ( n − f ) n , and M r n lies in a non-totally geodesic, totally umbilical hypersurface of S 1 n + p ( 1 ) as a minimal submanifold;
If f < n , then H → is timelike with H = n − f n , and M r n lies in a flat totally umbilical hypersurface of S 1 n + p ( 1 ) as a minimal submanifold.

Proof

Using the similar processing to the proof of Theorem 3.3, we obtain that f is a constant, which is not equal to n , and satisfies

(3.45) H 2 = − ε n + 1 ( n − f ) n .

If f > n , it follows from (3.45) that ε n + 1 = 1 , i.e., H → is a spacelike vector field, and H 2 = − ( n − f ) n , then it follows from (3.6) that D H → = 0 , which together with the hypothesis that M r n is pseudo-umbilical, we know from [25, Proposition 3.10] that M r n lies in a non-totally geodesic, totally umbilical hypersurface of S 1 n + p ( 1 ) as a minimal submanifold.

Conversely, we know from Proposition 3.10 in [25] that A H → = ⟨ H → , H → ⟩ I and D H → = 0 . Applying the same arguments as in the proof of Theorem 3.3, we know that M r n is f-biharmonic.

If f < n , we have from (3.45) that ε n + 1 = − 1 , i.e., H → is timelike, and H 2 = n − f n . By the similar way followed in the proof of Case f > n , we know from [25, Corollary 3.2] that M r n lies in a flat totally umbilical hypersurface of S 1 n + p ( 1 ) as a minimal submanifold.

We complete the proof of Theorem 3.4.□

Similarly, we have

Theorem 3.5

Let M r n be a pseudo-umbilical submanifold with parallel normalized mean curvature vector field in anti de-Sitter space H 1 n + p ( − 1 ) . Assume that M r n is f-biharmonic, then f is a constant which is not equal to − n . Furthermore,

If f > − n , then H → is spacelike with H = n + f n , and M r n lies in a flat totally umbilical hypersurface of H 1 n + p ( − 1 ) as a minimal submanifold;
If f < − n , then H → is timelike with H = − ( n + f ) n , and M r n lies in a non-totally geodesic, totally umbilical hypersurface of H 1 n + p ( − 1 ) as a minimal submanifold.

Remark 3.1

According to Theorems 3.3–3.5, we know from (3.6) that a pseudo-umbilical f-biharmonic submanifold with parallel normalized mean curvature vector field in Lorentz space forms has parallel mean curvature vector field.

Acknowledgements

The author would like to thank Dr. Juan Zhang for helpful discussions and valuable suggestions. Also, the author is grateful to reviewers for their numerous suggestions to improve the original version of the manuscript.

Funding information: This work was supported by the Natural Science Foundation of Chongqing (No. cstc2019jcyj-msxmX0172), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJQN201901128), and the Scientific Research Starting Foundation of Chongqing University of Technology (No. 2017ZD52).
Conflict of interest: Author states no conflict of interest.

References

[1] J. Eells and L. Lemaire , Selected Topics in Harmonic Maps, CBMS, vol. 50, American Mathematical Society, Rhode Island, 1983. 10.1090/cbms/050Search in Google Scholar

[2] J. E. Marsden and E. J. Tipler , Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Report. 66 (1980), no. 3, 109–139, https://doi.org/10.1016/0370-1573(80)90154-4 .10.1016/0370-1573(80)90154-4Search in Google Scholar

[3] X.-H. Mo , Harmonic maps from Finsler spaces, Illinois J. Math. 45 (2001), no. 4, 1331–1345, https://doi.org/10.1215/ijm/1258138069 .10.1215/ijm/1258138069Search in Google Scholar

[4] Z. Sinaei , Convergence of harmonic maps, J. Geom. Anal. 26 (2016), no. 1, 529–556, https://doi.org/10.1007/s12220-015-9561-2. Search in Google Scholar

[5] A. P. Whitman , R. J. Knill , and W. R. Stoeger , Some harmonic maps on pseudo-Riemannian manifolds, Int. J. Theo. Phys. 25 (1986), no. 10, 1139–1153, https://doi.org/10.1007/bf00671689. Search in Google Scholar

[6] G.-Y. Jiang , 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389–402. Search in Google Scholar

[7] P. Strzelecki , On biharmonic maps and their generalizations, Calc. Var. 18 (2003), 401–432, https://doi.org/10.1007/s00526-003-0210-4 . 10.1007/s00526-003-0210-4Search in Google Scholar

[8] H. Urakawa , Biharmonic maps on principal G-bundles over complete Riemannian manifolds of nonpositive Ricci curvature, Michigan Math. J. 68 (2019), no. 1, 19–31, https://doi.org/10.1307/mmj/1547089468. Search in Google Scholar

[9] G.-Y. Jiang , 2-harmonic isometric immersions between Riemannian manifolds, Chinese Ann. Math. Ser. A 7 (1986), no. 2, 130–144. Search in Google Scholar

[10] B.-Y. Chen , Surfaces with parallel normalized mean curvature vector, Monatsh. Math. 90 (1980), no. 3, 185–194, https://doi.org/10.1007/bf01295363. Search in Google Scholar

[11] A. Arvanitoyeorgos , F. Defever , G. Kaimakamis , and V. J. Papantoniou , Biharmonic Lorentz hypersurfaces in E14 , Pacific J. Math. 229 (2007), no. 2, 293–305, https://doi.org/10.2140/pjm.2007.229.293. Search in Google Scholar

[12] R. Caddeo , S. Montaldo , and C. Oniciuc , Biharmonic submanifolds of S3 , Int. J. Math. 12 (2001), no. 8, 867–876, https://doi.org/10.1142/S0129167X01001027. Search in Google Scholar

[13] B.-Y. Chen and S. Ishikawa , Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), no. 2, 323–347, https://doi.org/10.2206/kyushumfs.45.323 .10.2206/kyushumfs.45.323Search in Google Scholar

[14] L. Du and J. Zhang , Biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms, Bull. Malays. Math. Sci. Soc. 42 (2019), no. 4, 1469–1484, https://doi.org/10.1007/s40840-017-0556-y. Search in Google Scholar

[15] Y.-X. Dong and Y.-L. Ou , Biharmonic submanifolds of pseudo-Riemannian manifolds, J. Geom. Phys. 112 (2017), 252–262, https://doi.org/10.1016/j.geomphys.2016.11.019. Search in Google Scholar

[16] T. Sasahara , Biharmonic submanifolds in nonflat Lorentz 3-space forms, Bull. Aust. Math. Soc. 85 (2012), no. 3, 422–432, https://doi.org/10.1017/s0004972711002978. Search in Google Scholar

[17] A. Ferrández and P. Lucas , On surfaces in the 3-dimensional Lorentz-Minkowski space, Pacific J. Math. 152 (1992), no. 1, 93–100, https://doi.org/10.2140/pjm.1992.152.93. Search in Google Scholar

[18] A. Ferrández and P. Lucas , Classifying hypersurfaces in the Lorentz-Minkowski space with a characteristic eigenvector, Tokyo J. Math. 15 (1992), no. 2, 451–459, https://doi.org/10.3836/tjm/1270129470. Search in Google Scholar

[19] L. Du , Classification of η -biharmonic surfaces in non-flat Lorentz space forms, Mediterr. J. Math. 15 (2018), no. 5, 203, https://doi.org/10.1007/s00009-018-1250-5. Search in Google Scholar

[20] A. Arvanitoyeorgos , F. Defever , and G. Kaimakamis , Hypersurfaces of Es4 with proper mean curvature vector, J. Math. Soc. Japan 59 (2007), no. 3, 797–809, https://doi.org/10.2969/jmsj/05930797. Search in Google Scholar

[21] A. Arvanitoyeorgos , G. Kaimakamis , and M. Magid , Lorentz hypersurfaces in E14 satisfying ΔH→=αH , Illinois J. Math. 53 (2009), no. 2, 581–590, https://doi.org/10.1215/ijm/1266934794 .10.1215/ijm/1266934794Search in Google Scholar

[22] A. Arvanitoyeorgos and G. Kaimakamis , Hypersurfaces of type M23 in E24 with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106, https://doi.org/10.1016/j.geomphys.2012.09.011 .10.1016/j.geomphys.2012.09.011Search in Google Scholar

[23] J.-C. Liu and C. Yang , Hypersurfaces in Esn+1 satisfying ΔH→=λH→ with at most two distinct principal curvatures, J. Math. Anal. Appl. 451 (2017), no. 1, 14–33, https://doi.org/10.1016/j.jmaa.2017.01.090. Search in Google Scholar

[24] L. Du , J. Zhang , and X. Xie , Hypersurfaces satisfying τ2(ϕ)=ητ(ϕ) in pseudo-Riemannian space forms, Math. Phys. Anal. Geom. 20 (2017), no. 2, 17, https://doi.org/10.1007/s11040-017-9248-y. Search in Google Scholar

[25] B.-Y. Chen , Pseudo-Riemannian Geometry, δ-Invariants and Applications, World Scientific, Hackensack, NJ, 2011. 10.1142/8003Search in Google Scholar

[26] B. O’Neill , Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983. Search in Google Scholar

[27] H. I. Eliasson , Introduction to global calculus of variations, Glob. Anal. Appl. 2 (1974), 113–135. Search in Google Scholar

[28] E. Loubeau and S. Montaldo , Biminimal immersions, Proc. Edinb. Math. Soc. 51 (2008), no. 2, 421–437, https://doi.org/10.1017/s0013091506000393. Search in Google Scholar

[29] B.-Y. Chen , Total Mean Curvature and Submanifolds of Finite Type: 2nd Edition, World Scientific, Hackensack, NJ, 2015. 10.1142/9237Search in Google Scholar

[30] Y.-L. Ou , On f -biharmonic maps and f -biharmonic submanifolds, Pacific J. Math. 271 (2014), no. 2, 461–477, https://doi.org/10.2140/pjm.2014.271.461. Search in Google Scholar

Received: 2020-10-13

Revised: 2021-06-28

Accepted: 2021-07-19

Published Online: 2021-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0084

Keywords for this article

Lorentz space forms; f-biharmonic submanifolds; parallel normal mean curvature vector field; the shape operator; pseudo-umbilical

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