Let M r m and N q m + p be pseudo-Riemannian manifolds with indexs r and q ( q ≥ r ≥ 0 ) , respectively, and we consider a smooth map ϕ : M r m → N q m + p . The energy functional is defined by

where e ( ϕ ) = 1 2 | d ( ϕ ) | 2 is called the energy density of ϕ . Critical points of

E ( ϕ ) are called harmonic maps. The theory of harmonic maps has been applied to various fields in differential geometry, we refer to [1][2] for a review.

The study of biharmonic submanifolds began in the mid-1980s, proposed by Chen in his research on finite type submanifolds in Euclidean and pseudo-Euclidean spaces [3]. Meanwhile, In [4] and [5], Jiang introduced the concept of k -harmonic mapping proposed by Eells and Sampson in [6] to study the double harmonic isometric immersion between Riemannian manifolds. Harmonic mappings are characterized by the vanishing of the tension field as the criterion, corresponding to the critical points of the energy functional, and describe the smoothest mapping relationship between manifolds [6], while biharmonic mappings are defined by the vanishing of the biharmonic field and are the critical points of the bienergy functional [7].

In order to understand the geometric characteristics of biharmonic systems, some geometers have begun to focus on studying doubly conservative submanifolds [8]-[11]. For example, the general notion of biconservative submanifolds was introduced in [8]. Aslo, the complete classification of biconservative hypersurfaces in Euclidean spaces with three distinct principal curvatures is obtained by the second named author in [10].

The stress-energy tensor was intitated by G. Y. Jiang in [12] and afterwards developed by E. Loubeau, S. Montaldo and C. Oniciuc in [13], defining the stress-energy tensor S 2 with

Moreover, Jiang [12] demonstrated that φ is biharmonic if and only if it satisfies the Euler-Lagrange equation related to the dual energy functional, i.e., τ 2 ( φ ) = 0 . where τ 2 ( φ ) is the bitension field of φ defined by

where Δ is the Rough-Laplacian. If the condition 〈 τ 2 ( φ ) , d φ 〉 = 0 is satisfied, then φ is called a biconservative mapping. Pseudo-Riemannian manifolds are generalizations of Riemannian manifolds. The condition for a Riemannian submanifold to be biconservative can be used to obtain the condition for a pseudo-Riemannian submanifold to be bi-conservative as follows (cf [8][14])

where D is the normal connection, A ξ is the shape operator with respect to the normal vector field ξ and H is the mean curvature vector filed of M r m . In [11], authors studied geometrical properties of PNMCV surfaces of E 4 and proved that a biharmonic PNMCV surface in E 4 is minimal. Motivated by above paragraphs, in this paper, we will study 3-dimensional biconservative of E s 5 with parallel normal mean curvture vector (PNMCV). we prove

Theorem 1.1.If a proper biconservative submanifold M r 3 ( ‖ g r a d   f ≠ 0 ‖ ) in E s 5 has a diagonalizable shape operator and at most two distinct principalcurvatures in the direction of H , then its scalar curvature is a power-law polynomial in the mean curvature f with non-zero coefficients. Furthermore,the expression for the scalar curvature is derived from the Gauss equation and the results concerning the shape operator.

Theorem 1.2.Let M r 3 be a biconservative submanifold with constant scalar curvature in the pseudo-Euclidean space E s 5 , possessing a parallel normaliazed mean curvature vector field. If,in the direction of f ,it has at most two distinctprincipal curvatures and its shape operator is diagonalizable, then M r 3 necessarily has constant mean curvature.

Remark 1.3. It is easy to see that a PNMC submanifold is CMC if and only if it is PMC. As a corollary of Theorem 1.3, let M r 3 be a biconservative pnmc submanifold with constant scalar curvature in E s 5 . Then M r 3 is PMC provided that it has diagonalizable shape operator with two distinct principal curvatures in the direction of f .

Remark 1.4. Let M r 3 be a biconservative pnmc submanifold with constant scalar curvature in the pseudo-Euclidean space from E s 5 . Assume that M r 3 has two distinct principal curvatures in the direction of f . As an immediate consequence of Theorem 1.3, we know that M r 3 has constant mean curvature.

Let E s n denote the pseudo-Euclidean n-space with the metric tensor g ˜ given by

Let φ : M r 3 → E s 5 be an isometric immersion of an 3-dimensional pseudo-Riemannian manifold M r 3 into a pseudo-Euclidean 5-space. Denote the Levi-Civita connections of M r 3 and E s 3 by ∇ and ∇ ˜ , respectively. Then the Gauss and Weingarten formulae are given by (cf [15][16])

and

respectively, for any vectors X , Y tangent to M r 3 and ξ normal to M r 3 , where h and A ξ are the second fundamental form and the shape of M r 3 along the normal direction ξ , respectively and ∇ ⊥ is the normal connection, It is well known that h and A ξ are related by

If R and R ˜ stand for the curvature tensor of M r 3 and E s 3 respectively,

then, the Codazzi equation ( R ˜ ( X , Y ) Z ) ⊥ = 0 and the Gauss equation

( R ˜ ( X , Y ) Z ) ⊤ = 0 become

where ∇ ˜ h is defined by

where

Let e 1 , e 2 , e 3 , e 4 , e 5 be a pseudo-Euclidean orthonormal field on E s 3 such that e 1 , e 2 , e 3 are tangent to M r 3 and e 4 , e 5 are normal to M r 3 , and we denote the connection forms corresponding to this fram field by ω i j . Then, we have

The mean curvature vector f of M r 3 is defined by

where A 4 = A e 4 . The mean curvature f of M r 3 in E s 5 is expressed as f = 〈 H , H 〉 1 / 2 .

At a point p ∈ M , a 2-dimensional linear subspae π of the tangent space T p M is called a plane section. For a given basis ν , ω of the palne section π , we define a real number by

The plane section π is called nondegenerate if and only if Q ( ν , ω ) ≠ 0 .

Q ( ν , ω ) is positive when g | π is definite, and is negative when g | π is indefinte.

The absolute value ‖ Q ( μ , ν ) ‖ is the square of the area of the parallelogram with

sides μ and ν .

For a nondegenerate plane setion π at p , the number

is independent of the choice of basis ν , ω for π , which is called the setional curvature K ( π ) of π .

According to [17]-[19]:

Lemma 2.1 Let φ : M r 3 → E s 5 be an isometric immersion of an 3-dimensional pseudo-Riemannian manifold M r 3 into a pseudo-Euclidean space. φ is bicon-servative if and only if the equation

is satisfied, where m is the imension of M . By Lemma 2.1, we can obtain

Lemma 2.2 (cf [18]) Let M r 3 be a submanifolds with parallel normalized mean curvature vector field in E s 5 . Then M r 3 is biconservative if and only if the equation holds:

where

Proof. Essentially this lemma is a special case of Lemma 2.2 in [18] We can choose a pesudo-Riemannian orthonormal frame field e 1 , e 2 , e 3 , e 4 , e 5 , such that

e 1 is parallel to ∇ f , e 2 , e 3 are tangent to M r 3 , e 4 = H f , e 5 are normal to

Note that ∇ e i ⊥ e 4 = ∇ e i ⊥ ( H f ) = 0 , i = 1 , 2 , 3 , then it follows form (16) we have

which together with (15) and (16) we have

using (16), we can obtain

Combining with (13), (18) and (19) we can obtain A e 4 ( ∇ f ) = − 3 ε 4 f 2 ( ∇ f ) .

According to [17]-[20]:

Lemma 2.3 Assum that φ has parallel normalized mean curvature vector e 4 .

In this case, the Ricci equation ( R ˜ ( X , Y ) ξ ) ⊤ = 0 yields that all the shape

operators of φ can be diagonalized simulataneously (see [21]). Therefore, by abusung the terminology, we are going to call X as principal direction of φ , if A e 4 X = λ X , where the smooth function λ is going to be called as the corrrs-ponding principal curvature. Note that there exist an orthonormal fram field e 1 , e 2 , e 3 , e 4 , e 5 such that

for some smooth fuctions λ i , μ i satisfying λ 1 + λ 2 + λ 3 = 3 ε 4 f and

μ 1 + μ 2 + μ 3 = 0 . We are going to call a biconservative PNMCV immersion as proper if ‖ ∇ f ‖ does not vanish. Assum that φ is proper biconservative

PNMCV immersion. By calculating ∇ e i ⊥ 〈 e 4 , e 4 〉 = 0 and ∇ e i ⊥ 〈 e 4 , e 5 〉 = 0 , where

i = 1 , 2 , 3 . we obtain

where e 5 is a unit normal vector field orthogonal to e 4 . If e 1 is chosen to be proportional to ‖ ∇ f ‖ , then (14) implies

Lemma 2.4 Let φ : M r 3 → E s 5 be an isometric immersion with two distinct principal curvatures, if φ is proper biconservative PNMCV then there exists an orthonormal frame field e 1 , e 2 , e 3 , e 4 , e 5 such that the shape operators A e 4 has the form

and A e 5 has the form

where g 2 and g 3 are nonzero constants with g 2 + g 3 = 0 , or.

for c 1 and g 2 , g 3 are nonzero constants with g 2 + g 3 = 0 .

Proof. We have from (10) and (16) that

When M r 3 has the same principal curvatures in the direction of H ,

λ 1 = λ 2 = λ 3 = − 3 ε 4 f 2 , from (27), it follows that

which shows f = 0 , it is contradition. There for we are going to consider the case λ 1 ≠ λ 2 = λ 3 . from (20) and (22) we have λ 2 , λ 3 satisfy

Now, we start to derive the explicit expressions of the shape operator A e 5 of M r 3 Combing with (4) and (20) yields, for any i , j = 1 , 2 , 3

which means that

Calculating ( ∇ ˜ e 1 h ) ( e A , e 1 ) = ( ∇ ˜ e A h ) ( e 1 , e 1 ) , for A = 2 , 3 . Uising (6), (9), (22), (29), (30), we get

from (22) and (29) we have

Furthermore, we have

whih implies that

Similary, calculating ( ∇ ˜ e A h ) ( e 1 , e A ) = ( ∇ ˜ e A h ) ( e A , e A ) , from (6), (9), (22), (29), (30), we find

and

This together with (29) and (36) deduces to

Substituting (37) into (35), we have

Furthermore we have

using (21), (16) and (39), we get

case (1) when μ 1 = 0 , then (38) becomes

if μ 2 = μ 3 = 0 , then A e 5 has the form ( I ).

If μ 2 , μ 3 are not equal to 0, Then it follows from (36) that

Integrating (42), we have

where g A are nonzero constents. Substituting (43) into (20) we get

Thus A e 5 takes the form

Case (2) If μ 1 ≠ 0 for A = 2 , 3 , we have form (36) that

where c 1 is a nonzero constant, substituting (45) into (38) we have

By integrating (46). It fllows that

where e 1 ( g A ) = 0 and g A are smooth function. Also g 2 + g 3 = 0 . Thus

In the following, we will prove that g A ( A = 2 , 3 ) are constants.

If g 2 , g 3 are not constants, then we know from (47) that e A ( g 2 ) and e A ( g 3 ) are not equal to 0 for A = 2 , 3 .

Calculating the equation ( ∇ ˜ e 1 h ) ( e 2 , e 3 ) = ( ∇ ˜ e 2 h ) ( e 1 , e 3 ) and

( ∇ ˜ e 1 h ) ( e 3 , e 2 ) = ( ∇ ˜ e 2 h ) ( e 1 , e 2 ) , combing (6), (9), we have

Combing (30) we obtain

Note that λ 2 = λ 3 , then it follows from (46) that

A short calulation from (9) shows that

Then it follows from (8), (9), (34), (51)

Also using Guauss equation we from (30) that

Those two facts shows that

Calculating ( ∇ e 1 e 2 − ∇ e 2 e 1 ) f = [ e 1 , e 2 ] ( f ) = e 1 e 2 ( f ) − e 2 e 1 ( f ) and

( ∇ e 1 e 3 − ∇ e 3 e 1 ) f = [ e 1 , e 3 ] ( f ) = e 1 e 3 ( f ) − e 3 e 1 ( f ) , yeid

Calculating ( ∇ e 1 e 2 − ∇ e 2 e 1 ) ( e 1 f ) = [ e 1 , e 2 ] ( e 1 f ) = e 1 e 2 e 1 ( f ) − e 2 e 1 e 1 ( f ) and ( ∇ e 1 e 3 − ∇ e 3 e 1 ) ( e 1 f ) = [ e 1 , e 3 ] ( e 1 f ) = e 1 e 3 e 1 ( f ) − e 3 e 1 e 1 ( f ) , yeid

Differentiating (54) along e A , for A = 2 , 3 we have

Combing (45), (47), (57), we obtain

Noting that e A ( g A ) ≠ 0 and f ≠ 0 , then it follows from (22) and (56) that c 1 = 0 , which together with (45) proves μ 1 = 0 , a contradiction. There for g 2 and g 3 are constant function.

Proof of Theorem 1.1. we choose a pseudo-Riemannian orthonormal frame field e 1 , e 2 , ⋯ , e n . For i < j , we have from (11) that

Using the Gauss equation, it follows from (30) that

Combing (11), (12), (20) we have

When A e 5 has the form (I), Combining with (24) and (61) we obtain

When A e 5 has the form (II), Combining with (25) and (61) we obtain

where g 2 and g 3 are constants with g 2 + g 3 = 0 .

When A e 5 has the form (III), Combining with (26) and (61) we obtain

where g 2 and g 3 are constants satisfying g 2 + g 3 = 0 . From the above, it follows that R is expressed as a fractional power polynomial in the mean curvature with non-zero coefficients, which completes the proof of Theorem 1.1.

Assuming that f is non-constant, a contradiction can be derived by applying Theorem 1.2, thereby concluding the proof of Theorem 1.2.