Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nn+1 s ðcÞ with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces M2k−1 r in even-dimensional pseudo-Euclidean space E 2k s , M 2k−1 s−1 in even-dimensional de Sitter space S2k s ðcÞðc > 0Þ, and M2k−1 s in even-dimensional anti-de Sitter space H2k s ðcÞðc < 0Þ are minimal.


Introduction
Let N n+1 s ðcÞ be a (n + 1)-dimensional pseudo-Riemannian space form with index 1 ≤ s ≤ n + 1 and constant sectional curvature c. According to c = 0, c > 0, and c < 0, N n+1 s ðcÞ is isometric to pseudo-Euclidean space E n+1 s , de Sitter space S n+1 s ðcÞ, and anti-de Sitter space ℍ n+1 s ðcÞ. Suppose that ϕ : M n r → N n+1 s ðcÞ, r = s − 1, or s be an isometric immersion of a pseudo-Riemannian hypersurface M n r into N n+1 s ðcÞ. The hypersurface M n r is said to be biharmonic if its bitension field τ 2 ðϕÞ vanishes identically, i.e., where τðϕÞ ≔ div ðdϕÞ,R, ∇ ϕ , and ∇ are the curvature tensor of N n+1 s ðcÞ, the induced connection by ϕ on the bundle ϕ −1 TN n+1 s ðcÞ, and the connection of M n r , respectively. If the mean curvature of the hypersurface M n r is zero, then we call M n r as minimal. It is generally known that minimal hypersurfaces are biharmonic ones. Conversely, the natural question is whether any biharmonic hypersurface is minimal.
For biharmonic hypersurfaces in pseudo-Euclidean spaces, there is a conjecture in [2] that every biharmonic hypersurface of pseudo-Euclidean space E n+1 s is minimal. Up to now, this conjecture has been examined for many biharmonic hypersurfaces, such as M 2 r of E 3 s (cf. [2,3]), M 3 1 of E 4 1 (cf. [1]), M 3 2 of E 4 2 (cf. [10]), and M n r in E n+1 s with at most three distinct principal curvatures and diagonalizable shape operator (cf. [5,7]).
When the ambient space is de Sitter space S n+1 s ðcÞ, there are also some papers that studied the above problem. Sasahara in [11] considered biharmonic hypersurfaces M 2 r of S 3 1 ðcÞ and proved that it must be minimal when r = 0, but may not when r = 1. Investigators studied biharmonic hypersurfaces with at most two distinct principal curvatures in S n+1 s ðcÞ whose shape operator is diagonalizable in [6,8] and showed that such hypersurface M n s−1 is minimal, but the hypersurface M n s may not. Naturally, there is a question as to whether any biharmonic hypersurface M n s−1 in de Sitter space S n+1 s ðcÞ is minimal. The situation is quite different when the ambient space is anti-de Sitter space ℍ n+1 s ðcÞ. For biharmonic hypersurface M 2 r of ℍ 3 1 ðcÞ, it must be minimal when r = 1 and may not when r = 0 (cf. [11]). For biharmonic hypersurface M n r with at most two distinct principal curvatures in ℍ n+1 s ðcÞ whose shape operator is diagonalizable, it is minimal when r = s and may not when r = s − 1 (cf. [6,8]). A natural question is whether any biharmonic hypersurface M n s in anti-de Sitter space ℍ n+1 s ðcÞ is minimal. In this paper, we study biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space forms N n+1 s ðcÞ, without the restriction that the shape operator is diagonalizable. We proved such biharmonic hypersurfaces have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces where h is the scalar-valued second fundamental form and A is the shape operator of M n r associated to ξ ! . here A hypersurface M n r of N n+1 s ðcÞ is biharmonic if and only if its mean curvature H satisfies the following two equations (cf. [4]): where here fe 1 , e 2 ,⋯,e n g is a local orthonormal frame of T x M n r . 2.2. The Shape Operator of M n r in N n+1 s ðcÞ. According to [9] (exercise 18, pp. 260-261), the tangent space T x M n r at x ∈ M n r can be expressed as a direct sum of subspaces V k , 1 ≤ k ≤ m, that are mutually orthogonal and invariant under the shape operator A, and each Aj V k (the restriction of A on V k ) has form (a) or (b) as follows.
(a) Aj V k has the form with respect to a basis B k = fu k 1 , u k 2 ,⋯,u k α k g of V k . The inner products of the basis elements in B k are all zero except with respect to a basis inner products of the basis elements in B k are all zero except We denote by t the number of terms Aj V k having form (a). We adjust the order of Journal of Function Spaces With respect to this basis B, the shape operator A of the hypersurface M n r in N n+1 s ðcÞ can be expressed as an almost diagonal matrix: and the inner products of the elements in B are all zero except where Observe the forms (a) and (b); we see that It follows from the form of the shape operator A that M n r has principal curvatures So, under the assumption that M n r has at most two distinct principal curvatures, the shape operator A has the following two possible forms: (I) t = m, i.e., A = diag fA 1 , A 2 ,⋯,A m g, and there are at most two distinct values among fλ 1 , λ 2 ,⋯,λ m g And for the form (II), we have with t Proof. From Section 2, the shape operator A has the form (I) or (II). If A has the form (II), then its eigenvalues are not real and −ðn/2ÞεH is not an eigenvalue. It follows from (5) that ∇H = 0, which tells us H is a constant.

Theorems
For the form (I), if we assume that H is not a constant, then (5) implies that −ðn/2ÞεH is an eigenvalue of the shape operator A. When λ 1 = ⋯ = λ m , then trA = −ðn 2 /2ÞεH. On the other hand, trA = nεH. These two expressions imply H = 0, a contradiction. So, in the following, we need only to discuss the situation where there are two distinct values among fλ 1 ,⋯,λ m g. Expression (5) also informs us that ∇H is an eigenvector of A with corresponding eigenvalue −ðn/2ÞεH. In view of (16), Without loss of generality, we suppose ∇H is in the direction of u 1 α 1 ; it may be a light-like vector or not. We will follow different processes to lead contradictions for these two cases.
First of all, we give a lot of equations deduced from compatibility and symmetry of the connection, as well as the Codazzi equation.
Observe the inner products of the elements in basis B given in Section 2, we can express Since ∇H is in the direction of u 1 α 1 , the above equation implies that Applying compatibility condition to calculate we conclude for , and (19), we easily get We state that in the proof, if not otherwise specified, then for i a in u i a and the connection's coefficients, the ranges of i 3 Journal of Function Spaces and a are as follows: 1 ≤ i ≤ m and 1 ≤ a ≤ α i . For the equations about the connection's coefficients, when a = α i (or a = 1), then the terms about i a+1 (or i a−1 ) disappear. And when b = β j (or b = 1), then the terms about j b+1 (or j b−1 ) and j b+1 (orj b−1 ) disappear.
It follows from the Codazzi equation (3) that for any vector fields X, Y, Z tangent to M n r , Start with this equation, we can get a series of equations about the coefficients of connection. (24), then combining (19), we obtain Applying (21), (22), and (23), we get from the above equation that Note that if α 1 = 1, then (26) tells us nothing.

Journal of Function Spaces
Lemma 2. We have Proof.

Lemma 3. We have
Proof. For 1 ≤ k ≤ m, if α k is an even number, we put with 1 ≤ a ≤ α k /2. And if α k is an odd number, we take and e k ðα k +1Þ/2 = u k ðα k +1Þ/2 . We easily find E k = fe k 1 , e k 2 ,⋯,e k α k g is an orthonormal basis of V k with 1 ≤ k ≤ m. Thus, E = fe k a , 1 ≤ k ≤ m, 1 ≤ a ≤ α k g is an orthonormal basis of T x M n r .
with 1 ≤ α ≤ α k /2, and when α k is an odd number, ∇ e k a e k a H ð Þ Since the above, we obtain from (6) that