A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).


Introduction
Let M n+p (c) (c = 0) be an (n + p)-dimensional real space form with constant sectional curvature c and M n be an n(≥ 2)-dimensional immersed connected submanifold of M n+p (c). Denote by H the mean curvature of M n . The normalized scalar curvature ρ and the normal scalar curvature ρ ⊥ are defined by where R is the curvature tensor of the tangent bundle and R ⊥ is the normal curvature tensor of the normal bundle. In 1999, De Smet et al. [1] proposed the following well-known Normal Scalar Curvature Conjecture or DDVV Conjecture: DDVV Conjecture: (c.f. [1]) Let M n be an n(≥ 2)-dimensional immersed submanifold in a real space form M n+p (c). Then the inequality holds at every point p of M n . The formula (3) is called DDVV inequality. Submanifolds achieving the equality everywhere in (3) are called Wintgen ideal submanifolds which carry interesting geometry and are not classified completely so far, see [2]. In 2007, Dillen et al. [3] transferred the conjecture into an algebraic version inequality: (c.f. [3]) Let B 1 , B 2 , . . . , B p be symmetric (n × n)-matrices with trace zero. If then DDVV Conjecture is true.

Theorem 2.
(c.f. [5]) Let M n be an n(≥ 2)-dimensional compact minimal submanifold in S n+p (c). If then M n is a totally geodesic submanifold S n (c), or one of the Clifford torus M r,n−r (1 ≤ r ≤ n − 1), or the Veronese surface M 2 . Here σ is the squared length of the second fundamental form, and λ 2 is the second largest eigenvalue of the fundamental matrix as stated in Definition 1. The Clifford torus M r,n−r is a Riemannian product of the form S r ( nc r ) × S n−r ( nc n−r ), which is a minimal hypersurface immersed into S n+1 (c).
Besides, it would be interesting and important to study the similar problems in a product space M m (c) × R. Now we use ∂ ∂t to denote the unit R direction and write T for the projection of ∂ ∂t on M. With using the tensor T, Chen and Cui [22] proved the corresponding interesting DDVV type inequality and obtained a pinching theorem in M m (c) × R. Precisely, the authors obtained the following two theorems.
Inspired by the above results, the first author [23] further generalized Chen-Cui's work to a product manifold of a space form and a Euclidean space of higher dimension. Recently, Roth [24] extended Theorem 3 to the case that the ambient space is a Riemannian warped product I × f M m (c) by proving a new DDVV type inequality for submanifolds immersed in I × f M m (c), which is similar to (3) and (6).

Theorem 5.
(c.f. [24]) Let M n be an n-dimensional immersed submanifold in I × f M m (c) (m ≥ n ≥ 2). Then we have Hence, it seems natural and interesting to extend the classical pinching theorems (Theorems 2 and 4) obtained for submanifolds in real space forms, or in the product of a line with a real space form to warped product manifolds. In this article, we prove the following result: Main Theorem: Let M n be an n-dimensional compact minimal submanifold in I × f S m (c) (m ≥ n ≥ 2) with warping function satisfying c − ( f 2 − f f ) = c 1 f 4 for some c 1 > 0 at every t ∈ I. If So, we can see that f = ± c 1 f 4 + c 2 f 2 + c, which is a first-order separable equation. A trivial example can be obtained by taking I as R = (−∞, +∞) and f (t) = 1 in the Main Theorem. Then we recover Theorem 4. Moreover, we can see easily that f = e 2t −C e 2t +C provides a particular solution of the Equation (11) for c = c 1 = 1 and c 2 = −2, where C is a real constant. Another non-trivial example is M = (0, π 2 ) × f S m (c), where f : (0, π 2 ) −→ R + , f (t) = tan t. Needless to say, this function also satisfies (11) for c = c 1 = 1 and c 2 = 2. Hence our theorem can be view as a generalization of Theorem 4.

Preliminaries
Let t be an arc-length parameter of I and ∂ t = ∂ ∂t be the unit vector field tangent to I. We consider the Riemannian warped product M = I × f M m (c) endowed with the Riemannian warped metric defined by where , M denotes the standard Riemannian metric of M m (c) and f is called the warping function of the warped product I × f M m (c). Let M n be an n-dimensional immersed connected submanifold in I × f M m (c) with codimension p = m + 1 − n(≥ 1). We denote by ∇ and ∇ the Riemannian connections of M n and M, respectively. Moreover, we use ∇ ⊥ for the normal connection of M n . Throughout this paper, we will agree on the following index ranges and use the Einstein summation convention unless otherwise stated: 1 ≤ A, B, C, · · · ≤ m + 1; 1 ≤ i, j, k, · · · ≤ n; n + 1 ≤ α, β, γ, · · · ≤ m + 1. denote the Riemannian connection forms in TM and the normal connection forms in T ⊥ M. From Cartan's lemma, we get ii and the mean curvature H = | H|. Let A α be the shape operator with respective to e α . It is well known that h and A are related by h(e i , e j ), e α = A e α e i , e j .
We can certainly assume that λ 1 ≥ λ 2 ≥ · · · ≥ λ p be all the eigenvalues of the fundamental matrix F. In particular, λ 1 and λ 2 are the largest and the second largest eigenvalue of F, respectively.
Obviously, by (13) and the definition of σ, it follows that Recall that (c.f. [25], p. 74) the curvature tensor R of M is given by We write R(e A , e B )e C := R CDAB e D for any e A , e B , e C , e D ∈ T(M). From the properties of curvature tensor we find Similarly, it is convenient to write The first and the second covariant derivatives of h α ij are respectively defined by We also denote Then we have the well-known Codazzi equation and Ricci identity as below: Decompose ∂ t into the tangential and normal parts as follows: Obviously, we have |T| 2 + |N| 2 = |∂ t | 2 = 1. By Gauss-Weingarten formulae one has We define π : to be the projection map, and π(X) := X * = X − X, ∂ t ∂ t to be the orthogonal projection of X to the tangent space TM m (c). Using Proposition 35 of Chapter 7 in [25], it follows that Comparison of (20) and (21) shows that In [26], the authors deduced the structure equations for a semi-Riemannian submanifold immersed into a warped product ±I × f M m k (c), where I ⊆ R and M m k (c) is a semi-Riemannian space form of constant nonzero sectional curvature c and index k. In the Riemannian case, we shall now derive the following structure equations by the moving frame method.

Proof. A direct computation gives
From the expression of the curvature tensor R (c.f. [25], p. 210) one has On substituting these into (26) we have We conclude similarly that and Observe that Likewise, we can deduce that and Combining (30)-(32) with the standard Gauss, Codazzi and Ricci equations gives the proof of Proposition 1.
By Proposition 1, we can obtain a lower bound of |∇h| 2 , which extends Proposition 1 in [27].

Proposition 2. Let M n be an n-dimensional immersed connected submanifold in M
Proof. It follows from the definition of η α ijk and (24) that Squaring the both sides of the above equation, and summing over i, j, k, it turns out that The proof is completed.

Remark 2.
In Proposition 2, suppose that f 2 − f f = c and h α ijk = 0 for any i, j, k, α, then |T| = 0 or |N| = 0, i.e., M n is either contained in a slice M m (c), or ∂ t is everywhere tangent to M n . In the latter case, it is of the form M n = J × f P, where J is an open subinterval of I, P is an (n − 1)-dimentional submanifold of M m (c) and f is the restriction of f on I.

Proof of Main Theorem
In this section, we will give the proof of our Main Theorem. We shall adopt the similar procedure as in the proof of [22]. Firstly, we proceed to calculate 1 2 ∆ A α 2 . For arbitrary fixed α, we conclude from (17) and (18) that Note that By (22) one has Putting these expressions into (35) gives Using Codazzi Equation (24) again, we have First observe that Then by (22) again, we have By Ricci Equation (25) we have Substituting (39)-(41) into (34) gives Assume that M n is a minimal submanifold satisfying c − ( f 2 − f f ) = c 1 f 4 at each t ∈ I, where c 1 is a positive constant. We thus obtain and then (42) becomes

Proof of Main Theorem:
For fixed x ∈ M n , we can take a local coordinate system U; (x 1 , x 2 , . . . , x n ) and a suitable local orthonormal normal frame around x such that F(x) = diag λ 1 , λ 2 , . . . , λ p with λ 1 = · · · = λ r > λ r+1 ≥ · · · ≥ λ p . For simplicity, we denote A n+α briefly by A α for 1 ≤ α ≤ p, and define A r+1 = 0 if r = p. Furthermore, for arbitrary integer q ≥ 2, we define to be a smooth function on M n . A straightforward calculation gives rise to, at x ∈ M, where the covariant derivative of S αβ is given by Here and subsequently, we use λ α instead of λ α−n for n + 1 ≤ α ≤ n + p.
Applying the Cauchy-Schwarz inequality to (44) gives Combining the definition of f q and (43), we obtain Observe that Using the Cauchy-Schwarz inequality we get Applying the above estimates to (46), it follows that It is straightforward to show that Following Lu's paper ( [5], Lemma 2), we see that From Lemma 1 in [20] we have for any 1 ≤ α, β ≤ p. By applying (48) and (49) we conclude where the last step is based on Substituting the above estimates into (47), we thus obtain For fixed x ∈ M, we see that As q → ∞, applying the above estimates to (53), we have It follows from the hypothesis (9) that One thus obtains A 1 2 = 0, or σ + λ 2 = c 1 f 2 n − (2n + 1)|T| 2 − n| f | f . The first case shows that M is a totally geodesic submanifold in I × f S m (c). Since σ = 0, we infer that h α ijk = 0 for any i, j, k, α. It follows immediately from Remark 2 and the compactness of M that M lies in a slice S m (c). This is the desired conclusion.
Author Contributions: writing-review and editing, X.Z.; supervision, Z.H. Both authors equally contributed to this manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.