Abstract
In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold in the hyperbolic space satisfy various extrinsic restrictions, then has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.
1. Introduction and Main Results
For any Riemannian manifold , it is well known that any integral homology class in which is nontrivial is correlated to integral stable currents. This result was initially proved by Federer and Fleming [1]. Utilizing the method of variational calculus for the geometric measure concept combining with the method of Federer and Fleming, Lawson and Simons [2] obtained the optimization for the second fundamental form, which leads to the vanishing homology in a range of intermediate dimensions and the nonexistence of stable currents in the submanifold in the simply connected space form, and obtained the key theorem of that paper.
Theorem 1 (see [2–4]). Suppose is a compact submanifold of dimension in the space form of constant curvature . If the strict inequality for the second fundamental form is satisfied, then has no stable -currents with the vanished pth homology group, i.e.,where and is the signature of constant curvature .
The geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4–17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1). For example, as an application of the Ricci curvature and the ambient manifold is an Euclidean space, Vlachoas [18] proved that a compact oriented submanifold of dimension in Euclidean space of dimension satisfies the pinching condition , in which is any unit vector, is the shape operator regarding the mean curvature , and is a constant such that if is even and if is odd. Therefore, has no stable currents. Moreover, is homeomorphic to . Moreover, using Theorem 1 in [19], it was found that if a compact oriented submanifold of dimension in space form satisfied the second fundamental form pinching condition , for any integer in which and is a constant sectional curvature, then the th homology groups are vanishing, , for all , and if the fundamental group is finite and simply connected, then is homeomorphic to . Using Theorem 1, Xu and Gu [16] extended the pinching condition in terms of the Ricci curvature and showed that in satisfied ; then, is homeomorphic to . Motivated by the nonexistence of stable submanifolds or stable currents, a number of topological properties have been studied by many authors [3, 4, 14, 15, 17, 19–24] inspired by Theorem 1.
Inspired by the aforementioned results, we want to obtain some similar results of warped product submanifold theory where the second fundamental form pinching condition shall be replaced by the warping function. Using Theorem 1, we now give the first main result of this note.
Theorem 2. Let be an isometric embedding from a compact warped product submanifold into an -dimensional hyperbolic space in which the base manifold is minimal in and the following inequalityhold, where and are the gradient and Laplacian of the function , respectively. Then, we have the following:(a)The warped product submanifold does not have any stable integral -currents.(b)The th integral homology groups of with integer coefficients are vanished, which means(c)If , then the fundamental group is null, i.e., . Moreover, is a simply connected warped product manifold.
Motivated by the geometric rigidity (Theorem 2), the second goal of this approach is to prove a new vanishing theorem for compact warped product submanifolds in terms of the Ricci curvature and using the eigenvalue of Laplacian of the warping function. In particular, we can give the following theorem.
Theorem 3. Under the assumption of Theorem 2 and if the warping function is an eigenfunction of Laplacian of corresponding to the first positive eigenvalue and satisfying the strict inequalityfor integral Hessian tensor of the warping function and the integral Ricci curvature along the gradient , then we have the following:(a)The warped product submanifold does not have any stable integral -currents.(b)The the integral homology groups of with integer coefficients are vanished; that is,(c)The fundamental group is null, i.e., . Moreover, is a simply connected warped product manifold.
Now, we give a direct application of Theorem 3.
Theorem 4. Assume that is an isometric embedding from a compact warped product submanifold into an -dimensional hyperbolic space satisfying the following inequality:Then, statements (a), (b), and (c) in Theorem 2 hold.
Another interesting result obtained from Theorem 4 is the following:
Corollary 1. Under the same assumption as Theorem 4 and if with the following holds,then, statements (a), (b), and (c) in Theorem 2 are satisfied.
Remark 1. Theorem 2 is the main vanishing homology theorem for a compact warped product submanifold with no need for to be simply connected. Moreover, our result is of significance due to involving the new pinching conditions in terms of the warping function, the integral of the squared norm of the Hessian tensor, the integral Ricci curvature, and the first nontrivial eigenvalue of the warped function.
2. Preliminaries
Let denote the hyperbolic space with dimension and constant sectional curvature . We use the fact that has a canonical isometric embedding in Lorentz–Minkowski space which is the vector space with the metricwhere are canonical coordinates in . Therefore, the hyperbolic space with negative constant curvature is
Thus,, where is a tangent bundle of and is the Riemannian curvature tensor of . This means that is a manifold of constant sectional curvature −1.
Assume that admits isometric immersion into a Riemannian manifold with induced metric . Then, the formulais the Gauss equation for in which , and and are the curvature tensors on and , respectively. The mean curvature vector for a local orthonormal basis on is defined by
In addition, we set
In this connection, we shall define the scalar curvature of Riemannian submanifold , which is considered as a Riemannian intrinsic invariant, at some in as follows:where . The first equality (15) is equivalent to the following equation which will be frequently used in subsequent proofs:
In a similar manner, the scalar curvature of an plane is defined as
If the plane sections are spanned by and at , we give denotations and , respectively, for the sectional curvature of the Riemannian manifold and submanifold , which are considered as the extrinsic and intrinsic sectional curvature of the span at . Using Gauss (12) and (15), we have
On the contrary, the conception of warped product manifolds was originally introduced by Bishop and O’Neill [25] for manifolds of negative curvature. Assume that and are two Riemannian manifolds with Riemannian metrics and , respectively. Assume that is a differentiable function on . A warped product manifold is with and the Riemannian metric . Assume that is a warped product manifold. Hence, and , we attain that
Furthermore, is the gradient of , given by
Thus, from [26], we havewhere is a Hessian tensor of . Let be a local orthonormal basis of vector field ; thus, the squared norm of the gradient of the positive differential function for an orthonormal basis is
Now, if we replace in (20) by , we getand Laplacian of is defined as
Remark 2. It should be noted that we consider the opposite sign of Chen [27] of Laplacian of ; that is, .
Moreover, because unit vector fields and are tangent to and , respectively, we get
Let us sum up over the vector fields such thatwhich leads to
3. Proof of Main Results
We also need to use the following method which is an important tool in proving our result. In the first case, we assume that the warped product submanifold is isometrically immersed in a hyperbolic space ; we will give the proof of our main results.
3.1. Proof of Theorem 2
Let and , and consider and to be orthonormal frames of and , respectively. Then, from Gauss equation (12) for the standard hyperbolic space , we have
From in (21), we derive
Thus, from (28) and (29), we derive
Computing the Laplacian on , one obtains
Since is totally geodesic in , , and by utilizing the definition of the warped product, we obtain
Multiplying the above equation by , we get
After some computations, we find that
Thus, from (30) and (34), one obtains
Next, using Gauss equation (12) for hyperbolic space , we find that
By rewriting the above equation for and utilizing (13) and (21), we attain
Using (18) and (27) in the above equation, we derive
Thus, using the curvature equation and the sphere and rearranging the last equation, we attain
After some rearrangements of the above equation, we get
Utilizing the binomial theorem and the fact that the base of a warped product manifold is minimal, we can readily obtain that
Utilizing the assumption of the theorem and since is minimal, the fifth term of the right-hand side (RHS) in equation (41) is identically zero, and the first term of the left-hand side is equal to the seventh term of the RHS. Hence, we get
Then, from equation (11), one obtains
This follows that
From our assumption (3) and (45), we obtain
Applying Theorem 1 for constant holomorphic sectional curvature , we obtain that there are no stable -currents in and , which completes the proof of (a) and (b) of the theorem. In the other part, from (45), substituting , we have
If the pinching condition (22) for and holds, then we get
Then, there are no stable 1-currents in and . Let us assume that . The compactness property of ; it follows from the classical theorem, using the results of Cartan and Hadamard, which states that there is a minimal closed geodesic in any nontrivial homotopy class in , and this leads to a contradiction. Therefore, = 0. This is the third part of the theorem. If the finite fundamental group is null for any Riemannian manifold, this Riemannian manifold is simply connected. As a result, is simply connected.
3.2. Proof of Theorem 3
If , that is, is the first eigenfunction of Laplacian of corresponding to the first nonzero eigenvalue of h. Recall the Bochner formula for a differentiable function defined on a Riemannian manifold (see, e.g., [28]) which states that the following formulaholds. Then, by integrating the above equation and using Stokes’ theorem, we get
Now, using and making some rearrangements in equation (50), we derive
By assumption of the theorem, (5) holds, then we have
Then, substituting equation (52) in (51), we getwhich implies that
Now, using in the left-hand side of equation (54), we arrive at
Then, the above equation gives us
Then, using the above equation and also Theorem 2, we get the conclusion of our theorem.
3.3. Proof of Theorem 4
As we know that is -minimal compact warped product submanifold, from the Gauss equation, one obtainswhich implies the following:
Taking into account that is a minimal submanifold and using the argument of the Ricci curvature for the hyperbolic space, we get
The above equation yields that
Using equation (60), we get
Plugging the above equation in (51), we obtain
If our assumption (1.4) is satisfied, then
Rewrite the above equation using :
Inserting the above equation in (62), we derive thatwhich implies the following:
Thus, we get the conclusion of our theorem.
3.4. Proof of Corollary 1
Using the hypothesis of the corollary,, we get . Using this condition in (7), we can easily obtain the required result.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this study and finalized the manuscript.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Groups Program (Grant no. R.G.P.1/50/42). This work was also funded by the National Natural Science Foundation of China (Grant no. 12101168).