Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space H m (− 1 ) satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 ( Ω p + q ) 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.


Introduction and Main Results
For any Riemannian manifold Ω n , it is well known that any integral homology class in H q (Ω n , Z) which is nontrivial is correlated to integral stable currents. is 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][3][4] is satisfied, then Ω n has no stable p-currents with the vanished pth homology group, i.e., where p + q � n and sgn(c) is the signature of constant curvature c.
e geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4][5][6][7][8][9][10][11][12][13][14][15][16][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 Ω n of dimension n in Euclidean space E m of dimension m � n + k satisfies the pinching condition Ric(X) > δ 1 (n)g (A H X, X), in which X is any unit vector, A H is the shape operator regarding the mean curvature H, and δ 1 (n) is a constant such that δ 1 (n) � n − 2 if n is even and δ 1 (n) � n(n − 3)/(n − 1) if n is odd. erefore, Ω n has no stable currents. Moreover, Ω n is homeomorphic to S n . Moreover, using eorem 1 in [19], it was found that if a compact oriented submanifold of dimension n Ω n in space form F m c satisfied the second fundamental form pinching condition S < a(n, k, |H|, c), for any integer k in which 0 < k < n and c is a constant sectional curvature, then the pth homology groups are vanishing, H p (Ω n , Z) � 0, for all p ∈ k, . . . , { n − k}, and if the fundamental group π 1 (Ω n ) is finite and simply connected, then Ω n is homeomorphic to S n . Using eorem 1, Xu and Gu [16] extended the pinching condition in terms of the Ricci curvature and showed that Ω n in F m c satisfied Ric > (n − 2)(c + H 2 ); then, Ω n is homeomorphic to S n . 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][20][21][22][23][24] inspired by eorem 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 eorem 1, we now give the first main result of this note.
be an isometric embedding from a compact warped product submanifold Ω p+q into an m-dimensional hyperbolic space H m (− 1) in which the base manifold N p 1 is minimal in H m (− 1) and the following inequality (c) If p � 1, then the fundamental group π 1 (Ω) is null, i.e., π 1 (Ω) � 0. Moreover, Ω p+q is a simply connected warped product manifold.
Motivated by the geometric rigidity ( eorem 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.
Now, we give a direct application of eorem 3.

Theorem 4. Assume that
is an isometric embedding from a compact warped product submanifold Ω p+q into an m-dimensional hyperbolic space H m (− 1) satisfying the following inequality:

en, statements (a), (b), and (c) in eorem 2 hold.
Another interesting result obtained from eorem 4 is the following: Corollary 1. Under the same assumption as eorem 4 and if ∇h ∈ KerΠ with the following holds, then, statements (a), (b), and (c) in eorem 2 are satisfied. Remark 1. eorem 2 is the main vanishing homology theorem for a compact warped product submanifold with no need for Ω p+q 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.

Preliminaries
Let H m (c) denote the hyperbolic space with dimension (m) and constant sectional curvature c � − 1. We use the fact that H m (c) has a canonical isometric embedding in Lor- where x 1 · · · x n+1 are canonical coordinates in L m+1 us, is the Gauss equation for Ω n in which X, Y, Z, W ∈ X(TΩ), and R and R are the curvature tensors on Ω n and M, respectively. e mean curvature vector H for a local orthonormal basis e 1 , e 2 , . . . , e n on Ω is defined by In addition, we set In this connection, we shall define the scalar curvature τ(T x Ω n ) of Riemannian submanifold Ω n , which is considered as a Riemannian intrinsic invariant, at some x in Ω n as follows: where K βα � K(e β ∧e α ). e first equality (15) is equivalent to the following equation which will be frequently used in subsequent proofs: In a similar manner, the scalar curvature τ(L x ) of an L− plane is defined as If the plane sections are spanned by e β and e α at x, we give denotations K βα and K βα , respectively, for the sectional curvature of the Riemannian manifold M m and submanifold Ω n , which are considered as the extrinsic and intrinsic sectional curvature of the span e β , e α at x. 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 N p 1 and N q 2 are two Riemannian manifolds with Riemannian metrics g 1 and g 2 , respectively. Assume that h is a differentiable function on N p 1 . A warped product manifold is Ω p+q � N p 1 × h N q 2 with n � p + q and the Riemannian metric Hence, ∀W 1 ∈ Γ(TN 1 ) and W 2 ∈ Γ(TN 2 ), we attain that Furthermore, ∇(lnh) is the gradient of lnh, given by us, from [26], we have where H h is a Hessian tensor of h. Let e 1 , . . . , e n be a local orthonormal basis of vector field Ω p+q ; thus, the squared norm of the gradient of the positive differential function h for an orthonormal basis e 1 , . . . , e n is Now, if we replace p + q in (20) by n, we get Journal of Mathematics and Laplacian Δh of h is defined as g ∇ e β gradh, e β � trHess(h).

(24)
Remark 2. It should be noted that we consider the opposite sign of Chen [27] of Laplacian of h; that is, Δh � div(∇h).
Moreover, because unit vector fields W 1 and W 2 are tangent to N p 1 and N q 2 , respectively, we get Let us sum up over the vector fields such that which leads to

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 H m (− 1); we will give the proof of our main results.

Proof of eorem 2.
Let dim(N 1 ) � p and dim(N 2 ) � q, and consider e 1 , e 2 , . . . , e p and e * p+1 , . . . , e * n to be orthonormal frames of TN 1 and TN 2 , respectively. en, from Gauss equation (12) for the standard hyperbolic space us, from (28) and (29), we derive Computing the Laplacian Δh on Ω p+q , one obtains Since N p 1 is totally geodesic in Ω p+q , gradh ∈ X(TN 1 ), and by utilizing the definition of the warped product, we obtain Multiplying the above equation by (1/h), we get After some computations, we find that us, from (30) and (34), one obtains g R e β , e α e β , e α .
Next, using Gauss equation (12) for hyperbolic space H m (− 1), we find that By rewriting the above equation for Ω p+q and utilizing (13) and (21), we attain Using (18) and (27) (38) us, using the curvature equation and the sphere H m (− 1) and rearranging the last equation, we attain After some rearrangements of the above equation, we get Utilizing the assumption of the theorem and since N p 1 is minimal, the fifth term of the right-hand side (RHS) in equation (41) is identically zero, and the first term of the lefthand side is equal to the seventh term of the RHS. Hence, we get From (36) and (42), we get g R e β , e α e β , e α .
is follows that From our assumption (3) Journal of Mathematics Applying eorem 1 for constant holomorphic sectional curvature c � 1, we obtain that there are no stable p-currents in Ω p+q and H p (Ω p+q , Z) � H q (Ω p+q , Z) � 0, which completes the proof of (a) and (b) of the theorem. In the other part, from (45), substituting p � 1, we have If the pinching condition (22) for p � 1 and q � n − 1 holds, then we get n α�2 2 Π e 1 , e α � � � � � � � � 2 − g Π e α , e α , Π e 1 , e 1 <(n + 1). (48) en, there are no stable 1-currents in Ω 1+q and e compactness property of Ω 1+q ; 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 π 1 (Ω), and this leads to a contradiction. erefore, π 1 (Ω) � 0. is 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, Ω p+q is simply connected.