Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

: In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold M n of a complex space form (cid:101) M ( 4 (cid:101) ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i -the homology groups of M n are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold M n , in which M n is homeomorphic to a sphere S n if n ≥ 4 and if n = 3, then M 3 is homotopic to a sphere S 3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.


Introduction and Main Results
A traditional topic in Riemannian geometry is to find the geometrical and topological structures of submanifolds; there has been much progress in this field. For instance, the rigidity theorem was proved by Berger [1] for an even-dimensional complete simply connected manifold M with sectional curvature 1 4 ≤ K M ≤ 1. Further, Gauhmen [2] considered even n = 2m-dimensional submanifolds minimally immersed in the unit sphere S n+1 with a co-dimension equal to one, and showed that if ||h(u, u)|| 2 < 1 for any unit vector u of M n where h is the second fundamental form M n , then M n is totally geodesic in S n+1 . If max u∈M {||h(u, u)|| 2 } = 1, then M n is S m ( 1 2 ) × S m ( 1 2 ) minimally embedded in S 2m+1 , as described. A very famous result in this respect was formulated by Poincare [3], who stated that every simply connected closed 3-manifold is homeomorphic to a 3-sphere. Smale [4] generalized the Poincare conjecture and proved that for a closed C ∞ -manifold M n which has the homotopic types of an n-dimensional sphere greater than five, the manifold M n is homeomorphic to S n . The differentiable sphere theorem was proven by Brendle and Schoen [5] under Ricci flow. In recent years, much attention has been paid to the classification of geometric function theory, topological sphere theorems, and differentiable sphere theorems (see [6][7][8][9][10][11]). In the sequelae, the homology groups of a manifold are important topological invariants that provide algebraic information about the manifold. Federer-Fleming [7] showed that any non-trivial integral homology class in H p (M, G)) corresponds to a stable current. Motivated by the work of Federer and Fleming [7], Lawson and Simon [9], and Xin [11] proved the nonexistence of stable integral currents in a submanifold M n and vanishing homology groups of M n with non-negative sectional curvature according to the following theorem. Theorem 1 ([9,11]). Let M n be a compact n-dimensional submanifold isometrically immersed in the space form M(c) of curvature c ≥ 0 with the second fundamental form h. Let l 1 , l 2 be any positive integers such that l 1 + l 2 = n and 2||h(e α , e β )|| 2 − g h(e α , e α ), h(e β , e β ) < l 1 l 2 c, for any x ∈ M n and an orthonormal frame e i 1≤i≤n of the tangent space TM n . Then, there do not exist stable l 1 -currents in M n and where H i (M n , G) stands for i-the homology group of M n and G is a finite abelian group with integer coefficients.
Due to these previous studies on large scales, a particular case we consider here is that of warped product pointwise semi-slant submanifolds of complex space form where 4 is represented as a constant sectional curvature. In this regard, our motivation comes from the study of Sahin [12], where he discussed the warped product pointwise semi-slant submanifolds in a Kaehler manifold and showed that a warped product pointwise semi-slant submanifold of type N l 1 T × f N l 2 θ is nontrivial when angle θ is treated as a slant function. Furthermore, it was shown in [12] that the warped product pointwise semi-slant submanifold N l 1 T × f N l 2 θ of a Kaehler manifold is a natural generalization of CR-warped products [13]. Inspired by this notion, we define the extrinsic condition to prove nonexistence-stable integral l 1 -currents and vanishing homology groups in a warped product pointwise semi-slant submanifold of complex space forms M m (4 ). We use Theorem 1 on this basis to arrive at our first result. Theorem 2. Let M l 1 +l 2 = N l 1 T × f N l 2 θ be a compact warped product pointwise semi-slant submanifold of a complex space form M m (4 ). If the following condition is satisfied then there do not exist stable integral l 1 -currents in M l 1 +l @ and where H i (M l 1 +l 2 , G) stands for i-the homology group of M l 1 +l 2 with integer coefficients, ∇ f and ∆ f are the gradient and the Laplacian of the warped function f , respectively, and h µ represents the components of the second fundamental form h in an invariant subspace µ.
Our next result is in accordance with Lemma 3.1 in [12], which states that the inner product of the second fundamental form of N l 1 T and F-components of N l 1 θ is equal to zero. To be precise, we have the following result. Theorem 3. Let M l 1 +l 2 = N l 1 T × f N l 2 θ be a compact warped product pointwise semi-slant submanifold of a complex space form M m (4 ). If the inequality holds, then there do not exist stable integral l 1 -currents in M l 1 +l 2 and H l 1 (M l 1 +l 2 , G) = H l 2 (M l 1 +l 2 , G) = 0.
The notation is the same as in Theorem 2.
To apply Theorems 2 and 3 in [14], let the slant function θ become globally constant, setting θ = π 2 in Theorems 2 and 3. Then, the pointwise slant submanifold N l 1 θ is turned into a totally real submanifold N l 2 ⊥ . Thus, a warped product pointwise semi-slant submanifold M l 1 +l 2 = N l 1 T × f N l 2 θ becomes CR-warped products in a Kaehler manifold of type M n = N l 1 T × f N l 2 ⊥ . Therefore, following to the motivation of Chen [13], we deduce the following result from Theorem 2 for the nonexistence of stable integral l 1 -currents and vanishing homology in a CR-warped product submanifold of complex space forms M m (4 ).
⊥ be a compact CR-warped product submanifold of complex space form M 2m (4 ). If the following condition is satisfied then there do not exist stable integral l 1 -currents in M l 1 +l 2 and H l 1 (M l 1 +l 2 , G) = H l 2 (M l 1 +l 2 , G) = 0.
As an immediate consequence of Theorem 3, we have ⊥ be a compact CR-warped product submanifold of complex space form M 2m (4 ) satisfying the following inequality Then, there do not exist stable integral l 1 -currents in M l 1 +l 2 and we have the trivial homology groups, i.e., H l 1 (M l 1 +l 2 , G) = H l 1 (M l 1 +l 2 , G) = 0.
Our next motivation comes from Calin [15] who studied geometric mechanics on Riemannian manifolds and defined a positive differentiable function ϕ ( ϕ ∈ F (M n ) ) on a compact Riemannian manifold M n . The Dirichlet energy of a function ϕ is defined in [15] (see p. 41) as follows: In view of the kinetic energy formula (5) for a compact oriented manifold without boundary along with Theorem 2, we arrive at the following result. T × f N l 2 ⊥ be a compact warped product pointwise semi-slant submanifold of a complex space form M 2m (4 ) without boundary. If the following condition is satisfied where E( f ) is the Dirichlet energy of the warping function f with respect to the volume element dV, then there do not exist stable integral l 1 -currents in M l 1 +l 2 and H l 1 (M l 1 +l 2 , G) = An important concept relates to the geometrical and topological properties on Riemannian manifolds when considering the pinched condition on its metric. It is interesting to investigate the curvature and topology of submanifolds in a Riemannian manifold and the usual sphere theorems in Riemannian geometry. For instance, using the nonexistence of stable currents on compact submanifolds, Lawson and Simon [9] obtained their striking sphere theorem, which proved that for an n-dimensional compact-oriented submanifold M n in a unit sphere S n+k with the second fundamental form bounded above by a constant which depends on the dimension n, then M n is homeomorphic to a sphere S n when n = 3 and M 3 are homotopic to a sphere S 3 .
Making use of Lawson and Simon [9], Leung [16] proved that for a compact connected oriented submanifold M n in the unit sphere S n+k such that h(X, X) 2 < 1 3 , when n = 3 and M 3 are homotopic to a sphere S 3 , then M n is homeomorphic to a sphere S n . Recently, it has been shown in [17] that if the sectional curvature satisfies some pinching condition for n-dimensional compact oriented minimal submanifold M in the unit sphere S n+l 1 with co-dimension l 1 , then M is either a totally geodesic sphere, one of the Clifford minimal hyper-surfaces S k ( k n ) × S n−k ( n−k n ) in S n+1 for k = 1, . . . , n − 1, or a Veronese surface in S 4 . More recently, several results have been derived on topological and differentiable structures of submanifolds when imposing certain conditions on the second fundamental form, Ricci curvatures, and sectional curvatures in a series of articles [4,10,11,[18][19][20][21][22][23] by different geometers. For the warped product structure, we refer to [20,[24][25][26][27][28][29][30].
The second target of note is to establish topological sphere theorems from the viewpoint of warped product submanifold geometry with positive constant sectional curvature and pinching conditions in terms of the squared norm of the warping function and Laplacian of the warped function as extrinsic invariants. In this sense, we work with conditions on the extrinsic curvature (second fundamental form, warping function), which have the advantage of being invariant under rigid motions. Motivated by Lawson and Simon [9], (p. 441, Theorem 4), we consider a warped product pointwise semi-slant submanifold in a complex space form M 2m (4 ) such that the constant holomorphic sectional curvature is 4 , and state our main theorem of this paper.
θ be a compact warped product pointwise semi-slant submanifold in a complex space form M 2m (4 ) satisfying the condition (2). Then, M l 1 +l 2 is homeomorphic to sphere S l 1 +l 2 when l 1 + l 2 ≥ 4, while M 3 is homotopic to a sphere S 3 .

Remark 1.
As a consequence of Theorem 5, we obtain the following sphere theorem for a compact CR-warped product submanifold in a complex space form M 2m (4 ), thanks to Chen [13].
⊥ be a compact CR-warped product submanifold in a complex space form M 2m (4 ) satisfying the pinching condition (4). Then, M l 1 +l 2 is homeomorphic to a sphere S l 1 +l 2 when l 1 + l 2 ≥ 4, and M 3 is homotopic to a sphere S 3 .
Using Theorem 4 and 5, we can now obtain an important result.
θ be a compact warped product pointwise semi-slant submanifold of complex space form M 2m (4 ). If (6) is satisfied, then M l 1 +l 2 is homeomorphic to sphere S l 1 +l 2 when l 1 + l 2 ≥ 4 and M 3 is homotopic to a sphere S 3 .

Remark 2.
The principle behind Cheng's eigenvalue comparison theorem (see [31]) forms the basis of the following finding. With the help of the first non-zero eigenvalue of the Laplacian operator, Cheng has demonstrated that if M is complete and isometric to the sphere of the standard unit then the following theorem can be inferred using the maximum principle for the first non-zero eigenvalue λ 1 , provided that Ric(M)geq1 and d(M) = π.
θ be a compact warped product pointwise semi-slant submanifold of a complex space form M 2m (4 ) with f being a non-constant eigenfunction of the first non-zero eigenvalue λ 1 such that the following inequality is satisfied: Then, M l 1 +l 2 is homeomorphic to sphere S l 1 +l 2 when l 1 + l 2 ≥ 4 and M 3 is homotopic to a sphere S 3 when l 1 + l 2 = 3.
Motivated by Bochner's formula [32], we arrive at the following result.
Theorem 7. Let M l 1 +l 2 = N l 1 T × f N l 2 θ be a compact warped product pointwise semi-slant submanifold of a complex space form M 2m (4 ) such that following inequality holds: where ∇ 2 f 2 denotes the Hessian form of the warping function f and Ric denotes the Ricci curvature along the base manifold N l 1 T . Then, M l 1 +l 2 is homeomorphic to sphere S l 1 +l 2 when l 1 + l 2 ≥ 4 and M 3 is homotopic to a sphere S 3 when l 1 + l 2 = 3.

Preliminaries
Let M 2m (4 ) be a complex space form with the complex dimension dim R M = 2m. Then, the curvature tensor R of M 2m (4 ) with constant holomorphic sectional curvature 4 is expressed as The Gauss and Weingarten formulas for transforming submanifold M n into an almost Hermitian manifold M 2m are provided by for each X 2 , Y 2 ∈ X(TM) and N ∈ X(T ⊥ M) such that the second fundamental form and the shape operator are denoted by h and A N . They are connected as g(h(U, V), N) = g(A N U, V). Now, for any X 2 ∈ X(M) and N ∈ X(T ⊥ M), we have where TX 2 (tN) and FX 2 ( f N) are the tangential and normal components of JX 2 (JN), respectively.
The Gauss equation for a submanifold M n is defined as for any X 2 , Y 2 , Z 2 , W 2 ∈ X(TM), where R and R are the curvature tensors on M 2m and M n , respectively. The norm of second fundamental form h for an orthonormal frame {e 1 , e 2 , · · · e n } of the tangent space TM on M n is defined by Let {e 1 , . . . , e n } be an local orthonormal frame of vector field M n . Then, we have where ∇ϕ and ||∇ϕ|| 2 are the gradient of function ϕ and its squared norm.
The following classifications can be provided as: There are four types of submanifolds of a Kaehler manifold, namely, the CR-submanifold, slant submanifold, semi-slant submanifold, pointwise slant submanifold, and pointwise semislant submanifold. The definitions and classifications of such submanifolds are discussed in [12,13]. Moreover, for examples of a pointwise semi-slant submanifold in a Kaehler manifold and related problems, we refer to [12]. It follows from Definition 3.1 in [12] that if we denote as l 1 and l 2 the dimensions of a complex distribution D T and pointwise slant distribution D θ of a pointwise semi-slant submanifold in a Kaehler manifold M 2m , then the following remarks hold:

Remark 4.
If we consider the slant function θ : M n → R as globally constant on M n and θ = π 2 , then M n is a CR-submanifold.

Warped Product Submanifolds
A product manifold of the type M n = N l 1 1 × f N l 2 2 is a warped product manifold if the metric is defined as g = g 1 + f 2 g 2 , where N l 1 1 and N l 2 2 are two Riemannian manifolds and their Riemannian metrics are g 1 and g 2 , respectively. It was discovered by Bishop and O'Neill [33] that the warping function f is a smooth function defined on base N l 1 1 . The following properties are a direct consequence of the warped product manifold M n = N l 1 1 × f N l 2 2 : for any X, Y ∈ X(TN 1 ) and Z, W ∈ X(TN 2 ), where ∇ and ∇ denote the Levi-Civita connection on M n and N 2 , respectively.
The gradient ∇ f of f is written as The following relation is an interesting property of warped products: where H f is a Hessian tensor of f ; the remarks below follow as a consequence.

Remark 6.
A warped product manifold M n = N l 1 1 × f N l 2 2 is said to be trivial or simply a Riemannian product manifold if the warping function f is a constant function along N l 1 1 .

Remark 7.
If M n = N l 1 1 × f N l 2 2 is a warped product manifold, then N l 1 1 is totally geodesic and N l 2 2 is a totally umbilical submanifold of M n , respectively.

Non-Trivial Warped Product Pointwise Semi-Slant Submanifolds
It is well known that warped product submanifolds of types are called warped product pointwise semi-slant submanifolds, which were discovered in [12]. They contain holomorphic and pointwise slant submanifolds of a Kähler manifold. The first case, with M n = N l 2 θ × f N l 1 T in a Kähler manifold, is trivial. The second is nontrivial. Before proceeding to the second case, let us recall the following result [12].

Proof of Theorem 2
The crucial point of this paper is to derive an upper bound for in terms of ∆ f and ||∇ f || 2 .
The above equation can be expressed as components of N q θ from adapted orthonormal framel in this way, we obtain Benefiting from ∇ being a Levi-Civita connection on M n , we derive e j g grad f , e j − g ∇ e j e j , grad f .
Te j g grad f , Te j − g ∇ Te j Te j , grad f .
From the property of the gradient of function (14), we obtain After computation, we have Te j g(grad f , Te j ) − g(∇ Te j Te j , grad f ) .
Starting from the hypothesis of a warped product pointwise semi-slant submanifold, N l 1 T is totally geodesic in M n . This implies that grad f ∈ X(TN T ), and from (i)-(ii) in Section 3, we obtain By multiplying the above equation by 1 f , from (3.7) of Corollary 3.1 in [12] we obtain It is not difficult to check that This combines with (20) to yield On taking X = e i and Z = e j for 1 ≤ i ≤ l 1 and 1 ≤ j ≤ l 2 , respectively, we have In the above equation, the first term on the right hand side is the FD θ -component and the second term is the µ-component for the orthonormal frame for vector fields of N l 1 T and N l 2 θ . Summing over the vector fields of N l 1 T and N l 2 θ and using (16) and (17) from Lemma 1 in the last equation, we are able to find that From the adapted orthonormal frame for N T , the last equation can then be expressed as follows: Together with the definition of the squared norm of the gradient function f from (13), the above implies that Following (21) and (22), we arrive at 2||h(e i , e j )|| 2 − g h(e j , e j ), h(e i , e i ) Because we have the following relation for symmetry of the curvature tensor R, Next, we use the curvature tensor from Formula (9) for the complex space form M m (4 ), which can be simply written as As we know that e i ∈ X(TN T ) and e j ∈ X(TN θ ), then g(e i , e j ) = 0, and g(Je i , e i ) = 0(resp, g(Je j , e i ) = 0) by the fact that for Je i ⊥ e i (Je j ⊥ e j ), respectively. Similarly, from (10)i, we can derive that g(Je i , e j ) = g(Te i + Fe i , e j ) = 0 for Te i ∈ X(TN T ) and e j ∈ X(TN θ ); thus, (24) implies that g(e i , e i )g(e j , e j ).
After computation using the above equation, we can derive Therefore, following (23) and (25), we finally obtain 2||h(e i , e j )|| 2 − g h(e j , e j ), h(e i , e i ) .
This completes the proof of Theorem 2, as the assertion follows from Theorem 1.
Thus, the proof is complete from Theorem 1 and from (34).
Based on Theorem 3 and the similar proof of Theorem 5, we reach the following result.
Then, M p+q is homeomorphic to a sphere S p+q when p + q = 3, while M 3 is homotopic to a sphere S 3 .

Proof of Theorem 7
In this theorem, we replace our pinching condition (2) with the Hessian of the warping function and Ricci curvature by using the concept of the eigenvalue of the warped function. If f is a first eigenfunction of the Laplacian of M n associated with the first eigenvalue λ 1 , that is, ∆ f = λ 1 f , then we an recall Bochner's formula (see, e.g., [32]), which states that for a differentiable function f defined on a Riemannian manifold, the following relation holds: By integrating the above equation with the aid of Stokes' theorem, we obtain ∇ 2 f 2 dV + Ric(∇ f , ∇ f )dV + g ∇ f , ∇(∆ f ) dV = 0. Now, by using ∆ f = λ 1 f and slightly rearranging the above equation, we derive homologic theory. The contents of the present paper can be expected to attract researchers to the prospect of finding possible applications in various research areas of physics.