On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

: In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ω n = N lT × f N k φ in a complex projective space C P 2 m ( 4 ) . Additionally, we will ﬁnd certain restrictions on the warping function f , Dirichlet energy function E ( f ) , and ﬁrst non-zero eigenvalue λ 1 to prove that stable l -currents do not exist and also that the homology groups have vanished in Ω n . As an application of the non-existence of the stable currents in Ω n , we show that the fundamental group π 1 ( Ω n ) is trivial and Ω n is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.


Introduction and Main Results
A classical challenge in Riemannian geometry is to discuss the geometrical and topological structures of submanifolds. The stable currents and homology groups are the most important characterizations of the Riemannian submanifolds because they control the behavior of the topology of submanifolds. The notion of non-existence stable current and vanishing homology on pinching the second fundamental form was introduced by Lawson-Simons [1]. Xin proved in [2] as the following important form: Suppose Ω n is a compact n-dimensional submanifold in a space form Ω(c) of curvature c ≥ 0. Suppose l, k is any positive integer, that is, l + k = n, and the inequality l ∑ A=1 n ∑ B=l+1 2||h(e A , e B )|| 2 − g h(e A , e A ), h(e B , e B ) < lkc (1) holds for all orthonormal basis e i , · · · , e n } of the tangent space TΩ n ; then there are no stable l-currents in Ω n and H l (M n , G) = H n−l=k (Ω n , G) = 0, where H i (Ω n , G) stands for i integral homology groups of Ω n , while G is a finite abelian group with integer coefficients.
The generalized Poincaré conjecture for dimension n ≥ 5 was proved by Smale [3] by using the nonexistence for the stable currents over compact submanifolds on a sphere. Then, Lawson and Simons obtained the striking sphere theorem in [1], in which they showed that an n-dimensional compact-oriented submanifold Ω n in the unit sphere S n+k is homeomorphic to the sphere S n with n = 3, provided that the second fundamental form was bounded above by a constant that depends on the dimension n. Additionally, it was proved that Ω 3 is homotopic to the sphere S 3 . Using Theorem 1, Leung [4] proved that for a compact connected oriented submanifold Ω n in the unit sphere S n+k with h(X, X) 2 < 1 3 thus Ω n is homeomorphic to the sphere S n in the case n = 3, and also that Ω 3 is homotopic to a sphere S 3 . More recently, geometric, topological, and differentiable rigidity theorems of the Riemannian submanifold connecting to parallel mean curvature in space forms such that c + H 2 > 0 have been obtained in terms of Ricci curvature in [5]. In some articles such as [3,[6][7][8][9][10][11][12][13][14][15][16][17], several results have been derived on topological and differentiable structures of singular submanifolds and submanifolds with specific effective conditions for the second fundamental form, sectional curvatures, and Ricci curvatures.
However, very few topological obstructions to warped product submanifolds with positive sectional curvature are known; for example, Sahin et al. [13] verified some outcomes for the non-existence of the stable current and vanishing homology groups into a contact CR-warped product which immersed in a sphere with an odd dimension, by putting suitable restrictions on the Laplacian and the gradient of the warping function. Taking the benefits of the constant section curvature which could be zero or one, Sahin [13,14] extended this study on a class of CR-warped product in an Euclidean space and in the nearly Kaehler six-sphere. By assuming negative constant section curvature, Ali et al. [18][19][20] obtained various results on CR-warped product, especially on the complex hyperbolic spaces, and many structures about this subject remain open.
Therefore, we shall study the warped product pointwise semi-slant submanifolds of complex projective spaces where the constant sectional curvature c = 4 > 0 is positive. More specifically, our motivation comes from the studies of Sahin [21]. In that paper, Sahin investigated the warped product pointwise semi-slant submanifolds in a Kähler manifold, and also showed that the warped product pointwise semi-slant of form N l T × f N k φ is nontrivial. It was shown by the Ref. [21] that the warped product pointwise semi-slant submanifold N l T × f N k φ of Kähler manifold generalized the CR-warped products [22] and the angle φ is treated as a slant function. In this case, suppose C * = C − {0} and C m+1 * = C m+1 − {0}. Additionally, assuming that the action C * on C m+1 * can be expressed using γ, which means (z 0 , z 1 , . . . , z m ) = (γz 0 , γz 1 , . . . , γz m ), then all equivalent classes set are produced from this idea are represented using CP m . If we denote with π(z), the equivalent classes which contains z, then C m+1 * → CP m is a surjection, and it is well-known that CP m endowed a complex structure derived by the complex construction of C m+1 with a Kähler metric such that the constant holomorphic sectional curvature is equal to 4 [23]. We can observe that the almost complex J on CP m (4) is determined by the almost complex construction of C m+1 via the Hopf fibration. Let us now recall introduce the following Theorem 1.

Theorem 2.
Let Ω n = N l T × f N k φ be a compact warped product pointwise semi-slant submanifold with regard to the complex projective space CP 2m (4), which satisfies the following condition where ∇ f and ∆ f are the gradient and the Laplacian of the warped function f , respectively. Then we have the following: (a) The warped product submanifold Ω n does not exist for any stable integral l-currents. (b) The i integral homology groups of Ω n with integer coefficients vanish; that is, (c) The finite fundamental group π 1 (Ω) is null, that is, π 1 (Ω) = 0. Moreover, Ω n is a simply connected warped product manifold.

Remark 1.
To apply Theorem 2, suppose the slant function φ becomes globally constant, setting φ = π 2 from [24]. Then, the pointwise slant submanifold N k φ turns into a totally real submanifold N k ⊥ . Thus, a warped product pointwise semi-slant submanifold Ω n = N l T × f N k φ turns to CRwarped products within a Kähler manifold of the type Ω n = N l T × f N k ⊥ such that N l T , as well as N k ⊥ are holomorphic and totally real submanifolds, respectively [22].
Therefore, we deduce the following result from Theorem 2 and Remark 1 for the nonexistence of stable integrable l-currents and homology groups in the CR-warped product submanifolds of the complex projective space CP 2m (4).
⊥ be a compact CR-warped product submanifold of the complex projective space CP 2m (4). In this case, the following conditions occur: Then we have the following: (a) For the CR-warped product submanifold, Ω n does not have any stable integral l-currents.
(b) The i integral homology groups of Ω n with integer coefficients vanish; that is, (c) The finite fundamental group π 1 (Ω) is null, that is, π 1 (Ω) = 0. Moreover, Ω n is a simply connected warped product manifold.
Other important motivation for our study comes from the Ref. [25], where some geometric mechanics on Riemannian manifolds were studied. From that study, we found that for a positive differentiable function ϕ ( ϕ ∈ F (Ω n ) ) defined at a compact Riemannian manifold Ω, the Dirichlet energy of that function ϕ is given as in see [25] (p. 41), as follows: Using the Dirichlet energy Formula (4) for a compact manifold without a boundary, as well as Theorem 2, we give the next theorem: Thus, the following properties hold: (a) For the warped product submanifold Ω n , there are no stable integral l-currents. (b) The i integral homology groups of Ω n with integer coefficients vanished; that is, (c) The finite fundamental group π 1 (Ω) is null, that is, π 1 (Ω) = 0. Moreover, Ω n is a simply connected warped product manifold.
Using the result of Theorem 3, we can now recall the next sphere theorem for the compact oriented CR-warped product submanifold of a complex projective space CP 2m (4) due to Chen [22], that is, Then, the following properties are satisfied: (a) For the warped product submanifold Ω n , there are no stable integral l-currents.
(b) The i integral homology groups of Ω n with integer coefficients vanished; that is, (c) The finite fundamental group π 1 (Ω) is null, that is, π 1 (Ω) = 0. Moreover, Ω n is simply connected warped product manifold.
Let Ω n be an n-dimensional compact Riemannian manifold, and therefore, the Laplacian is a second-order quasilinear operator on Ω n , given as For such a Laplacian, we can found many applications in mathematics as well as in physics, and this is possible due to the eigenvalue problem of ∆. The corresponding Laplace eigenvalue equation is defined as follows: a real number λ is named eigenvalue if it is a non-vanishing function ϕ, which satisfies the following equation: with appropriate boundary conditions. Considering a Riemannian manifold Ω n with no boundary, the first nonzero eigenvalue of ∆, defined as λ 1 , includes variational properties (cf. [26]): Inspired by the above characterization, using the first non-zero eigenvalue of the Laplace operator and the maximum principle for the first non-zero eigenvalue λ 1 , we deduce the following: (4); that is, f is a non-constant eigenfunction of the first non-zero eigenvalue λ 1 . Assume that holds. Then the properties (a), (b), and (c) of Theorem 2 are satisfied.

Remark 2.
Some important applications of this theory can be found for the singularity structure in liquid crystals, in the system in statistical mechanics with low dimensions, and physical phase transitions (see [27]). In addition, general relativity contains warped product manifolds as a model of space-times. There are two famous warped product spaces. One is the generalization of Robertson-Walker space-times, and the other is the standard static space-times [17,[28][29][30][31]. General relativity depends heavily on the differential topological methods, especially in mathematical physics, and particularly regarding the way that the space-time homology is used in in quantum gravity [13,17]. On the other hand, the formulation of a theory which unifies quantum mechanics and the special theory of relativity, performed by Dirac nearly a century ago, required introduction of new mathematical and physical concepts which led to models that, on one hand, have been very successful in terms of the interpretation of physical reality but, on the other, still creates some challenges, both conceptual and computational. A central notion of relativistic quantum mechanics is a construct known as the Dirac operator. It may be defined as the result of factorization of a second-order differential operator in the Minkowski space. The eigenvalues of the Dirac operator on a curved spacetime are diffeomorphism-invariant functions of the geometry. They form an innite set of observables for general relativity. Some recent work suggests that they can be taken as variables for an invariant description of the gravitational field's dynamics. Because this paper is connected to both warped product manifold and homotopy-homology theory, its results can be used as physical applications.

Some Important Background
This part includes some notations and definitions that is important to the work relay essentially on [18,21,22]. Suppose CP m is a m-dimensional complex projective space among the Fubini-Study metric g FS with J being its almost complex structure. In the case where the Levi-Civita connection is defined using ∇, the Fubini-Study metric is Kähler, that is, ∇J = 0. A Kähler manifold Ω m for a positive constant sectional curvature c = 4 > 0 is named a complex projective space CP m (4) and can be endowed with the Fubini-Study metric g FS . Therefore, the curvature tensor R of CP m (4) is given as: (4)). Assume that Ω n is an isometrically immersed to an almost Hermitian manifold Ω m among the induced metric g. The Gauss equation of the submanifold Ω n is determined by: where R and R are curvature tensors at M m and Ω n , in the same order. The definition of mean curvature vector H of the orthonormal frame {e 1 , e 2 , · · · e n } of the tangent space TM on Ω n is given as where n = dim Ω.
The gradient positive function ϕ defined on Ω n and its squared norm is written as: We will provide some short definitions of different classes of submanifold Ω n according to J conserves all tangent spaces of Ω n , such that (i) Ω n is holomorphic submanifold if J(T x Ω) ⊆ T x Ω [22]. (ii) Ω n is named totally real submanifold in the case where J(T x Ω) ⊆ T ⊥ x Ω [22]. (iii) Combining (i) and (ii) such that TΩ = D T ⊕ D ⊥ , then Ω n is a CR-submanifold [22].
(iv) In the case where the angle φ(X) is enclosed by JX and the tangent space is T x Ω for any vector field X of Ω n that is not equal to zero, it is a real-valued function such that φ : Ω → R, then Ω n is called a pointwise slant submanifold (more details in [24]). In the same paper, the authors provided a necessary and sufficient condition for Ω n to be a pointwise slant T 2 X = − cos 2 φX, where T is tangential (1, 1) tensor field [24]. (v) If the tangent space TΩ is introduced as a decomposition in the form of TΩ = D T ⊕ D φ for J(D T ) ⊆ D T and pointwise slant distribution D φ , then Ω n is classified as a pointwise semi-slant submanifold [21]. For some examples of pointwise semi-slant submanifolds in a Kähler manifold, and related problems, we recommend the Ref. [18,21].
Regarding the above study, we give some remarks as follows.
Remark 3. If we consider a slant function φ : Ω n → R that is globally constant on Ω n and φ = π 2 , thus, Ω n is named a CR-submanifold.

Warped Product Submanifolds
Warped product manifolds Ω n = N l 1 × f N k 2 were originally initiated by Bishop and O'Neill [28], where N l 1 and N k 2 are two Riemannian manifolds and their Riemannain metrics are g 1 and g 2 in the same order. f is also a smooth function defined on N l 1 . The warped product manifold Ω n = N l 1 × f N k 2 is the manifold N l 1 × N k 2 furnished by the Riemannian metric g = g 1 + f 2 g 2 , and the function f is named a warping function of Ω n . The following important consequences of the warped product manifolds are given in [28,29]. For all U 1 , U 2 ∈ X(TN 1 ) and V 1 , V 2 ∈ X(TN 2 ), where we have where H f is a Hessian tensor of f . Furthermore, we have

The Non-Trivial Warped Product Pointwise Semi-Slant Submanifolds
Based on the pointwise semi-slant submanifold definition, it is possible to define the warped product pointwise semi-slant submanifolds of a Kähler manifold as follows: We will consider the second type because the first type of N k φ × f N l T is trivial (see Theorem 4.1 in [21]). Additionally for the non-trivial case N l T × f N k φ with examples, see the Ref. [21]. This warped product pointwise semi-slant submanifold is interesting because it is a generalized CR-warped product [22]. The proofs of the main results are ready to be introduced as follows.

Proof of Theorem 2
Let Ω n = N l T × f N k φ be an n = l + k−dimensional warped product pointwise semislant submanifold with dimN l T = l = 2A and dimN k φ = k = 2B, where N k φ and N l T are integral manifolds of D φ and D, in the same order. Then, {e 1 , e 2 , · · · e A , e A+1 = Je 1 , · · · e 2A = Je A } and {e 2A+1 = e * 1 , · · · e 2A+B = e * B , e 2A+B+1 = e * B+1 = sec φPe * 1 , · · · e l+k = e * k = sec φPe * B } will be orthonormal frames of TN T and TN φ , in the same order. Therefore, the orthonormal basis of FD φ and µ are {e n+1 =ē 1 = csc φFe * 1 , · · · e n+B =ē B = csc φFe * 1 , e n+B+1 =ē B+1 = csc φ sec φFPe * 1 , · · · e n+2B =ē 2B = csc φ sec φFPe * B } and {e n+2B+1 , · · · e 2m }, respectively. Then, we arrange the terms Then, from the Gauss Equation (12), we have Using the orthonormal frames e i 1≤A≤p as well as e B 1≤B≤q of N l T and N k φ , respectively, in (17), we derive Summing up, with respect to the orthonormal frame e B 1≤B≤q in addition to taking into account the adoption of the opposite of the usual sign convention for the Laplacian, one obtains: Thus, from Equations (19) and (20), we derive Firstly, the term ∆ f for Ω n is computed, which is originally the Laplacian of f .
The previous equation will be rewritten using components of N k φ for an adapted orthonormal frame. One obtains: It is noted that ∇ is a Levi-Civita connection on Ω n , and N l T is also totally geodesic in M n . It leads to grad f ∈ X(TN T ), and then we have It is clear that the next equation is satisfied This result, combined with (21)  At this point, suppose X = e A and Z = e B for 1 ≤ A ≤ l and 1 ≤ B ≤ k, in the same order. Thus, by the use of the bilinear form h definition according to an orthonormal basis, we can write In the previous equation, the first term at the right-hand side is a FD φ -component, while the second term is a µ invariant subspace. From the viewpoint of an adapted orthonormal basis, vector fields of N l T and N k φ are summed up over the vector fields of N l T and N k φ . Then, using Lemma 5.2 from [21] and (Equation (5.8) of Lemma 5.3 in [21]), we conclude that Using the squared norm definition of the gradient function f (15) (ii), one obtains: From (23) and (24), we get: =k∆(ln f ) + k(k − 1)||∇(ln f )|| 2 (25) For the symmetry of the curvature tensor R, the following relation holds: Next, we remark that the curvature tensor Formula (11) for the complex projective space CP 2m (4) is easily given as As we know, if e A ∈ X(TN T ) and e B ∈ X(TN φ ), then g(e A , e B ) = 0, and g(Je A , e A ) = 0(resp, g(Je B , e B ) = 0), using Je A ⊥ e A (Je B ⊥ e B ), respectively. Similarly, from (Equation (2.6) in [21]), we derive that g(Je A , e B ) = g(Te A + Fe A , e B ) = 0 for Te A ∈ X(TN T ) and Fe A ∈ X(FN φ ). Thus, (27) implies that g(e A , e A )g(e B , e B ) = −lk (28) Therefore, using (26) and (28), we finally get Now, computing ∆ ln f , we get: Then, from (29) and (30), we find that Let the pinching condition (2) be satisfied. Then, from (31), we get 2||h(e A , e B )|| 2 −g h(e B , e B ), h(e A , e A ) < 4lk.
It well-known that constant sectional curvature for the complex projective spaces CP 2m (4) is equal to c = 4. Then, the last equation is implied: 2||h(e A , e B )|| 2 −g h(e B , e B ), h(e A , e A ) < lkc.
Therefore, using Theorem 1, we reached our promised results (a) and (b). For the third part, let us assume that π 1 (Ω) = 0. From the compactness of Ω n , it follows from the classical theorem of Cartan and Hadamard that there is a minimal closed geodesic in any non-trivial homotopy class in π 1 (Ω), which leads to a contradiction. Therefore, π 1 (Ω) = 0. This is the third part of the theorem. If the finite fundamental group is null of any Riemannian manifold, this Riemannian manifold is simply connected. As a result, Ω n is simply connected. Therefore, the assertion follows from Theorem 2. The proof is completed. Hamiltonian at the point x ∈ M n for the local orthonormal frame, is given as (see [25]): Substituting p = dϕ into the previous equation, and since d is a differentiable operator, we use (15) to have: Using the previous equation leads to the next result from inequality (2), as the following: Corollary 3. Under the same assumption in Theorem 2, it satisfies the following inequality: where H(d f , x) is the Hamiltonian of the warping function f , so no stable integral l-currents exist in Ω n and H l (Ω n , G) = H k (Ω n , G) = 0.
Proof. Combining the Hamiltonian formula (39) and inequality (2), we have the result.

Proof of the Corollarys 1 and 2
The proof of Corollarys 1 and 2 can be obtained directly from the Theorems 2 and 3 by substituting φ = π 2 to derive a totally real submanifold from a pointwise slant submanifold.