Homology of Warped Product Semi-Invariant Submanifolds of a Sasakian Space Form with Semisymmetric Metric Connection

Tis paper focuses on the investigation of semi-invariant warped product submanifolds of Sasakian space forms endowed with a semisymmetric metric connection. We delve into the study of these submanifolds and derive several fundamental results. Additionally, we explore the practical implications of our fndings by applying them to the homology analysis of these sub-manifolds. Notably, we present a proof demonstrating the absence of stable currents for these submanifolds under a specifc condition.


Introduction
Te homology groups of a manifold provide an algebraic representation that captures important topological characteristics.Tese groups contain extensive topological information about the various components, voids, tunnels, and overall structure of the manifold.Consequently, homology theory fnds numerous applications in diverse felds such as root construction, molecular anchoring, image segmentation, and genetic expression analysis.Te relationship between submanifold theory and homological theory is widely recognized for its signifcance.In the seminal work by Federer and Fleming [1], it was demonstrated that any nontrivial integral homological group H p (M, Z) is connected through stable currents.Building upon this, Lawson and Simons [2] extended the investigation to submanifolds of a sphere, where they established that the existence of an integral current is precluded under a pinching condition imposed on the second fundamental form.Notably, Leung [3] and Xin [4] expanded the scope of these results from spheres to Euclidean space.Additionally, Zhang [5] conducted a study on the homology of tori, further contributing to this line of research.In a recent development, Liu and Zhang [6] presented a proof demonstrating that stable integral currents cannot exist for specifc types of hypersurfaces in Euclidean spaces.
Te examination of warped products in submanifold theory was initiated by Chen [7].Chen introduced the concept of CR-warped product submanifolds within the framework of almost Hermitian manifolds.He also provided an approximation for the norm of the second fundamental form by incorporating a warping function into the expressions.Tis pioneering work by Chen served as an inspiration for further research.Expanding on Chen's ideas, Hasegawa and Mihai [8] explored the contact form associated with these submanifolds.Tey derived a similar approximation for the second fundamental form in the context of contact CR-warped product submanifolds immersed in Sasakian space forms.In a related study [9], it was concluded that the homology groups of a contact CR-warped product submanifold, immersed in an odd-dimensional sphere, are trivial.Tis result was attributed to the nonexistence of stable integral currents and the vanishing of homology.Taking a step forward, Sahin [10,11] made notable progress by demonstrating that CR-warped product submanifolds in both R n and S 6 yield identical outcomes.However, diferent scholars have arrived at distinct fndings concerning the topological and diferentiable structures of submanifolds when imposing specifc constraints on the second fundamental form [4,6,9,12,13].
Te concept of a semisymmetric linear connection on a Riemannian manifold was initially proposed by Friedmann and Schouten [27].Subsequently, Hayden [28] defned a semisymmetric connection as a linear connection ∇ existing on an n-dimensional Riemannian manifold (M, g), where the torsion tensor Here, π represents a 1-form, and ω 1 , ω 2 ∈ TM.Further exploration of semisymmetric connections was undertaken by Yano [29], who investigated semisymmetric metric connections and analyzed their properties.It was demonstrated that a conformally fat Riemannian manifold possessing a semisymmetric connection exhibits a vanishing curvature tensor.Additionally, Sular and Ozgur [30] focused on investigating warped product manifolds equipped with a semisymmetric metric connection.Tey specifcally considered Einstein warped product manifolds featuring a semisymmetric metric connection.Teir work in [24] also provided additional insights into warped product manifolds with a semisymmetric metric connection.Motivated by these previous studies, our current interest lies in examining the infuence of a semisymmetric metric connection on semi-invariant warped product submanifolds and their homology within a Sasakian space form.

Preliminaries
Suppose (M, g) is an odd-dimensional Riemannian manifold.In that case, M is considered an almost contact metric manifold if there exists a tensor feld ϕ of type (1, 1) and a global vector feld ξ defned on M such that the following conditions are satisfed: where η represents the dual 1-form of ξ.It is widely recognized that an almost contact metric manifold is classifed as a Sasakian manifold if and only if the following conditions hold: It is straightforward to observe on a Sasakian manifold where ω 1 , ω 2 ∈ TM and ∇ is the Riemannian connection with respect to g.Now, defning a connection ∇ as such that ∇ g � 0 for any ω 1 , ω 2 ∈ TM, where ∇ is the Riemannian connection with respect to g.
Using ( 4) in ( 2), we have and A Sasakian manifold M is referred to as a Sasakian space form when it possesses a constant ϕ-holomorphic sectional curvature denoted by c, and it is denoted as M(c).Te curvature tensor R of a Sasakian space form M(c) with a semi-symmetric metric connection can be expressed as follows [32]: By performing a routine calculation, we can derive the Gauss and Weingarten formulas for a submanifold M that is isometrically immersed in a diferentiable manifold M equipped with a semisymmetric metric connection.Tese formulas are as follows: where ∇ is the induced semisymmetric metric connection on M, N ∈ T ⊥ M h is the second fundamental form of M, ∇ ⊥ is the normal connection on the normal bundle T ⊥ M, and A N is the shape operator.Te second fundamental form h and the shape operator are associated by the following formula: For the vector felds ω 1 ∈ TM and ω 3 ∈ T ⊥ M, we have the following decomposition: and where Pω 1 (tω 3 ) and Fω 1 (fω 3 ) are the tangential and normal parts of ϕω 1 (ϕω 3 ), respectively.Let R denote the Riemannian curvature tensor of the submanifold M. In the case of a semisymmetric connection, the equation of Gauss can be expressed as follows: for ω 1 , ω 2 , ω 3 , ω 4 ∈ TM.
Sular and Ozgur investigated the warped product structures of the form M 1 × f M 2 in their work [30].Tey examined these structures under the assumption of a semisymmetric metric connection and an associated vector feld P on the product manifold M 1 × f M 2 .Here, M 1 and M 2 denote Riemannian manifolds, and f represents a positive diferentiable function defned on M 1 known as the warping function.In this context, we present several key fndings from [30] in the form of the following lemma.Tese results hold signifcant relevance for the subsequent investigation.
Lemma 1.Given a warped product manifold M 1 × f M 2 with a semisymmetric metric connection ∇ , the following results hold: (1) If the associated vector feld P ∈ TM 1 , then (2) If P ∈ TN 2 , then where ω 1 ∈ TM 1 , ω 3 ∈ TM 2 , and π is the 1-form associated with the vector feld P.
Suppose R is the curvature tensor of the semisymmetric metric connection ∇ , then from equation ( 5) and part (ii) of Lemma 3.2 of [30], we derive where ω 1 , ω 2 ∈ TM 1 , ω 3 ∈ TM 2 , P ∈ TM 1 , and H f is the Hessian of the warping function f.In (4), we defned the semisymmetric connection by setting P � ξ.Tus, for a warped product submanifold M � M 1 × f M 2 of a Riemannian manifold M, we can derive the following relation using part (i) of Lemma 1: and In addition, the ( 14) with (6) yields for ξ, ω 1 , ω 2 ∈ TM 1 , and Journal of Mathematics

Semi-Invariant Warped Product Submanifolds and Their Homology
In 1981, Bejancu [32] gave the idea of semi-invariant submanifolds in an almost contact metric manifold.An m− dimensional Riemannian submanifold M of a Sasakian manifold M is called a semi-invariant submanifold if ξ is tangent to M, and there exists on M a diferentiable distribution D: x ⟶ D x ⊂ T x M such that D x is invariant under ϕ.Te orthogonal complementary distribution D ⊥ x of D x on M is anti-invariant, i.e., ϕD ⊥ ⊆ T ⊥ x M, where T x M and T ⊥ x M are the tangent space and normal space at x ∈ M. In [8], Hesigawa and Mihai considered the warped product submanifold of the type M T × f M ⊥ of a Sasakian manifold M, where M T is an invariant submanifold and N ⊥ is an antiinvariant submanifold, and ξ ∈ TN T .Tey called these types of submanifolds to contact CR-submanifold and provided some basic results.Troughout this study, we consider the warped products of the type N T × f N ⊥ of a Sasakian manifold M with semisymmetric metric connection and ξ ∈ TN T .We refer to these submanifolds as semi-invariant warped product submanifolds.Now, we start with the following initial results: product submanifold of a Sasakian manifold M endowed with a semisymmetric metric connection, then Proof.Using the Gauss formula and (5), we get (19) Now, by formula ( 16), we have this is part (i).Again using ( 5) and ( 15), and the Gauss formula, part (ii) is proved immediately.Now, by equation ∇ ω 3 ξ � ω 3 − η(ω 3 ) − Pω 3 , and applying ( 16), we have ξ lnf + η(ξ)ω 3 � ω 3 or ξ lnf � 0, which is the part (iii) of the Lemma.
Next, we investigate the existence of stable currents on semi-invariant warped product submanifolds.Specifcally, we establish a proof demonstrating that under certain conditions, stable currents do not exist.In this context, we present the well-known results established by Simons, Xin, and Lang.□ Lemma 3 (see [2,6]).Consider a compact submanifold M n of dimension n immersed in a space form M(c) with positive curvature c.If the second fundamental form satisfes the following inequality, where n 1 and n 2 are positive integers satisfying is an orthonormal basis in T x M for any x ∈ M, h denotes the second fundamental form of M, g represents the metric tensor, and ‖h(u i , u j )‖ is the norm of the second fundamental form evaluated at u i and u j .Under these conditions, it can be proven that there are no stable currents in M n .Moreover, it is observed that  H n 1 (M n , Z) � 0 and  H n 2 (M n , Z) � 0, where  H j (M n , Z) denotes the j-th homology group of M n with integer coefcients.Now, we have the following theorem.
⊥ be a compact semiinvariant warped product submanifold of a Sasakian space form M 2((n 1 /2)+n 2 )+1 (c), with semisymmetric metric connection.If the following inequality holds, then the (n 1 + 1)− stable currents are absent in M n 1 +1+n 2 .In addition, ⊥ .Ten, by equation ( 7) for unit odd-dimensional sphere and (11), we have Terefore, we get On the other hand, from (17) and part (iii) of Lemma 2, we have where H f is the Hessian form of f.Tus, we derive Putting ( 26) in (24), we have Journal of Mathematics using ( 16), we derive Using ( 29) in (27), we arrive at 6 Journal of Mathematics By using part (i) of Lemma 2, we conclude Putting the above value in (31), we fnd or Te proof is derived from (36) and Lemma 3.

Conclusion
Tis paper has provided an in-depth investigation of semiinvariant warped product submanifolds of Sasakian space forms equipped with a semisymmetric metric connection.
Trough our study, we have derived several fundamental results that contribute to the understanding of these submanifolds.Furthermore, we have explored the practical implications of our fndings by applying them to the homology analysis of these semi-invariant warped product submanifolds.Our analysis has revealed important insights into the homology properties of these submanifolds within the context of Sasakian space forms.One notable result we established is the proof of the absence of stable currents for these submanifolds under a specifc condition.Tis fnding has signifcant implications for the understanding of the geometric and topological behavior of semi-invariant warped product submanifolds in Sasakian space forms.Overall, this research contributes to the broader understanding of semi-invariant warped product submanifolds and their interaction with Sasakian space forms endowed with a semisymmetric metric connection.Te outcomes of this study pave the way for further research in this area, as well as potential applications in related felds such as diferential geometry and topology [33][34][35][36].
homology group of M and n 1 + 1, n 2 are the dimensions of the invariant submanifold M Let ξ, z 1 , . . ., z n 1 , s 1 , . . ., s n 2   be an orthonormal basis of TM, such that ξ, z 1 , . . ., z n 1   is an orthonormal basis of TM T and s 1 , . . ., s n 2   be the basis of TM