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

Abstract

This 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 findings by applying them to the homology analysis of these submanifolds. Notably, we present a proof demonstrating the absence of stable currents for these submanifolds under a specific condition.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The homology groups of a manifold provide an algebraic representation that captures important topological characteristics. These groups contain extensive topological information about the various components, voids, tunnels, and overall structure of the manifold. Consequently, homology theory finds numerous applications in diverse fields such as root construction, molecular anchoring, image segmentation, and genetic expression analysis. The relationship between submanifold theory and homological theory is widely recognized for its significance. 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 specific types of hypersurfaces in Euclidean spaces.

The 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. This 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. They 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. This 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, different scholars have arrived at distinct findings concerning the topological and differentiable structures of submanifolds when imposing specific constraints on the second fundamental form [4, 6, 9, 12, 13].

Nevertheless, multiple researchers have arrived at contrasting conclusions regarding the topological and differentiable characteristics of submanifolds through the utilization of advanced theories such as submanifold theory and soliton theory, among others [6, 12, 14–18]. Further inspiration for our work can be gleaned from the papers cited as references [5, 9, 19–26].

The concept of a semisymmetric linear connection on a Riemannian manifold was initially proposed by Friedmann and Schouten [27]. Subsequently, Hayden [28] defined a semisymmetric connection as a linear connection $\nabla$ existing on an $n$-dimensional Riemannian manifold $M,g$, where the torsion tensor $T$ satisfies $T{\omega }_1,{\omega }_2=\pi {\omega }_2{\omega }_1-\pi {\omega }_1{\omega }_2$. Here, $\pi$ represents a 1-form, and ${\omega }_1,{\omega }_2\in \text{TM}$. 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 flat 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. They specifically considered Einstein warped product manifolds featuring a semisymmetric metric connection. Their 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 influence of a semisymmetric metric connection on semi-invariant warped product submanifolds and their homology within a Sasakian space form.

2. Preliminaries

Suppose $\overline{M},g$ is an odd-dimensional Riemannian manifold. In that case, $\overline{M}$ is considered an almost contact metric manifold if there exists a tensor field $\varphi$ of type $\mathrm{1,1}$ and a global vector field $\xi$ defined on $\overline{M}$ such that the following conditions are satisfied:$$\begin{array}{c}\text{(1)}& {\varphi }^2{\omega }_1=-{\omega }_1+\eta {\omega }_1\xi ,g{\omega }_1,\xi =\eta {\omega }_1,g\varphi {\omega }_1,\varphi {\omega }_2=g{\omega }_1,{\omega }_2-\eta {\omega }_1\eta {\omega }_2,\end{array}$$where $\eta$ represents the dual 1-form of $\xi$. It is widely recognized that an almost contact metric manifold is classified as a Sasakian manifold if and only if the following conditions hold:$$\begin{array}{}\text{(2)}& {\overline{\overline{\nabla }}}_{{\omega }_1}\varphi {\omega }_2=g{\omega }_1,{\omega }_2\xi -\eta {\omega }_2{\omega }_1.\end{array}$$

It is straightforward to observe on a Sasakian manifold $\overline{M}$ that$$\begin{array}{}\text{(3)}& {\overline{\overline{\nabla }}}_{{\omega }_1}\xi =-\varphi {\omega }_1,\end{array}$$where ${\omega }_1,{\omega }_2\in \mathrm{T}\overline{\mathrm{M}}$ and $\overline{\overline{\nabla }}$ is the Riemannian connection with respect to $g$.

Now, defining a connection $\overline{\nabla }$ as$$\begin{array}{}\text{(4)}& {\overline{\nabla }}_{{\omega }_1}{\omega }_2={\overline{\overline{\nabla }}}_{{\omega }_1}{\omega }_2+\eta {\omega }_2{\omega }_1-g{\omega }_1,{\omega }_2\xi ,\end{array}$$such that $\overline{\nabla }g=0$ for any ${\omega }_1,{\omega }_2\in \text{TM}$, where $\overline{\overline{\nabla }}$ is the Riemannian connection with respect to $g$. The connection $\overline{\nabla }$ is semisymmetric because $T{\omega }_1,{\omega }_2=\eta {\omega }_2{\omega }_1-\eta {\omega }_1{\omega }_2$. Using (4) in (2), we have$$\begin{array}{}\text{(5)}& {\overline{\nabla }}_{{\omega }_1}\varphi {\omega }_2=g{\omega }_1,{\omega }_2\xi -g{\omega }_1,\varphi {\omega }_2\xi -\eta {\omega }_2{\omega }_1-\eta {\omega }_2\varphi {\omega }_1,\end{array}$$and$$\begin{array}{}\text{(6)}& {\overline{\nabla }}_{{\omega }_1}\xi ={\omega }_1-\eta {\omega }_1\xi -\varphi {\omega }_1.\end{array}$$

A Sasakian manifold $\overline{M}$ is referred to as a Sasakian space form when it possesses a constant $\varphi$-holomorphic sectional curvature denoted by $c$, and it is denoted as $\overline{M}c$. The curvature tensor $\overline{R}$ of a Sasakian space form $\overline{M}c$ with a semi-symmetric metric connection can be expressed as follows [32]:$$\begin{array}{c}\text{(7)}& \overline{R}{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4=\frac{c+3}{4}g{\omega }_2,{\omega }_{3}g{\omega }_1,{\omega }_4-g{\omega }_1,{\omega }_{3}g{\omega }_2,{\omega }_4+\frac{c-1}{4}\eta {\omega }_1\eta {\omega }_{3}g{\omega }_2,{\omega }_4-\eta {\omega }_2\eta {\omega }_{3}g{\omega }_1,{\omega }_4\\ +g{\omega }_1,{\omega }_3\eta {\omega }_2\eta {\omega }_4-g{\omega }_2,{\omega }_3\eta {\omega }_1\eta {\omega }_4\\ +g\varphi {\omega }_2,{\omega }_{3}g\varphi {\omega }_1,{\omega }_4+g\varphi {\omega }_3,{\omega }_{1}g\varphi {\omega }_2,{\omega }_4\\ -2g\varphi {\omega }_1,{\omega }_{2}g\varphi {\omega }_3,{\omega }_4+\beta {\omega }_1,{\omega }_{3}g{\omega }_2,{\omega }_4\\ -\beta {\omega }_2,{\omega }_{3}g{\omega }_1,{\omega }_4+\beta {\omega }_2,{\omega }_{4}g{\omega }_1,{\omega }_3\\ -\beta {\omega }_1,{\omega }_{4}g{\omega }_2,{\omega }_3,\end{array}$$for all ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\in \mathrm{T}\overline{\mathrm{M}}$, where $\beta {\omega }_1,{\omega }_2={\overline{\nabla }}_{{\omega }_1}{\omega }_2-\eta {\omega }_1\eta {\omega }_2+1/2g{\omega }_1,{\omega }_2$.

By performing a routine calculation, we can derive the Gauss and Weingarten formulas for a submanifold $M$ that is isometrically immersed in a differentiable manifold $\overline{M}$ equipped with a semisymmetric metric connection. These formulas are as follows: ${\overline{\nabla }}_{{\omega }_1}{\omega }_2={\nabla }_{{\omega }_1}{\omega }_2+h{\omega }_1,{\omega }_2$ and ${\overline{\nabla }}_{{\omega }_1}N=-A_N{\omega }_1+{\nabla }_{{\omega }_1}^{\perp }N+\eta N{\omega }_1$, where $\nabla$ is the induced semisymmetric metric connection on $M$, $N\in T^{\perp }Mh$ is the second fundamental form of $M$, ${\nabla }^{\perp }$ is the normal connection on the normal bundle $T^{\perp }M$, and $A_N$ is the shape operator. The second fundamental form $h$ and the shape operator are associated by the following formula:$$\begin{array}{}\text{(8)}& gh{\omega }_1,{\omega }_2,N=g{A}_N{\omega }_1,{\omega }_2.\end{array}$$

For the vector fields ${\omega }_1\in \text{TM}$ and ${\omega }_3\in T^{\perp }M$, we have the following decomposition:$$\begin{array}{}\text{(9)}& \varphi {\omega }_1=P{\omega }_1+F{\omega }_1,\end{array}$$and$$\begin{array}{}\text{(10)}& \varphi {\omega }_3=t{\omega }_3+f{\omega }_3,\end{array}$$where $P{\omega }_{1}t{\omega }_3$ and $F{\omega }_{1}f{\omega }_3$ are the tangential and normal parts of $\varphi {\omega }_1\varphi {\omega }_3$, 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:$$\begin{array}{}\text{(11)}& \overline{R}{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4=R{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4-gh{\omega }_1,{\omega }_4,h{\omega }_2,{\omega }_3+gh{\omega }_2,{\omega }_4,h{\omega }_1,{\omega }_3,\end{array}$$for ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\in \text{TM}$.

Sular and Oz̈gur investigated the warped product structures of the form $M_1{\times }_f{M}_2$ in their work [30]. They examined these structures under the assumption of a semisymmetric metric connection and an associated vector field $P$ on the product manifold $M_1{\times }_f{M}_2$. Here, $M_1$ and $M_2$ denote Riemannian manifolds, and $f$ represents a positive differentiable function defined on $M_1$ known as the warping function. In this context, we present several key findings from [30] in the form of the following lemma. These results hold significant relevance for the subsequent investigation.

Lemma 1.

Given a warped product manifold $M_1{\times }_f{M}_2$ with a semisymmetric metric connection $\overline{\nabla }$, the following results hold:

(1) If the associated vector field $P\in {\text{TM}}_1$, then$$\begin{array}{c}\text{(12)}& {\overline{\nabla }}_{{\omega }_1}{\omega }_3=\frac{{\omega }_{1}f}{f}{\omega }_3,{\overline{\nabla }}_{{\omega }_3}{\omega }_1=\frac{{\omega }_{1}f}{f}{\omega }_3+\pi {\omega }_1{\omega }_3.\end{array}$$

(2) If $P\in {\text{TN}}_2$, then

$$\begin{array}{c}\text{(13)}& {\overline{\nabla }}_{{\omega }_1}{\omega }_3=\frac{{\omega }_{1}f}{f}{\omega }_3,{\overline{\nabla }}_{{\omega }_3}{\omega }_1=\frac{{\omega }_{1}f}{f}{\omega }_3,\end{array}$$where ${\omega }_1\in {\text{TM}}_1,{\omega }_3\in {\text{TM}}_2,$ and $\pi$ is the 1-form associated with the vector field $P$.

Suppose $R$ is the curvature tensor of the semisymmetric metric connection $\overline{\nabla }$, then from equation (5) and part (ii) of Lemma 3.2 of [30], we derive$$\begin{array}{c}\text{(14)}& R{\omega }_1,{\omega }_3{\omega }_2=\frac{H^f{\omega }_1,{\omega }_2}{f}+\frac{Pf}{f}g{\omega }_1,{\omega }_2{\omega }_3+\pi Pg{\omega }_1,{\omega }_2{\omega }_3+g{\omega }_2,{\overline{\nabla }}_{{\omega }_1}P{\omega }_3-\pi {\omega }_1\pi {\omega }_2{\omega }_3,\end{array}$$where ${\omega }_1,{\omega }_2\in {\text{TM}}_1,{\omega }_3\in {\text{TM}}_2,P\in {\text{TM}}_1$, and $H^f$ is the Hessian of the warping function $f$.

In (4), we defined the semisymmetric connection by setting $P=\xi$. Thus, for a warped product submanifold $M=M_1{\times }_f{M}_2$ of a Riemannian manifold $\overline{M}$, we can derive the following relation using part (i) of Lemma 1:$$\begin{array}{}\text{(15)}& {\nabla }_{{\omega }_1}{\omega }_3={\omega }_1\text{lnf}{\omega }_3,\end{array}$$and$$\begin{array}{}\text{(16)}& {\nabla }_{{\omega }_3}{\omega }_1={\omega }_1\text{lnf}{\omega }_3+\eta {\omega }_1{\omega }_3.\end{array}$$

In addition, the (14) with (6) yields$$\begin{array}{c}\text{(17)}& R{\omega }_1,{\omega }_3{\omega }_2=\frac{H^f{\omega }_1,{\omega }_2}{f}{\omega }_3+\frac{\xi f}{f}g{\omega }_1,{\omega }_2{\omega }_3+2g{\omega }_1,{\omega }_2{\omega }_3-2\eta {\omega }_1\eta {\omega }_2{\omega }_3-g{\omega }_2,\varphi {\omega }_1{\omega }_3,\end{array}$$for $\xi ,{\omega }_1,{\omega }_2\in {\text{TM}}_1$, and ${\omega }_3\in {\text{TM}}_2$.

3. 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 $\overline{M}$ is called a semi-invariant submanifold if $\xi$ is tangent to $M$, and there exists on $M$ a differentiable distribution $D:x⟶D_x\subset T_{x}M$ such that $D_x$ is invariant under $\varphi$. The orthogonal complementary distribution $D_x^{\perp }$ of $D_x$ on $M$ is anti-invariant, i.e., $\varphi D^{\perp }\subseteq T_x^{\perp }M$, where $T_{x}M$ and $T_x^{\perp }M$ are the tangent space and normal space at $x\in M$. In [8], Hesigawa and Mihai considered the warped product submanifold of the type $M_T{\times }_f{M}_{\perp }$ of a Sasakian manifold $\overline{M}$, where $M_T$ is an invariant submanifold and $N_{\perp }$ is an anti-invariant submanifold, and $\xi \in {\text{TN}}_T$. They called these types of submanifolds to contact CR-submanifold and provided some basic results. Throughout this study, we consider the warped products of the type $N_T{\times }_f{N}_{\perp }$ of a Sasakian manifold $\overline{M}$ with semisymmetric metric connection and $\xi \in {\text{TN}}_T$. We refer to these submanifolds as semi-invariant warped product submanifolds.

Now, we start with the following initial results:

Lemma 2.

Let $M=M_T{\times }_f{M}_{\perp }$ be a semi-invariant warped product submanifold of a Sasakian manifold $\overline{M}$ endowed with a semisymmetric metric connection, then

(i) $gh\varphi {\omega }_1,{\omega }_3,\varphi {\omega }_4={\omega }_1\text{lnf\hspace{0.17em}}g{\omega }_3,{\omega }_4+\eta {\omega }_{1}g{\omega }_3,{\omega }_4$,

(ii) $gh{\omega }_1,{\omega }_2,\varphi {\omega }_3=0$,

(iii) $\xi \text{\hspace{0.17em}lnf}=0$,

$$\begin{array}{c}\text{(18)}& \forall {\omega }_1,{\omega }_2\in {\text{TN}}_T,{\omega }_3,{\omega }_4\in {\text{TN}}_{\perp },\\ \xi \in {\text{TN}}_T.\end{array}$$

Proof.

Using the Gauss formula and (5), we get$$\begin{array}{c}\text{(19)}& gh\varphi {\omega }_1,{\omega }_3,\varphi {\omega }_4=g{\overline{\nabla }}_{{\omega }_3}\varphi {\omega }_1,\varphi {\omega }_4,=g{\overline{\nabla }}_{{\omega }_3}{\omega }_1,{\omega }_4.\end{array}$$

Now, by formula (16), we have$$\begin{array}{c}\text{(20)}& gh\varphi {\omega }_1,{\omega }_3,\varphi {\omega }_4=g{\nabla }_{{\omega }_3}{\omega }_1,{\omega }_4={\omega }_1\text{lnf\hspace{0.17em}}g{\omega }_3,{\omega }_4+\eta {\omega }_{1}g{\omega }_3,{\omega }_4,\end{array}$$this is part (i). Again using (5) and (15), and the Gauss formula, part (ii) is proved immediately. Now, by equation ${\nabla }_{{\omega }_3}\xi ={\omega }_3-\eta {\omega }_3-P{\omega }_3$, and applying (16), we have $\xi \text{\hspace{0.17em}lnf}+\eta \xi {\omega }_3={\omega }_3$ or $\xi \text{\hspace{0.17em}lnf}=0$, which is the part (iii) of the Lemma.

Next, we investigate the existence of stable currents on semi-invariant warped product submanifolds. Specifically, 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 $\overline{M}c$ with positive curvature $c$. If the second fundamental form satisfies the following inequality,$$\begin{array}{}\text{(21)}& {\displaystyle \sum }_{i=1}^{n_1}{\displaystyle \sum }_{s=n_1+1}^{n}2{h{u}_i,u_j}^2-gh{u}_i,u_i,h{u}_i,u_s<n_1{n}_{2}c,\end{array}$$where $n_1$ and $n_2$ are positive integers satisfying $n_1+n_2=n$,$u_1,\dots ,u_n$ is an orthonormal basis in $T_{x}M$ for any $x\in M$,$h$ 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 ${\tilde{H}}_{n_1}{M}^n,\mathbb{Z}=0$ and ${\tilde{H}}_{n_2}{M}^n,\mathbb{Z}=0$, where ${\tilde{H}}_j{M}^n,\mathbb{Z}$ denotes the $j$-th homology group of $M^n$ with integer coefficients.

Now, we have the following theorem.

Theorem 4.

Let $M^{n_1+n_2+1}=M_T^{n_1+1}{\times }_f{M}_{\perp }^{n_2}$ be a compact semi-invariant warped product submanifold of a Sasakian space form $M^{2n_1/2+n_2+1}c$, with semisymmetric metric connection. If the following inequality holds,$$\begin{array}{c}\text{(22)}& ∆f-{\nabla f}_{D-\xi }^2+\frac{n_2}{f}{\nabla f}^2+\eta \nabla f{n}_2>\frac{2}{f}1+n_1+\frac{c}{4}2-3n_{1}f+\frac{11}{4}{n}_{1}f+\frac{3}{2}f+\frac{f{n}_1+1}{n_2}{\displaystyle \sum }_{k=1}^{n_2}\beta n_k,n_k+f{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i,\end{array}$$then the $n_1+1-$ stable currents are absent in $M^{n_1+1+n_2}$. In addition, $H_{n_1+1}{M}^n,Z=0,H_{n_2}{M}^n,Z=0$, where $H_{j}M,Z$ is the $j-$th homology group of $M$ and $n_1+1,n_2$ are the dimensions of the invariant submanifold $M_T^{n_1+1}$ and the anti-invariant submanifold $M_{\perp }^{n_2}$, respectively.

Proof.

Let $\xi ,z_1,\dots ,z_{n_1},s_1,\dots ,s_{n_2}$ be an orthonormal basis of $\text{TM}$, such that $\xi ,z_1,\dots ,z_{n_1}$ is an orthonormal basis of ${\text{TM}}_T$ and $s_1,\dots ,s_{n_2}$ be the basis of ${\text{TM}}_{\perp }^{n_2}$. Then, by equation (7) for unit odd-dimensional sphere and (11), we have$$\begin{array}{c}\text{(23)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}gR{z}_i,s_k{z}_i,s_k=-n_1+1n_2\frac{c+3}{4}-\frac{c-1}{4}{n}_2-n_1+1{\displaystyle \sum }_{k=1}^{n_2}{\beta }_1{s}_k,s_k-n_2{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i+{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}-gh{s}_k,s_k,h{z}_i,z_i+{h{z}_i,s_k}^2.\end{array}$$

Therefore, we get$$\begin{array}{c}\text{(24)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}gR{z}_i,n_k{z}_i,s_k+{h{z}_i,s_k}^2=-n_1+1n_2\frac{c+3}{4}-\frac{c-1}{4}{n}_2-n_1+1{\displaystyle \sum }_{k=1}^{n_2}{\beta }_1{s}_k,s_k-n_2{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i+{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}-gh{s}_k,s_k,h{z}_i,z_i\\ +2{h{z}_i,s_k}^2.\end{array}$$

On the other hand, from (17) and part (iii) of Lemma 2, we have$$\begin{array}{}\text{(25)}& R{z}_i,s_k{z}_i=\frac{H^f{z}_i,z_i}{f}{s}_k+2g{z}_i,z_i{s}_k-2\eta z_i\eta z_i{s}_k,\end{array}$$where $H^f$ is the Hessian form of $f$. Thus, we derive$$\begin{array}{c}\text{(26)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}gR{z}_i,s_k{z}_i,s_k=\frac{n_2}{f}{\displaystyle \sum }_{i=1}^{n_1+1}g{\nabla }_{z_i}\nabla f,z_i+2n_1+1n_2-2n_2=\frac{n_2}{f}{\displaystyle \sum }_{i=1}^{n_1+1}g\nabla z_i\nabla f,z_i+2n_1{n}_2.\end{array}$$

Putting (26) in (24), we have$$\begin{array}{c}\text{(27)}& \frac{n_2}{f}{\displaystyle \sum }_{i=1}^{n_1+1}g{\nabla }_{z_i}\nabla f,z_i+2n_1{n}_2+{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}{h{z}_i,s_k}^2=-n_1+1n_2\frac{c+3}{4}-n_2\frac{c-1}{4}\\ -n_1+1{\displaystyle \sum }_{k=1}^{n_2}\beta s_k,n_k-n_2{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i\\ +{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}-gh{s}_k,s_k,h{z}_i,z_i+2{h{z}_i,s_k}^2.\end{array}$$

Since$$\begin{array}{}\text{(28)}& ∆f=-\mathrm{div}\nabla f=-{\displaystyle \sum }_{i=1}^{n_1+1}g{\nabla }_{z_i}\nabla f,z_i-{\displaystyle \sum }_{k=1}^{n_2}g{\nabla }_{s_k}\nabla f,s_k,\end{array}$$using (16), we derive$$\begin{array}{}\text{(29)}& ∆f=-{\displaystyle \sum }_{i=1}^{n_1+1}g{\nabla }_{z_i}\nabla f,z_i-\frac{n_2}{f}{\nabla f}^2-\eta \nabla f{n}_2.\end{array}$$

Using (29) in (27), we arrive at$$\begin{array}{c}\text{(30)}& ∆f-\frac{f}{n_2}{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}{h{z}_i,s_k}^2=n_1+2f\frac{c}{4}+\frac{11}{4}{\text{fn}}_1+\frac{1}{2}f+\frac{f{n}_1+1}{n_2}{\displaystyle \sum }_{k=1}^{n_2}\beta s_k,s_k+f{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i\\ +\frac{f}{n_2}{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}gh{s}_k,s_k,h{z}_i,z_i\\ -2{h{z}_i,s_k}^2-\frac{n_2}{f}{\nabla f}^2-\eta \nabla f{n}_2,\end{array}$$or$$\begin{array}{c}\text{(31)}& ∆f-\frac{f}{n_2}{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}{h{z}_i,s_k}^2=\frac{c}{4}{n}_1+2f+\frac{11}{4}{n}_{1}f+\frac{1}{2}f+\frac{f{n}_1+1}{n_2}{\displaystyle \sum }_{k=1}^{n_2}\beta s_k,s_k+f{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i\\ -\frac{f}{n_2}{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}2{h{z}_i,s_k}^2\\ -gh{s}_k,s_k,h{z}_i,z_i-\frac{n_2}{f}{\nabla f}^2-\eta \nabla f{n}_2.\end{array}$$

Moreover, if $s_1,\dots ,s_k$ is an orthonormal frame of $N_{\perp }$, then $\varphi s_1,\dots ,\varphi s_k$ is the orthonormal frame of $T^{\perp }M$, then$$\begin{array}{}\text{(32)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}{h{z}_i,s_k}^2={\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}g{h{z}_i,s_k,\varphi s_k}^2.\end{array}$$

By using part (i) of Lemma 2, we conclude$$\begin{array}{}\text{(33)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}{h{z}_i,s_k}^2={\nabla \text{lnf}}_{D-\xi }^2{n}_2+n_2,\end{array}$$or$$\begin{array}{}\text{(34)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}{h{z}_i,s_k}^2=\frac{n_2}{f}{\nabla f}_{D-\xi }^2+n_2.\end{array}$$

Putting the above value in (31), we find$$\begin{array}{c}\text{(35)}& \frac{n_2}{f}∆f-\frac{n_2}{f}{\nabla f}_{D-\xi }^2=\frac{c}{4}{n}_2{n}_1+2+\frac{11}{4}{n}_1{n}_2+\frac{3}{2}{n}_2+n_1+1{\displaystyle \sum }_{k=1}^{n_2}\beta s_k,s_k+n_2{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i-{\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}2{h{z}_i,s_k}^2-gh{s}_k,s_k,h{z}_i,z_i\\ -\frac{{n_2}^2}{f^2}{\nabla f}^2-\eta \nabla f\frac{q^2}{f},\end{array}$$or$$\begin{array}{c}\text{(36)}& {\displaystyle \sum }_{i=1}^{n_1+1}{\displaystyle \sum }_{k=1}^{n_2}2{h{z}_i,s_k}^2-gh{s}_k,s_k,h{z}_i,z_i-n_1{n}_{2}c=-\frac{n_2}{f}∆f+\frac{n_2}{f}{\nabla f}_{D-\xi }^2-\frac{n_2^2}{f^2}{\nabla f}^2-\eta \nabla f\frac{q^2}{f}\\ +\frac{c}{4}{n}_{2}2-3n_1+\frac{11}{4}{n}_1{n}_2+\frac{3}{2}{n}_2\\ +n_1+1{\displaystyle \sum }_{k=1}^{n_2}\beta s_k,s_k+n_2{\displaystyle \sum }_{i=1}^{n_1}\beta z_i,z_i.\end{array}$$

The proof is derived from (36) and Lemma 3.

4. Conclusion

This paper has provided an in-depth investigation of semi-invariant warped product submanifolds of Sasakian space forms equipped with a semisymmetric metric connection. Through 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 findings 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 specific condition. This finding has significant 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. The outcomes of this study pave the way for further research in this area, as well as potential applications in related fields such as differential geometry and topology [33–36].

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23003).

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