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

Al-Dayel, Ibrahim; Khan, Meraj Ali; Shuaib, Mohammad

doi:https://doi.org/10.1155/2023/5035740

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5035740 | https://doi.org/10.1155/2023/5035740

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

Ibrahim Al-Dayel,¹Meraj Ali Khan,¹and Mohammad Shuaib²

Academic Editor: Qingkai Zhao

Received10 Jul 2023

Revised09 Sept 2023

Accepted22 Sept 2023

Published10 Oct 2023

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.

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 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 and 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 existing on an -dimensional Riemannian manifold , where the torsion tensor satisfies . Here, represents a 1-form, and . 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 is an odd-dimensional Riemannian manifold. In that case, is considered an almost contact metric manifold if there exists a tensor field of type and a global vector field defined on such that the following conditions are satisfied:where represents the dual 1-form of . It is widely recognized that an almost contact metric manifold is classified as a Sasakian manifold if and only if the following conditions hold:

It is straightforward to observe on a Sasakian manifold thatwhere and is the Riemannian connection with respect to .

Now, defining a connection assuch that for any , where is the Riemannian connection with respect to . The connection is semisymmetric because . Using (4) in (2), we haveand

A Sasakian manifold is referred to as a Sasakian space form when it possesses a constant -holomorphic sectional curvature denoted by , and it is denoted as . The curvature tensor of a Sasakian space form with a semi-symmetric metric connection can be expressed as follows [32]:for all , where .

By performing a routine calculation, we can derive the Gauss and Weingarten formulas for a submanifold that is isometrically immersed in a differentiable manifold equipped with a semisymmetric metric connection. These formulas are as follows: and , where is the induced semisymmetric metric connection on , is the second fundamental form of , is the normal connection on the normal bundle , and is the shape operator. The second fundamental form and the shape operator are associated by the following formula:

For the vector fields and , we have the following decomposition:andwhere and are the tangential and normal parts of , respectively.

Let denote the Riemannian curvature tensor of the submanifold . In the case of a semisymmetric connection, the equation of Gauss can be expressed as follows:for .

Sular and Oz̈gur investigated the warped product structures of the form in their work [30]. They examined these structures under the assumption of a semisymmetric metric connection and an associated vector field on the product manifold . Here, and denote Riemannian manifolds, and represents a positive differentiable function defined on 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 with a semisymmetric metric connection , the following results hold:(1)If the associated vector field , then(2)If , thenwhere and is the 1-form associated with the vector field .

Suppose is the curvature tensor of the semisymmetric metric connection , then from equation (5) and part (ii) of Lemma 3.2 of [30], we derivewhere , and is the Hessian of the warping function .

In (4), we defined the semisymmetric connection by setting . Thus, for a warped product submanifold of a Riemannian manifold , we can derive the following relation using part (i) of Lemma 1:and

In addition, the (14) with (6) yieldsfor , and .

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 dimensional Riemannian submanifold of a Sasakian manifold is called a semi-invariant submanifold if is tangent to , and there exists on a differentiable distribution such that is invariant under . The orthogonal complementary distribution of on is anti-invariant, i.e., , where and are the tangent space and normal space at . In [8], Hesigawa and Mihai considered the warped product submanifold of the type of a Sasakian manifold , where is an invariant submanifold and is an anti-invariant submanifold, and . 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 of a Sasakian manifold with semisymmetric metric connection and . We refer to these submanifolds as semi-invariant warped product submanifolds.

Now, we start with the following initial results:

Lemma 2. Let be a semi-invariant warped product submanifold of a Sasakian manifold endowed with a semisymmetric metric connection, then(i),(ii),(iii),

Proof. Using the Gauss formula and (5), we getNow, by formula (16), we havethis is part (i). Again using (5) and (15), and the Gauss formula, part (ii) is proved immediately. Now, by equation , and applying (16), we have or , 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 of dimension immersed in a space form with positive curvature . If the second fundamental form satisfies the following inequality,where and are positive integers satisfying , is an orthonormal basis in for any , denotes the second fundamental form of , represents the metric tensor, and is the norm of the second fundamental form evaluated at and . Under these conditions, it can be proven that there are no stable currents in . Moreover, it is observed that and , where denotes the -th homology group of with integer coefficients.

Now, we have the following theorem.

Theorem 4. Let be a compact semi-invariant warped product submanifold of a Sasakian space form , with semisymmetric metric connection. If the following inequality holds,then the stable currents are absent in . In addition, , where is the th homology group of and are the dimensions of the invariant submanifold and the anti-invariant submanifold , respectively.

Proof. Let be an orthonormal basis of , such that is an orthonormal basis of and be the basis of . Then, by equation (7) for unit odd-dimensional sphere and (11), we haveTherefore, we getOn the other hand, from (17) and part (iii) of Lemma 2, we havewhere is the Hessian form of . Thus, we derivePutting (26) in (24), we haveSinceusing (16), we deriveUsing (29) in (27), we arrive atorMoreover, if is an orthonormal frame of , then is the orthonormal frame of , thenBy using part (i) of Lemma 2, we concludeorPutting the above value in (31), we findorThe 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].

Data Availability

No data are used in this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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).

References

H. Federer and W. Fleming, “Normal and integral currents,” Annals of Mathematics, vol. 72, pp. 458–520, 1960.
View at: Publisher Site | Google Scholar
H. B. Lawson and J. Simons, “On stable currents and their application to global problems in real and complex geometry,” Annals of Mathematics, vol. 98, no. 3, pp. 427–450, 1973.
View at: Publisher Site | Google Scholar
P. F. Leung, “On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold,” Proceedings of the Edinburgh Mathematical Society, vol. 28, no. 3, pp. 305–311, 1985.
View at: Publisher Site | Google Scholar
Y. L. Xin, “An application of integral currents to the vanishing theorems,” Scientia Sinica, vol. 27, pp. 233–241, 1984.
View at: Google Scholar
X. S. Zhang, “Nonexistence of stable currents in submanifolds of a product of two spheres,” Bulletin of the Australian Mathematical Society, vol. 44, no. 2, pp. 325–336, 1991.
View at: Publisher Site | Google Scholar
J. Liu and Q. Y. Zhang, “Non-existence of stable currents in submanifolds of the Euclidean spaces,” Journal of Geometry, vol. 96, no. 1-2, pp. 125–133, 2009.
View at: Publisher Site | Google Scholar
B. Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds,” Monatshefte for Mathematik, vol. 133, no. 3, pp. 177–195, 2001.
View at: Publisher Site | Google Scholar
I. Hasegawa and I. Mihai, “Contact CR-warped product submanifolds in Sasakian manifolds,” Geometriae Dedicata, vol. 102, no. 1, pp. 143–150, 2003.
View at: Publisher Site | Google Scholar
B. Sahin and F. Sahin, “Homology of contact CR-warped product submanifolds of an odd-dimensional unit sphere,” Bulletin of the Korean Mathematical Society, vol. 52, no. 1, pp. 215–222, 2015.
View at: Publisher Site | Google Scholar
F. Sahin, “On the topology of CR-warped product submanifolds,” International Journal of Geometric Methods in Modern Physics, vol. 15, no. 2, Article ID 1850032, 2018.
View at: Publisher Site | Google Scholar
F. Sahin, “Homology of submanifolds of six dimensional sphere,” Journal of Geometry and Physics, vol. 145, Article ID 103471, 2019.
View at: Publisher Site | Google Scholar
A. Ali, F. Mofarreh, C. Ozel, and W. A. M. Othman, “Homology of warped product submanifolds in the unit sphere and its applications,” International Journal of Geometric Methods in Modern Physics, vol. 17, no. 8, Article ID 2050121, 2020.
View at: Publisher Site | Google Scholar
H. W. Xu and F. Ye, “Differentiable sphere theorems for submanifolds of positive K-th ricci curvature,” Manuscripta Mathematica, vol. 138, no. 3-4, pp. 529–543, 2012.
View at: Publisher Site | Google Scholar
Y. Li, M. K. Gupta, S. Sharma, and S. K. Chaubey, “On ricci curvature of a homogeneous generalized matsumoto finsler space,” Mathematics, vol. 11, p. 3365, 2023.
View at: Google Scholar
Y. Li, S. Bhattacharyya, A. Azami, A. Saha, and S. K. Hui, “Harnack estimation for nonlinear, weighted, heat-type equation along geometric flow and applications,” Mathematics, vol. 11, p. 2516, 2023.
View at: Google Scholar
Y. Li and A. Caliskan, “Quaternionic shape operator and rotation matrix on ruled surfaces,” Aximos, vol. 12, p. 486, 2023.
View at: Google Scholar
Y. Li and E. Guler, “Hypersurfaces of revolution family supplying in pseudo-euclidean space ,” AIMS Math, vol. 8, no. 10, pp. 24957–24970, 2023.
View at: Google Scholar
Y. Li, H. A. Kumara, M. S. Siddesha, and D. M. Naik, “Characterization of ricci almost soliton on lorentzian manifolds,” Symmetry, vol. 15, p. 1175, 2023.
View at: Google Scholar
Y. Li and E. Guler, “A hypersurfaces of revolution family in the five-dimensional pseudo-euclidean space ,” Mathematics, vol. 11, p. 3427, 2023.
View at: Google Scholar
Y. Li, K. Eren, and S. Ersoy, “On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space,” AIMS Math, pp. 22256–22273, 2023.
View at: Google Scholar
Y. Li and M. Mak, “Framed natural mates of framed curves in euclidean 3-space,” Mathematics, vol. 2023, 3571 pages, 2013.
View at: Google Scholar
S. Smale, “Generalized Poincares conjecture in dimensions greater than four,” Annals of Mathematics, vol. 74, no. 2, pp. 391–406, 1961.
View at: Publisher Site | Google Scholar
B. Sahin, “Warped product pointwise semi-slant submanifolds of Kähler manifolds,” Portugaliae Mathematica, vol. 70, no. 3, pp. 251–268, 2013.
View at: Publisher Site | Google Scholar
C. Özgür and Ozgur, “Warped warped products with a semi-symmetric non-metric Connectionroducts with a semi-symmetric non-metric connection,” Arabian Journal for Science and Engineering, vol. 36, no. 3, pp. 461–473, 2011.
View at: Publisher Site | Google Scholar
H. W. Xu and E. T. Zhao, “Topological and differentiable sphere theorems for complete submanifolds,” Communications in Analysis and Geometry, vol. 17, no. 3, pp. 565–585, 2009.
View at: Publisher Site | Google Scholar
T. Imai, “Notes on semi-symmetric metric connections,” Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, Vol. I. Tensor (N.S.), vol. 24, pp. 293–296, 1972.
View at: Google Scholar
A. Friedmann and J. A. Schouten, “Über die Geometrie der halbsymmetrischen Übertragungen,” Mathematische Zeitschrift, vol. 21, no. 1, pp. 211–223, 1924.
View at: Publisher Site | Google Scholar
H. A. Hayden, “Sub-spaces of a space with torsion,” Proceedings of the London Mathematical Society, vol. 2–34, no. 1, pp. 27–50, 1932.
View at: Publisher Site | Google Scholar
K. Yano, “On semi-symmetric metric connections,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 15, pp. 1579–1586, 1970.
View at: Google Scholar
S. Sular and C. Ozgur, “Warped products with a semi-symmetric metric connection,” Taiwanese Journal of Mathematics, vol. 15, no. 4, pp. 1701–1719, 2011.
View at: Google Scholar
Y. Wang, “Chen inequalities for submanifolds of complex space forms and Sasakian space forms with quarter symmetric connections,” International Journal of Geometric Methods in Modern Physics, vol. 16, no. 8, Article ID 1950118, 2019.
View at: Publisher Site | Google Scholar
A. Bejancu, “CR submanifolds of a Kaehler manifold I,” Proceedings of the American Mathematical Society, vol. 69, no. 1, pp. 135–142, 1978.
View at: Publisher Site | Google Scholar
B. Y. Chen and O. J. Garay, “Pointwise slant submanifolds in almost Hermitian manifolds,” Turkish Journal of Mathematics, vol. 36, pp. 630–640, 2012.
View at: Publisher Site | Google Scholar
H. P. Fu and H. W. Xu, “Vanishing and topological sphere theorems for submanifolds in a hyperbolic space,” International Journal of Mathematics, vol. 19, no. 7, pp. 811–822, 2008.
View at: Publisher Site | Google Scholar
T. Imai, “Hypersurfaces of a Riemannian manifold with semi-symmetric metric connection,” Tensor, vol. 23, pp. 300–306, 1972.
View at: Google Scholar
K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore, 1984.

Copyright

Copyright © 2023 Ibrahim Al-Dayel et al. 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.

PDF Download Citation

Download other formats

Order printed copies