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Estimation of inequalities for warped product semi-slant submanifolds of Kenmotsu space forms
- Misbah Liaqat^{1},
- Piscoran Laurian-Ioan^{2},
- Wan Ainun Mior Othman^{3},
- Akram Ali^{3}Email authorView ORCID ID profile,
- Abdullah Gani^{1} and
- Cenap Ozel^{4, 5}
https://doi.org/10.1186/s13660-016-1177-y
© Liaqat et al. 2016
- Received: 22 May 2016
- Accepted: 16 September 2016
- Published: 29 September 2016
Abstract
In this paper, we construct the geometric inequalities for the squared norm of the mean curvature and warping functions of warped product semi-slant submanifolds in Kenmotsu space forms. The equality cases are also discussed.
Keywords
- warped products
- semi-slant submanifold
- inequalities
- Kenmotsu space forms
MSC
- 53C40
- 58C35
- 53C55
- 53C42
- 53C15
- 53D15
1 Introduction
The approach of such type inequalities for warped products in almost Hermitian and almost contact metric manifolds has been an important field for a few decades. Especially, Chen in [2] obtained the sharp relationship between norm of the squared mean curvature and the warping function f of the warped product \(M_{1}\times _{f}M_{2}\) isometrically immersed in a real space form, i.e., we have the following.
Theorem 1.1
However, on the base of the literature, we find that several inequalities have been extended to various structures for warped products by many geometers in [3–9]. Therefore other inequalities also appear, in [4, 5, 10–19] for slant submanifolds and semi-slant submanifolds in different curvature forms, which are called Chen inequalities. In addition, it is well known that Atceken [20] studied the non-existence of the warped product semi-slant submanifolds of a Kenmotsu manifold such that structure vector ξ is tangent to the fiber. Meanwhile, Uddin in [21] and Srivastava in [22] proved that the warped product semi-slant submanifold of a Kenmotsu manifold exists in the forms \(M=M_{T}\times_{f}M_{\theta}\) and \(M=M_{\theta}\times_{f}M_{T}\), except in the case when the structure vector field ξ is tangent to \(M_{T}\) and \(M_{\theta}\), respectively. Moreover, we have studied some inequalities that Cioroboiu [13] and Aktan et al. [12] obtained for semi-slant submanifolds by constructing its orthonormal frame but overlooking the suitable conditions for inequalities of a warped semi-slant product. Therefore, one needs to derive the inequalities for the mean curvature and warping functions with slant angles of a warped semi-slant product in Kemotsu space form. In the current paper, we are extending studies like [13] for warped product semi-slant submanifolds in a Kenmotsu space form. We also generalize some other inequalities for CR-warped product submanifolds in special cases because of the warped product of semi-slant generalized CR-warped products in Kenmotsu manifolds. Moreover, the equality cases and geometric inequalities applications related to Wireless Sensor Network are also discussed.
2 Preliminaries
Recently, Cabrerizo et al. [24] extended of the mentioned definition into a characterization for a slant submanifold in a contact metric manifold. In fact, they have obtained the following theorem.
Theorem 2.1
There is another class, which is called a semi-slant submanifold. The notion of semi-slant submanifolds were defined and studied by Papaghiuc in [25] as a natural generalization of CR-submanifolds of almost Hermitian manifolds in terms of the slant distribution and it was later extended to the setting of contact manifolds by Cabrerizo [26]. One defined these submanifolds as follows.
Definition 2.2
- (i)
\(TM=\mathcal{D}\oplus\mathcal{D}^{\theta}\oplus\langle\xi\rangle\) where \(\langle\xi\rangle\) is a 1-dimensional distribution spanned by ξ:
- (ii)
\(\mathcal{D}\) is invariant, i.e., \(\varphi (\mathcal{D})\subseteq\mathcal{D}\);
- (iii)
\(\mathcal{D}^{\theta}\) is slant distribution with slant angle \(\theta\neq0, \pi/2\).
Lemma 2.3
[27]
3 Main inequalities
In this section, as applications of very famous studied of Nolker in [28], we obtain the following inequality for warped product semi-slant submanifolds of Kenmotsu space form such that ξ is tangent to the first factor of the warped product, i.e., we have the following.
Theorem 3.1
- (i)The relation between warping function and the squared norm of mean curvature is obtainedwhere \(n_{i}=\operatorname{dim} M_{i}, i=T,\theta\), and Δ is the Laplacian operator on \(M_{T}\).$$\begin{aligned} \frac{\Delta f}{f}\leq\frac{n^{2}}{4n_{2}}\|H\|^{2}+ \frac{c-3}{4}n_{1}-\frac {c+1}{4n_{2}} \bigl[3d_{1}+d_{2} \bigl(2+3\cos^{2}\theta\bigr) \bigr], \end{aligned}$$(3.1)
- (ii)
The equality case holds in (3.1) if and only if \(n_{1}\cdot H_{T}=n_{2}\cdot H_{\theta}\), where \(H_{T}\) and \(H_{\theta}\) are partially mean curvature vectors on \(M_{T}\) and \(M_{\theta}\), respectively. Moreover, ϕ is a mixed totally geodesic immersion.
Proof
Theorem 3.2
- (i)The relation between warping function and the norm of the squared mean curvature is given bywhere \(n_{i}=\operatorname{dim} M_{i}, i=T,\theta\), and Δ is the Laplacian operator on \(M_{\theta}\).$$\begin{aligned} \frac{\Delta f}{f}\leq\frac{n^{2}}{4n_{2}}\|H\|^{2}+ \frac{c-3}{4}n_{1}-\frac {c+1}{4n_{2}} \bigl(3d_{2}+d_{1} \bigl\{ 2+3\cos^{2}\theta\bigr\} \bigr), \end{aligned}$$(3.17)
- (ii)
The equality case holds in (3.17) if and only if \(n_{1}\cdot H_{T}=n_{2}\cdot H_{\theta}\), where \(H_{T}\) and \(H_{\theta}\) are partially mean curvature vector fields on \(M_{T}\) and \(M_{\theta}\), respectively, and ϕ is a mixed totally geodesic immersion.
Proof
The proof of Theorem 3.2 is similar to Theorem 3.1 by reversing and considering that the structure vector field ξ is normal to the fiber. □
In the sense of Papaghiuc, i.e., the generalization of semi-slant submanifolds, we directly obtain the following corollaries by using Theorem 3.1, Theorem 3.2, and \(\theta =\frac{\pi}{2}\).
Corollary 3.3
Corollary 3.4
4 Applications
Any geometric inequality reflects a free or constrained optimum problem with suitable strategies for improved bandwidth management in wireless communications due to the dynamically changing traffic conditions and network performance. Except, some applications of geometric inequalities can be found in Wireless Sensor Networks related to Power balanced coverage-time optimization and Coverage by randomly deployed sensors [29]. Therefore, some applications of geometric inequalities can be found in computer sciences.
Declarations
Acknowledgements
This work is funded by of the Malaysian Ministry of Education under the High Impact Research Grant no. UM.C/HIR/625/1/MOE/FSIT/03. The corresponding author supported by UMRG grant no. FP074-2015A.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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