Some inequalities for warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds
- Akram Ali^{1}Email authorView ORCID ID profile,
- Wan Ainun Mior Othman^{1} and
- Cenap Ozel^{2}
https://doi.org/10.1186/s13660-015-0802-5
© Ali et al. 2015
Received: 10 April 2015
Accepted: 28 August 2015
Published: 18 September 2015
Abstract
In this paper, we study non-trivial warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds. In the beginning, we obtain some lemmas and then develop the general sharp inequalities for mixed totally geodesic warped products pseudo-slant submanifolds. The equality cases are also considered.
Keywords
MSC
1 Introduction
The geometric inequalities of warped product submanifolds have been studied actively since Chen [1] introduced the notion of a CR-warped product submanifold in a Kaehler manifold and obtained inequalities for the second fundamental form in terms of warping functions. In fact, different types of warped product submanifolds of different structures are studied for the last 14 years (see [2]). Recently, Sahin [3] established a general inequality for warped product pseudo-slant (also named hemi-slant) isometrically immersed in a Kaehler manifold for mixed totally geodesic. Later on, Uddin, et al. [4–6] obtained some inequalities of warped product submanifolds in different structures. In the present paper, we extend this idea into a nearly Kenmotsu manifold and derive the geometric inequalities of non-trivial warped product pseudo-slant submanifolds which are the natural extensions of CR-warped product submanifolds. Every CR-warped product submanifold is a non-trivial warped product pseudo-slant submanifold of the forms \(M_{\perp}\times_{f}M_{\theta}\) and \(M_{\theta}\times_{f}M_{\perp}\) with slant angle \(\theta=0\). First of all we consider non-trivial warped product pseudo-slant submanifolds of the form \(M=M_{\perp}\times_{f}M_{\theta}\) and \(M_{\theta}\times_{f}M_{\perp}\) such that \(M_{\theta}\) and \(M_{\perp}\) are proper-slant and anti-invariant submanifolds. Next we establish inequalities involving the second fundamental form, slant angle, and warping functions.
The paper is organized as follows: In Section 2, we review some preliminary formulas, definitions and address the study of pseudo-slant submanifolds of nearly Kenmotsu manifolds. In Section 3, we study warped product pseudo-slant submanifolds of a nearly Kenmotsu manifold and obtain some lemmas. In Section 4, we define an orthonormal frame for warped product pseudo-slant submanifolds and then obtain general sharp inequalities for the second fundamental form in terms of warping functions and slant immersions.
2 Preliminaries
There is another class of submanifolds, which is called the class of slant submanifold. For each non-zero vector U tangent to M at p, such that U is not proportional to \(\xi_{p}\), we denote by \(0\leq\theta(U)\leq\pi/2\), the angle between φU and \(T_{p}M\), which is called the Wirtinger angle. If the angle \(\theta(U)\) is constant for all \(U\in{T_{p}M-\langle\xi(p)\rangle}\) and \(p\in M\), then M is called a slant submanifold [8] and the angle θ is called the slant angle of M. Obviously if \(\theta=0\), M is invariant and if \(\theta =\pi/2\), M is anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant.
In an almost contact metric manifold, in fact, Cabrerizo et al. [9] obtained the following theorem.
Theorem 2.1
Moreover, we define pseudo-slant submanifolds of almost contact manifolds by using the slant distribution given in [10]. However, pseudo-slant submanifolds were defined by Carriazo [11] under the name anti-slant submanifolds as a particular class of bi-slant submanifolds. The definition of pseudo-slant is as follows.
Definition 2.1
- (i)
\(TM=\mathcal{D}^{\theta}\oplus\mathcal{D}^{\perp}\oplus\langle \xi\rangle\), where \(\langle\xi\rangle\) is a 1-dimensional distribution spanned by ξ.
- (ii)
\(\mathcal{D}^{\perp}\) is an anti-invariant distribution under φ i.e., \(\varphi\mathcal{D}^{\perp}\subseteq T^{\perp}M\).
- (iii)
\(\mathcal{D}^{\theta}\) is a slant distribution with slant angle \(\theta\neq0,\frac{\pi}{2}\).
3 Warped product submanifolds
Lemma 3.1
[12]
- (i)
\(\nabla_{X}Y\in\Gamma(TM_{1})\),
- (ii)
\(\nabla_{Z}X=\nabla_{X}Z=(X\ln f)Z\),
- (iii)
\(\nabla_{Z}W=\nabla'_{Z}W-g(Z, W)\nabla\ln f\),
Lemma 3.2
Proof
Lemma 3.3
- (iii)
\(g(h(X, X), \varphi Z)=g(h(Z, X), FX)\),
- (iv)
\(g(h(PX, PX), \varphi Z)=g(h(Z, PX), FPX)\),
Proof
Lemma 3.4
- (i)
\(g(h(Z, Z), FPX)=g(h(Z, PX), \varphi Z)+\{\eta(X)-(X\ln f)\} \cos^{2}\theta\|Z\|^{2}\),
- (ii)
\(g(h(Z, Z), FX)=g(h(Z, X), \varphi Z)-(PX\ln f)\|Z\|^{2}\),
Proof
4 Inequalities of warped product submanifolds
4.1 Inequality for a warped product pseudo-slant submanifold of the form \(M_{\perp}\times_{f}M_{\theta}\)
In this section, we obtain a geometric inequality of warped product pseudo-slant submanifold in terms of the second fundamental form such that ξ is tangent to the anti-invariant submanifold and the mixed totally geodesic submanifold. First of all we define an orthonormal frame for later use.
Let \(M=M_{\perp}\times_{f}M_{\theta}\) be a m-dimensional warped product pseudo-slant submanifold of a \({2n+1}\)-dimensional nearly Kenmotsu manifold M̃ with \(M_{\theta}\) of dimension \(d_{1}=2\beta\) and \(M_{\perp}\) of dimension \(d_{2}={\alpha+1}\), where \(M_{\theta}\) and \(M_{\perp}\) are the integral manifolds of \(\mathcal{D}^{\theta}\) and \(\mathcal{D}^{\perp}\), respectively. Then we consider that \(\{ e_{1}, e_{2} ,\ldots,e_{\alpha}, e_{d_{2}=\alpha+1}=\xi\}\) and \(\{ e_{\alpha+2}=e^{*}_{1},\ldots,e_{\alpha +\beta +1}=e^{*}_{\beta}, e_{\alpha+\beta+2}=e^{*}_{\beta+1}=\operatorname{sec}{\theta }Pe^{*}_{1},\ldots,e_{\alpha+1+2\beta}=e^{*}_{2\beta}=\operatorname{sec}\theta Pe^{*}_{\beta}\}\) are orthonormal frames of \(\mathcal{D}^{\perp}\) and \(\mathcal{D}^{\theta}\), respectively. Thus the orthonormal frames of the normal subbundles \(\varphi\mathcal{D}^{\perp}\), \(F\mathcal{ D}^{\theta}\), and μ, respectively, are \(\{e_{m+1}=\bar{e}_{1}=\varphi e_{1},\ldots,e_{m+\alpha }=\bar {e}_{\alpha}=\varphi e_{\alpha}\}\), \(\{e_{m+\alpha+1}=\bar{e}_{\alpha +1}=\tilde{e}_{1}=\operatorname{csc}{\theta}Fe^{*}_{1},\ldots,e_{m+\alpha+\beta}=\bar {e}_{\alpha +\beta}=\tilde{e}_{\beta}=\operatorname{csc}{\theta}Fe^{*}_{\beta}, e_{m+\alpha+\beta+1}=\bar {e}_{\alpha+\beta+1}=\tilde{e}_{\beta+1}=\operatorname{csc}{\theta}\operatorname{sec}{\theta }FPe^{*}_{1},\ldots,e_{m+\alpha+2\beta}=\bar{e}_{\alpha+2\beta}=\tilde {e}_{2\beta}=\operatorname{csc}{\theta}\operatorname{sec}{\theta}FPe^{*}_{\beta}\}\), and \(\{ e_{2m-1}=\bar{e}_{m},\ldots,e_{2n+1}=\bar{e}_{2(n-m+1)}\}\).
Theorem 4.1
- (i)The squared norm of the second fundamental form of M is given by$$ \|h\|^{2}\geq\frac{2\beta}{9}\cos^{2}{\theta}\bigl\{ \bigl\| \nabla^{\perp}\ln f\bigr\| ^{2}-1\bigr\} . $$(4.1)
- (ii)
The equality holds in (4.1), if \(M_{\perp}\) is totally geodesic and \(M_{\theta}\) is a totally umbilical submanifold into M̃.
Proof
4.2 Inequality for a warped product pseudo-slant submanifold of the form \(M_{\theta}\times_{f}M_{\perp}\)
In this section, we obtain an inequality of warped product pseudo-slant subamnifolds such that the structure vector field ξ is tangent to the slant submanifold \(M_{\theta}\). Taking ξ tangent to \(\mathcal {D}^{\theta}\), then we use the last frame.
Theorem 4.2
- (i)The squared norm of the second fundamental form of M is given by$$ \|h\|^{2}\geq\alpha\Biggl\{ \operatorname{csc}^{2}{\theta}\bigl(\bigl\| \nabla^{\theta}\ln f\bigr\| ^{2}-1\bigr)-\sum _{i=1}^{\beta}\bigl(e^{*}_{i}\ln f \bigr)^{2}\Biggr\} . $$(4.5)
- (ii)
If equality holds identically in (4.5), then \(M_{\theta}\) is a totally geodesic submanifold and \(M_{\perp}\) is a totally umbilical submanifold of M̃, respectively.
Proof
Declarations
Acknowledgements
The authors would like to express their appreciation to the referees for their comments and valuable suggestions. This work is supported by the University of Malaya research grants RG278-14AFR and RG270-13AFR.
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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