An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds
- Falleh R Al-Solamy^{1}Email author
https://doi.org/10.1186/s13660-015-0825-y
© Al-Solamy 2015
Received: 7 May 2015
Accepted: 15 September 2015
Published: 30 September 2015
Abstract
Warped product submanifolds of nearly cosymplectic manifolds were studied in Uddin et al. (Math. Probl. Eng. 2011, doi:10.1155/2011/230374), Uddin and Khan (J. Inequal. Appl. 2012:304, 2012) and Uddin et al. (Rev. Unión Mat. Argent. 55:55-69, 2014). In this paper, we study warped product submanifolds of nearly cosymplectic manifolds in which the base manifold is slant and thus we derive a sharp relation for the squared norm of the second fundamental form. The equality case is also considered.
Keywords
MSC
1 Introduction
The almost contact manifolds with Killing structures tensors were defined in [1] as nearly cosymplectic manifolds. Later, these manifolds were studied by Blair and Showers from the topological point of view [2]. A totally geodesic hypersurface \(S^{5}\) of a 6-dimensional sphere \(S^{6}\) is a nearly cosymplectic manifold. A normal nearly cosymplectic manifold is cosymplectic (see [3]).
On the other hand, pseudo-slant submanifolds of almost contact metric manifolds were studied by Carriazo [4] under the name of anti-slant submanifolds. Later on, Sahin studied these submanifolds for their warped products [5].
Recently, Uddin et al. studied warped product semi-invariant and semi-slant submanifolds of nearly cosymplectic manifolds [6–8]. In this paper, we study the warped product pseudo-slant submanifolds of the type \(N_{\theta}\times{}_{f}N_{\perp}\) of a nearly cosymplectic manifold, where \(N_{\perp}\) and \(N_{\theta}\) are anti-invariant and proper slant submanifolds of a nearly cosymplectic manifold, respectively. We derive an inequality for the second fundamental form of such warped product immersions in terms of the warping function and the slant angle. The equality case is also discussed.
2 Preliminaries
- (i)
\(TM=\mathcal{D}\oplus\mathcal{D}^{\perp}\oplus\langle\xi\rangle\), where \(\langle\xi\rangle\) is the 1-dimensional distribution spanned by the structure vector field ξ.
- (ii)
\(\mathcal{D}\) is invariant, i.e., \(\varphi\mathcal{D}=\mathcal{D}\).
- (iii)
\(\mathcal{D}^{\perp}\) is anti-invariant, i.e., \(\varphi {\mathcal{D}^{\perp}}\subseteq T^{\perp}M\).
There is another class of submanifolds that is called the slant submanifold. For each non-zero vector X tangent to M at x, such that X is not proportional to \(\xi_{x}\), we denote by \(0\leq\theta (X)\leq\frac{\pi}{2}\), the angle between φX and \(T_{x}M\) is called the Wirtinger angle. If the angle \(\theta(X)\) is constant for all nonzero \(X\in T_{x}M-\langle\xi_{x}\rangle\) and \(x\in M\), then M is said to be a slant submanifold [10] and the angle θ is the slant angle of M. Obviously if \(\theta=0\), M is invariant and if \(\theta=\frac{\pi}{2}\), M is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant.
We recall the following result for a slant submanifold of an almost contact metric manifold.
Theorem 2.1
[10]
Now, we give the brief introduction of pseudo-slant submanifolds introduced by Carriazo in [4] under the name of anti-slant submanifolds, which are the generalization of contact CR-submanifolds and slant submanifolds [10]. He defined these submanifolds as follows.
Definition 2.1
- (i)
TM admits the orthogonal direct decomposition \(TM={\mathcal{D}}^{\perp}\oplus{\mathcal{D}}^{\theta}\oplus\langle\xi\rangle\).
- (ii)
The distribution \({\mathcal{D}}^{\perp}\) is anti-invariant, i.e., \(\varphi({\mathcal{D}}^{\perp})\subset T^{\perp}M\).
- (iii)
The distribution \({\mathcal{D}}^{\theta}\) is slant with angle \(\theta\neq\frac{\pi}{2}\).
3 Warped product pseudo-slant submanifolds
Now, we discuss the warped product pseudo-slant submanifolds of the type \(N_{\theta}\times{}_{f}N_{\perp}\) of a nearly cosymplectic manifold M̃. We consider the structure vector field ξ tangent to the base manifold \(N_{\theta}\) of the warped products. If ξ is tangential to \(N_{\perp}\) then the warped product is trivial [6]. We have the following results for later use.
Lemma 3.1
- (i)
\(\xi\ln f=0\),
- (ii)
\(2g(h(X, Y), \varphi Z)=g(h(X, Z), FY)+g(h(Y, Z), FX)\).
Proof
Lemma 3.2
- (i)
\(2g(h(Z, W), FX)=g(h(X, Z), \varphi W)+g(h(X, W), \varphi Z)+2(PX\ln f)g(Z, W)\),
- (ii)
\(2g(h(Z, W), FPX)=g(h(PX, Z), \varphi W)+g(h(PX, W), \varphi Z)-2\cos^{2}\theta(X\ln f)g(Z, W)\)
Proof
Now, we construct the following frame for a warped product pseudo-slant submanifold \(M=N_{\theta}\times{}_{f}N_{\perp}\) of a \((2n+1)\)-dimensional nearly cosymplectic manifold.
Let \(M=N_{\theta}\times{}_{f} N_{\perp}\) be a m-dimensional warped product pseudo-slant submanifold of a \((2n+1)\)-dimensional nearly cosymplectic manifold M̃ such that \(N_{\perp}\) is a q-dimensional anti-invariant submanifold and \(N_{\theta}\) is a \((2p+1)\)-dimensional slant submanifold tangent to the structure vector field ξ of M̃, respectively. Then the orthonormal frame fields of the tangent spaces of \(N_{\perp}\) and \(N_{\theta}\), respectively, are \(\{e_{1},\ldots,e_{q}\}\) and \(\{ e_{q+1}=e_{1}^{*},\ldots,e_{q+p}=e_{p}^{*}, e_{q+p+1}=e_{p+1}^{*}=\sec\theta Pe_{1}^{*},\ldots, e_{q+2p}=e_{2p}^{*}=\sec\theta Pe_{p}^{*}, e_{q+2p+1}=e_{m}=\xi \}\). The orthonormal frames of \(\varphi(TN_{\perp})\), \(F(TN_{\theta})\), and μ, respectively, are \(\{e_{m+1}=\varphi e_{1},\ldots, e_{m+q}=\varphi e_{q}\}\), \(\{e_{m+q+1}=\tilde{e}_{1}=\csc\theta Fe_{1}^{*},\ldots, e_{m+p+q}=\tilde{e}_{p}^{*}=\csc\theta Fe_{p}^{*}, e_{m+p+q+1}=\tilde{e}_{p+1}=\csc\theta\sec\theta FPe_{1}^{*},\ldots, e_{m+2p+q}=\tilde{e}_{2p}=\csc\theta\sec\theta FPe_{p}^{*}\}\) and \(\{e_{2m},\ldots, e_{2n+1}\}\). The dimensions of \(\varphi(TN_{\perp})\), \(F(TN_{\theta})\), and μ, respectively, are q, 2p, and \(2(n-m+1)\).
Theorem 3.1
- (i)The squared norm of the second fundamental form h of M satisfieswhere \(\nabla^{\theta}\ln f\) is the gradient of lnf over \(N_{\theta}\) and q is the dimension of \(N_{\perp}\).$$\|h\|^{2} \geq q\cot^{2}\theta\bigl\| \nabla^{\theta}\ln f \bigr\| ^{2} $$
- (ii)
If the equality holds in (i), then \(h(Z, W)\) lies in \(F(TN_{\theta})\) for any \(Z, W\in\Gamma(TN_{\perp})\) and \(h(X, Y)\) lies in \(\varphi(TN_{\perp})\), for any \(X, Y\in\Gamma(TN_{\theta})\).
Proof
Declarations
Acknowledgements
The author is thankful to Dr. Siraj Uddin for the discussion as regards the constructed frame which improved the inequality. Also he is thankful to the reviewers for their valuable suggestions, which really improved the quality of the manuscript.
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Authors’ Affiliations
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