 Research
 Open access
 Published:
An inequality for warped product pseudoslant submanifolds of nearly cosymplectic manifolds
Journal of Inequalities and Applications volume 2015, Article number: 306 (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. 55:5569, 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.
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 6dimensional sphere \(S^{6}\) is a nearly cosymplectic manifold. A normal nearly cosymplectic manifold is cosymplectic (see [3]).
On the other hand, pseudoslant submanifolds of almost contact metric manifolds were studied by Carriazo [4] under the name of antislant submanifolds. Later on, Sahin studied these submanifolds for their warped products [5].
Recently, Uddin et al. studied warped product semiinvariant and semislant submanifolds of nearly cosymplectic manifolds [6–8]. In this paper, we study the warped product pseudoslant submanifolds of the type \(N_{\theta}\times{}_{f}N_{\perp}\) of a nearly cosymplectic manifold, where \(N_{\perp}\) and \(N_{\theta}\) are antiinvariant 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
Let M̃ be a \((2n+1)\)dimensional \(C^{\infty}\) manifold with almost contact structure \((\varphi, \xi, \eta)\) i.e., a \((1,1)\) tensor field φ, a vector field ξ and a 1form η on M̃ such that
There always exists a Riemannian metric g on an almost contact manifold M̃ satisfying the following compatibility condition:
where X and Y are vector fields on M̃ [2].
An almost contact structure \((\varphi, \xi, \eta)\) is said to be nearly cosymplectic if φ is Killing, i.e., if
or any X, Y tangent to M̃, where ∇̃ denotes the Riemannian connection of the metric g. Equation (2.3) is equivalent to \((\widetilde{\nabla}_{X}\varphi)X=0\), for each X tangent to M̃. A normal nearly cosymplectic structure is cosymplectic. It is well known that an almost contact metric manifold is cosymplectic if and only if ∇̃φ vanishes identically, i.e., \((\widetilde{\nabla}_{X}\varphi)Y=0\) and \(\widetilde{\nabla}_{X}\xi=0\).
On a nearly cosymplectic manifold the structure vector field ξ is Killing [9], that is,
for any Y, Z tangent to M̃.
Let M be submanifold of an almost contact metric manifold M̃ with induced metric g and let ∇ and \(\nabla^{\perp}\) be the induced connections on the tangent bundle TM and the normal bundle \(T^{\perp}M\) of M, respectively. Denote by \({\mathcal{F}}(M)\) the algebra of smooth functions on M and by \(\Gamma(TM)\) the \({\mathcal {F}}(M)\)module of smooth sections of TM over M. Then the Gauss and Weingarten formulas are given by
for each \(X, Y\in\Gamma(TM)\) and \(N\in\Gamma(T^{\perp}M)\), where h and \(A_{N}\) are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively, for the immersion of M into M̃. They are related as
where g denotes the Riemannian metric on M̃ as well as the one induced on M. The mean curvature vector H of M is given by \(H=\frac{1}{m} \sum_{i=1} ^{m} h(e_{i},e_{i}) \), where n is the dimension of M and \(\{e_{1},e_{2},\ldots,e_{m}\}\) is a local orthonormal frame of vector fields on M. A submanifold M of an almost contact metric manifold M̃ is said to be totally umbilical if the second fundamental form satisfies \(h(X,Y)=g(X,Y)H\), for all \(X, Y\in\Gamma(TM)\). The submanifold M is totally geodesic if \(h(X, Y)=0\), for all \(X, Y\in\Gamma(TM)\) and minimal if \(H=0\).
Now, let \(\{e_{1},\ldots, e_{m}\}\) be an orthonormal basis of tangent space TM and \(e_{r}\) belong to the orthonormal basis \(\{ e_{m+1},\ldots, e_{2n+1}\}\) of the normal bundle \(T^{\perp}M\), we put
For a differentiable function φ on M, the gradient ∇⃗φ is defined by
for any \(X\in\Gamma(TM)\). As a consequence, we have
For any \(X\in\Gamma(TM)\), we write
where PX is the tangential component and FX is the normal component of φX. A submanifold M of an almost contact metric manifold M̃ is said to be invariant if F is identically zero, that is, \(\varphi X\in\Gamma(TM)\) and antiinvariant if P is identically zero, that is, \(\varphi X\in\Gamma(T^{\perp}M)\), for any \(X\in\Gamma(TM)\).
Let M be a submanifold tangent to the structure vector field ξ isometrically immersed into an almost contact metric manifold M̃. Then M is said to be a contact CRsubmanifold if there exists a pair of orthogonal distributions \({\mathcal{D}}:p\to{\mathcal{D}}_{p}\) and \({\mathcal{D}}^{\perp}:p\to {\mathcal{D}}^{\perp}_{p}\), \(\forall p\in M\) such that:

(i)
\(TM=\mathcal{D}\oplus\mathcal{D}^{\perp}\oplus\langle\xi\rangle\), where \(\langle\xi\rangle\) is the 1dimensional 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 antiinvariant, i.e., \(\varphi {\mathcal{D}^{\perp}}\subseteq T^{\perp}M\).
Invariant and antiinvariant submanifolds are the special cases of a contact CRsubmanifold. If we denote the dimensions of the distributions \(\mathcal{D}\) and \({\mathcal{D}^{\perp}}\) by \(d_{1}\) and \(d_{2}\), respectively. Then M is invariant (resp. antiinvariant) if \(d_{2}=0\) (resp. \(d_{1}=0\)).
There is another class of submanifolds that is called the slant submanifold. For each nonzero 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 antiinvariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor antiinvariant.
We recall the following result for a slant submanifold of an almost contact metric manifold.
Theorem 2.1
[10]
Let M be a submanifold of an almost contact metric manifold M̃, such that ξ is tangent to M. Then M is slant if and only if there exists a constant \(\lambda \in[0, 1]\) such that
Furthermore, if θ is slant angle of M, then \(\lambda= \cos ^{2}\theta\).
The following relations are straightforward consequences of (2.12):
for all \(X,Y\in\Gamma(TM)\).
Now, we give the brief introduction of pseudoslant submanifolds introduced by Carriazo in [4] under the name of antislant submanifolds, which are the generalization of contact CRsubmanifolds and slant submanifolds [10]. He defined these submanifolds as follows.
Definition 2.1
A submanifold M of an almost contact metric manifold M̃ is said to be a pseudoslant submanifold if there exists a pair of orthogonal distributions \({\mathcal {D}}^{\perp}\) and \({\mathcal{D}}^{\theta}\) on M such that:

(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 antiinvariant, i.e., \(\varphi({\mathcal{D}}^{\perp})\subset T^{\perp}M\).

(iii)
The distribution \({\mathcal{D}}^{\theta}\) is slant with angle \(\theta\neq\frac{\pi}{2}\).
The normal bundle \(T^{\perp}M\) of a pseudoslant submanifold is decomposed as
where μ is an invariant normal subbundle under φ.
3 Warped product pseudoslant submanifolds
In this section, we discuss the warped product submanifolds of a nearly cosymplectic manifold. These manifolds were studied by Bishop and O’Neill [11]. They defined these manifolds as follows: Let \((N_{1}, g_{1})\) and \((N_{2}, g_{2})\) be two Riemannian manifolds and f a positive differentiable function on \(N_{1}\). Then their warped product \(M=N_{1}\times{}_{f}N_{2}\) is the product manifold \(N_{1}\times N_{2}\) equipped with the Riemannian structure such that
The function f is called the warping function on M. It was proved in [11] that for any \(X\in\Gamma(TN_{1})\) and \(Z\in\Gamma(TN_{2})\), the following holds:
where ∇ denote the LeviCivita connection M. A warped product manifold \(M=N_{1}\times{}_{f}N_{2}\) is said to be trivial if the warping function f is constant. If \(M=N_{1}\times{}_{f} N_{2}\) is a warped product manifold then the base manifold \(N_{1}\) is totally geodesic and the fiber \(N_{2}\) is a totally umbilical submanifold of M, respectively [11].
Now, we discuss the warped product pseudoslant 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
Let \(M=N_{\theta}\times{}_{f}N_{\perp}\) be a warped product pseudoslant submanifold of a nearly cosymplectic manifold M̃, then:

(i)
\(\xi\ln f=0\),

(ii)
\(2g(h(X, Y), \varphi Z)=g(h(X, Z), FY)+g(h(Y, Z), FX)\).
for any \(X, Y\in\Gamma(TN_{\theta})\) and \(Z\in\Gamma(TN_{\perp})\).
Proof
For any \(Z, W\in\Gamma(TN_{\perp})\) and ξ tangential to \(N_{\theta}\), we have
Then from (3.1), we obtain
By the polarization identity, we derive
Thus the first part follows from (3.2) and (3.3) by using (2.4). For the second part, consider \(X, Y\in\Gamma(TN_{\theta})\) and \(Z\in\Gamma (TN_{\perp})\), we have
Then by the covariant derivative property of φ, we derive
From (3.1), we obtain
The second term of right hand side is identically zero by the orthogonality of vector fields, thus we have
Then by the polarization identity, we obtain
Then from (3.4) and (3.5), we get
The first term of right hand side is identically zero by (2.3), thus we get (ii), which proves the lemma completely. □
Lemma 3.2
Let \(M=N_{\theta}\times{}_{f}N_{\perp}\) be a warped product pseudoslant submanifold of a nearly cosymplectic manifold M̃, where \(N_{\perp}\) and \(N_{\theta}\) are antiinvariant and proper slant submanifolds of M̃, respectively. Then:

(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)
\(2g(h(Z, W), FPX)=g(h(PX, Z), \varphi W)+g(h(PX, W), \varphi Z)2\cos^{2}\theta(X\ln f)g(Z, W)\)
for any \(X\in\Gamma(TN_{\theta})\) and \(Z, W\in\Gamma(TN_{\perp})\).
Proof
For any \(Z, W\in\Gamma(TN_{\perp})\) and \(X\in\Gamma (TN_{\theta})\), we have
Using (2.5) and (2.6), we obtain
Then from (2.7) and (3.1), we get
Then by the polarization identity we derive
Then, from the above relation, (i) holds by using (2.3). If we interchange X by PX in (i) we get (ii) by using Theorem 2.1 and Lemma 3.1(i). Thus, the proof is complete. □
Now, we construct the following frame for a warped product pseudoslant 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 mdimensional warped product pseudoslant submanifold of a \((2n+1)\)dimensional nearly cosymplectic manifold M̃ such that \(N_{\perp}\) is a qdimensional antiinvariant 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 \(2(nm+1)\).
Theorem 3.1
Let \(M=N_{\theta}\times{}_{f} {N}_{\perp}\) be a mixed geodesic warped product pseudoslant submanifold of a nearly cosymplectic manifold M̃ such that \(N_{\perp}\) and \(N_{\theta}\) are antiinvariant and proper slant submanifolds of M̃, respectively. Then:

(i)
The squared norm of the second fundamental form h of M satisfies
$$\h\^{2} \geq q\cot^{2}\theta\bigl\ \nabla^{\theta}\ln f \bigr\ ^{2} $$where \(\nabla^{\theta}\ln f\) is the gradient of lnf over \(N_{\theta}\) and q is the dimension of \(N_{\perp}\).

(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
From (2.8), we have
Then using the frame fields of \(TN_{\perp}\) and \(TN_{\theta}\), we get
Since M is mixed geodesic, the second term of right hand side is identically zero and break the above relation for the frames of \(F(TN_{\theta})\), \(\varphi(TN_{\perp})\), and μ. Then we derive
The first term of right hand side is identically zero by Lemma 3.1(ii) for a mixed geodesic warped product submanifold. Also, we have no relation for the μ components with h and \(g(h(Z, W), FW^{\prime})\), for any \(Z, W, W^{\prime}\in\Gamma(TN_{\perp})\) in terms of the warping function. Thus, we shall leave all positive terms except the fifth term, then we have
Using Lemma 3.2 for mixed geodesic warped products, we derive
Since \(e^{*}_{2p+1}\ln f=\xi\ln f=0\), from (2.10), we derive
Then by Theorem 2.1, we obtain
Then from (2.9), (2.13), and the trigonometric identities, finally, we get
which is inequality (i). If the equality holds in (i), then from the second and third remaining terms
and
Then from (3.9) and (3.10), we get
Similarly, from the remaining fourth and sixth terms, we conclude that
and
Then from (3.12) and (3.13), we get
Thus (ii) follows from (3.11) and (3.14). This completes the proof of the theorem. □
References
Blair, DE: Almost contact manifolds with Killing structure tensors I. Pac. J. Math. 39, 285292 (1971)
Blair, DE, Showers, DK: Almost contact manifolds with Killing structures tensors II. J. Differ. Geom. 9, 577582 (1974)
Blair, DE, Yano, K: Affine almost contact manifolds and fmanifolds with affine Killing structure tensors. Kodai Math. Semin. Rep. 23, 473479 (1971)
Carriazo, A: New Developments in Slant Submanifolds Theory. Narosa Publishing House, New Delhi (2002)
Sahin, B: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Pol. Math. 95, 207226 (2009)
Uddin, S, Kon, SH, Khan, MA, Singh, K: Warped product semiinvariant submanifolds of nearly cosymplectic manifolds. Math. Probl. Eng. (2011). doi:10.1155/2011/230374
Uddin, S, Khan, KA: An inequality for contact CRwarped product submanifolds of nearly cosymplectic manifolds. J. Inequal. Appl. 2012, 304 (2012)
Uddin, S, Mustafa, A, Wong, BR, Ozel, C: A geometric inequality for warped product semislant submanifolds of nearly cosymplectic manifolds. Rev. Unión Mat. Argent. 55, 5569 (2014)
Endo, H: On the curvature tensor of nearly cosymplectic manifolds of constant φsectional curvature. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 51, 439554 (2005)
Cabrerizo, JL, Carriazo, A, Fernandez, LM, Fernandez, M: Slant submanifolds in Sasakian manifolds. Glasg. Math. J. 42, 125138 (2000)
Bishop, RL, O’Neill, B: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 149 (1969)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
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.
About this article
Cite this article
AlSolamy, F.R. An inequality for warped product pseudoslant submanifolds of nearly cosymplectic manifolds. J Inequal Appl 2015, 306 (2015). https://doi.org/10.1186/s136600150825y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s136600150825y