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Some inequalities for warped product pseudoslant submanifolds of nearly Kenmotsu manifolds
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 291 (2015)
Abstract
In this paper, we study nontrivial warped product pseudoslant 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 pseudoslant submanifolds. The equality cases are also considered.
1 Introduction
The geometric inequalities of warped product submanifolds have been studied actively since Chen [1] introduced the notion of a CRwarped 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 pseudoslant (also named hemislant) isometrically immersed in a Kaehler manifold for mixed totally geodesic. Later on, Uddin, et al. [4â€“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 nontrivial warped product pseudoslant submanifolds which are the natural extensions of CRwarped product submanifolds. Every CRwarped product submanifold is a nontrivial warped product pseudoslant 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 nontrivial warped product pseudoslant 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 properslant and antiinvariant 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 pseudoslant submanifolds of nearly Kenmotsu manifolds. In Section 3, we study warped product pseudoslant submanifolds of a nearly Kenmotsu manifold and obtain some lemmas. In Section 4, we define an orthonormal frame for warped product pseudoslant submanifolds and then obtain general sharp inequalities for the second fundamental form in terms of warping functions and slant immersions.
2 Preliminaries
Let MÌƒ be a \((2m+1)\)dimensional almost contact manifold with almost contact structure \((\varphi,\xi,\eta)\) where Ï† is a \((1,1)\) tensor field, Î¾ is the structure vector field and Î· is a dual 1form satisfying the following property:
On an almost contact manifold there exists a Riemannian metric g which satisfies the following:
for any U, V tangent to MÌƒ. Then an almost contact manifold MÌƒ equipped with a Riemannian metric g is called an almost contact metric manifold \((\widetilde{M}, g)\). Furthermore, an almost contact metric manifold is known to be a Kenmotsu manifold [7] if
and
for any vector fields U, V on MÌƒ, where âˆ‡Ìƒ denote the Riemannian connection with respect to g. Then an almost contact metric manifold MÌƒ is said to be a nearly Kenmotsu manifolds [7], if
for any U, V tangent to MÌƒ. We shall use the symbol \(\Gamma(T\widetilde{M})\) to denote the Lie algebras of vector fields on a manifold MÌƒ.
Let M be a Riemannian manifold that is isometrically immersed in an almost contact metric manifold MÌƒ and denote by the same symbol g the Riemannian metric induced on M. Let \(\Gamma(TM)\) and \(\Gamma(T^{\perp}M)\) be the Lie algebras of vector fields tangent to M and normal to M, respectively, and \(\nabla^{\perp}\) the induced connection on \(T^{\perp}M\). 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. Denote by âˆ‡ the LeviCivita connection of M then the Gauss and Weingarten formulas are given by
for each \(U, V\in\Gamma(TM)\) 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 MÌƒ. They are related as
Now for any \(U\in\Gamma(TM)\), we write
where PU and FU are the tangential and normal components of Ï†U, respectively. Similarly for any \(N\in\Gamma(T^{\perp}M)\), we have
where tN (resp. fN) are the tangential (resp. normal) components of Ï†N. A submanifold M is said to be totally geodesic and totally umbilical, if \(h(U, V)=0\) and \(h(U,V)=g(U, V)H\), respectively.
There is another class of submanifolds, which is called the class of slant submanifold. For each nonzero 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 Ï†U 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 Î¸ is called the slant angle of M. Obviously if \(\theta=0\), M is invariant and if \(\theta =\pi/2\), M is antiinvariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor antiinvariant.
In an almost contact metric manifold, in fact, Cabrerizo et al. [9] obtained the following theorem.
Theorem 2.1
Let M be a submanifold of an almost contact metric manifold MÌƒ such that \(\xi\in TM\). Then M is slant if and only if there exists a constant \(\lambda\in[0,1]\) such that
Furthermore, in such a case, Î¸ is the slant angle and it satisfies \(\lambda=\cos^{2}\theta\).
Hence, for a slant submanifold M of an almost contact metric manifold MÌƒ, we have the following relations which are consequences of Theorem 2.1:
for any \(U, V\in\Gamma(TM)\).
Moreover, we define pseudoslant submanifolds of almost contact manifolds by using the slant distribution given in [10]. However, pseudoslant submanifolds were defined by Carriazo [11] under the name antislant submanifolds as a particular class of bislant submanifolds. The definition of pseudoslant is as follows.
Definition 2.1
A submanifold M of an almost contact manifold MÌƒ is said to be pseudoslant submanifold, if there exist two orthogonal distributions \(\mathcal{D}^{\perp}\) and \(\mathcal{D}^{\theta}\) such that:

(i)
\(TM=\mathcal{D}^{\theta}\oplus\mathcal{D}^{\perp}\oplus\langle \xi\rangle\), where \(\langle\xi\rangle\) is a 1dimensional distribution spanned by Î¾.

(ii)
\(\mathcal{D}^{\perp}\) is an antiinvariant distribution under Ï† i.e., \(\varphi\mathcal{D}^{\perp}\subseteq T^{\perp}M\).

(iii)
\(\mathcal{D}^{\theta}\) is a slant distribution with slant angle \(\theta\neq0,\frac{\pi}{2}\).
Let \(m_{1}\) and \(m_{2}\) be the dimensions of the distributions \(\mathcal{D}^{\perp}\) and \(\mathcal{D}^{\theta}\), respectively. If \(m_{2}\)=0, then M is an antiinvariant submanifold. If \(m_{1}\)=0 and \(\theta=0\), then M is an invariant submanifold. If \(m_{1}\)=0 and \(\theta\neq0,\frac{\pi}{2}\), then M is a properslant submanifold, or if \(\theta=\frac{\pi}{2}\), then M is an antiinvariant submanifold and if \(\theta=0\), then M is a semiinvariant submanifold. Since Î¼ is an invariant subspace of a normal bundle \(T^{\perp}M\), then in the case of a pseudoslant submanifold, the normal bundle \(T^{\perp}M\) can be decomposed as follows:
where Î¼ is an even dimensional invariant subbundle of \(T^{\perp}M\). A pseudoslant submanifold is said to be mixed totally geodesic if \(h(X, Z)=0\), for all \(X\in\Gamma(\mathcal{D}^{\perp})\) and \(Z\in\Gamma (\mathcal{D}^{\theta})\). Now let \(\{e_{1}, e_{2},\ldots,e_{n}\}\) be an orthonormal basis of the tangent space TM and \(e_{r}\) belong to the orthonormal basis \(\{ e_{n+1}, e_{n+2},\ldots,e_{m}\}\) of a normal bundle \(T^{\perp}M\), then we define
As a consequence, for a differentiable function Î» on M, we have
where the gradient gradÎ» is defined by \(g(\nabla\lambda, X)=X\lambda\), for any \(X\in\Gamma(TM)\).
3 Warped product submanifolds
In [12], Bishop and Oâ€™Neill, defined the notion of warped product manifolds to construct examples of Riemannian manifolds with a negative curvature. These manifolds are natural generalizations of Riemannian product manifolds. They defined these manifolds as follows: Let \((M_{1}, g_{1})\) and \((M_{2}, g_{2})\) be two Riemannian manifolds and \(f:M_{1}\rightarrow (0, \infty)\) a positive differentiable function on \(M_{1}\). Consider the product manifold \(M_{1}\times M_{2}\) with its canonical projections \(\pi _{1}:M_{1}\times M_{2}\rightarrow M_{1}\), \(\pi_{2}:M_{1}\times M_{2}\rightarrow M_{2}\) and the projection maps given by \(\pi_{1}(p, q)=p\) and \(\pi_{2}(p, q)=q\) for every \(t=(p, q)\in M_{1}\times M_{2}\). The warped product \(M=M_{1}\times _{f}M_{2}\) is the product manifold \(M_{1}\times M_{2}\) equipped with the Riemannian structure such that
for any tangent vector \(U\in\Gamma(T_{t}M)\), where âˆ— is the symbol of the tangent maps. Thus we have \(g=g_{1}+f^{2}g_{2}\). The function f is called the warping function on M. It was defined in [12] and we have the following.
Lemma 3.1
[12]
Let \(M=M_{1}\times_{f}M_{2}\) be warped product manifolds. For any \(X, Y\in\Gamma(TM_{1})\) and \(Z, W\in\Gamma(TM_{2})\):

(i)
\(\nabla_{X}Y\in\Gamma(TM_{1})\),

(ii)
\(\nabla_{Z}X=\nabla_{X}Z=(X\ln f)Z\),

(iii)
\(\nabla_{Z}W=\nabla'_{Z}Wg(Z, W)\nabla\ln f\),
where âˆ‡ and \(\nabla'\) denote the LeviCivita connections on M and \(M_{2}\), respectively. On the other hand, \(\nabla\ln f\), the gradient of lnf, is defined as \(g(\nabla\ln f, U)=U\ln f\). A warped product manifold \(M=M_{1}\times_{f}M_{2}\) is said to be trivial if the warping function f is constant. If \(M=M_{1}\times_{f}M_{2}\) is a warped product manifold then \(M_{1}\) is a totally geodesic and \(M_{2}\) is a totally umbilical submanifold of M. First of all we give some preparatory lemmas.
Lemma 3.2
Let \(M=M_{\perp}\times_{f}M_{\theta}\) be a warped product pseudoslant submanifold of a nearly Kenmotsu manifold MÌƒ such that the structure vector field Î¾ is tangent to \(M_{\perp}\). Then
for any \(X\in\Gamma(TM_{\theta})\) and \(Z\in\Gamma(TM_{\perp})\).
Proof
Let \(M=M_{\perp}\times_{f}M_{\theta}\) be a warped product pseudoslant submanifold of a nearly Kenmotsu manifold MÌƒ, then by (2.6), we have
Thus using the covariant derivatives of Ï†, we obtain
From the nearly Kenmotsu structure (2.5) and Theorem 2.1, we derive
Thus from Lemma 3.1(ii) and the covariant derivative of an endomorphism Ï†, we obtain
Using (2.9) and (2.7), we arrive at
Finally, using Lemma 3.1(ii) and (2.12), we get
Again for any \(X\in\Gamma(TM_{\theta})\) and \(Z\in\Gamma(TM_{\perp})\), we have
From the fact that Î¾ is tangent to \(M_{\perp}\), (2.6), and (2.9), we obtain
Then from the definition of the covariant derivative of Ï† and Lemma 3.1(ii), we can derive
Thus by using the structure equation (2.5) and (2.5) and (2.12), the above equation takes the form
Using Lemma 3.1(ii), (2.7), and (2.12), we arrive at
Equation (2.7) and Theorem 2.1 for a slant submanifold give us
Finally, from Lemma 3.1(ii) and (2.6), we derive
Thus from (3.1) and (3.2), we get
which is our final result. This completes the proof of the lemma.â€ƒâ–¡
Lemma 3.3
Let \(M=M_{\perp}\times_{f}M_{\theta}\) be a warped product pseudoslant submanifold of a nearly Kenmotsu manifold MÌƒ. Then:

(iii)
\(g(h(X, X), \varphi Z)=g(h(Z, X), FX)\),

(iv)
\(g(h(PX, PX), \varphi Z)=g(h(Z, PX), FPX)\),
for any \(X\in\Gamma(TM_{\theta})\) and \(Z\in\Gamma(TM_{\perp})\).
Proof
For \(X\in\Gamma(TM_{\theta})\) and \(Z\in\Gamma (TM_{\perp})\), we have
From the definition of the covariant derivative of a tensor field Ï†, we get
Then from (2.5) and (2.2), we obtain
Thus from Lemma 3.1(ii) and (2.6), the above equation can be written as
Since X and PX are orthogonal vector fields and considering (2.8), we arrive at
which is the first result of the lemma. Interchanging X by PX in (3.3), we get the last result of the lemma. This completes the proof of the lemma.â€ƒâ–¡
Lemma 3.4
Assume we have a \(M=M_{\theta}\times _{f}M_{\perp}\) nontrivial warped product pseudoslant submanifold of a nearly Kenmotsu manifold MÌƒ. Then:

(i)
\(g(h(Z, Z), FPX)=g(h(Z, PX), \varphi Z)+\{\eta(X)(X\ln f)\} \cos^{2}\theta\Z\^{2}\),

(ii)
\(g(h(Z, Z), FX)=g(h(Z, X), \varphi Z)(PX\ln f)\Z\^{2}\),
for any \(X\in\Gamma(TM_{\theta})\) and \(Z\in\Gamma(TM_{\perp})\), where the structure vector field Î¾ is tangent to \(M_{\theta}\).
Proof
From Theorem 2.1, we obtain
Using the property of a Riemannian connection and the covariant derivative of an endomorphism, we derive
Thus from the fact that in a nearly Kenmotsu manifolds \((\xi\ln f)=1\) [13] and from Lemma 3.1(ii), we arrive at
which is the first part of the lemma. The second part of the lemma can easily be found by interchanging X by PX in the above equation. This completes the proof of the lemma.â€ƒâ–¡
4 Inequalities of warped product submanifolds
4.1 Inequality for a warped product pseudoslant submanifold of the form \(M_{\perp}\times_{f}M_{\theta}\)
In this section, we obtain a geometric inequality of warped product pseudoslant submanifold in terms of the second fundamental form such that Î¾ is tangent to the antiinvariant 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 mdimensional warped product pseudoslant submanifold of a \({2n+1}\)dimensional nearly Kenmotsu manifold MÌƒ 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 Î¼, 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 \(\{ e_{2m1}=\bar{e}_{m},\ldots,e_{2n+1}=\bar{e}_{2(nm+1)}\}\).
Theorem 4.1
Let \(M=M_{\perp}\times_{f}M_{\theta}\) be a mdimensional mixed totally geodesic warped product pseudoslant submanifold of a \({2n+1}\)dimensional nearly Kenmotsu manifold MÌƒ such that \(\xi\in\Gamma(TM_{\perp})\), where \(M_{\perp}\) is an antiinvariant submanifold of dimension \(d_{2}\) and \(M_{\theta}\) is a properslant submanifold of dimension \(d_{1}\) of MÌƒ. Then:

(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 MÌƒ.
Proof
The squared norm of the second fundamental form is defined as
Since M is mixed totally geodesic,
Then with (2.15), we obtain
Since the above equation can be expressed as in the components of \(\varphi\mathcal{D}^{\perp}\), \(F\mathcal{D}^{\theta}\), and Î½, we derive
Leaving all the terms except the first, we get
Using another adapted frame for \(\mathcal{D}^{\theta}\), we derive
Then for a mixed totally geodesic submanifold, the first and last terms of the right hand side in the above equation vanish identically by using Lemma 3.3, and we obtain
Thus from Lemma 3.2, for a mixed totally geodesic submanifold and using the fact that \(\eta(e_{i})=0\), \(i=1, 2,\ldots,d_{2}1\) for an orthonormal frame, we arrive at
Now we add and subtract the same term \(\xi\ln f\) in (4.4), getting
It well known that \(\xi\ln f=1\) [13] for a warped product submanifold of a nearly Kenmotsu manifold. Thus the above equation gives
If the equality holds above, then from the terms left in (4.2), we obtain the following condition from the first term:
which means that \(M_{\perp}\) is totally geodesic in MÌƒ. Similarly, from the second and third terms in (4.3), we derive
which implies that
We have Lemma 3.2, which shows that \(M_{\theta}\) is a totally umbilical into MÌƒ due to being totally umbilical in M. So equality holds. This completes the proof of the theorem.â€ƒâ–¡
4.2 Inequality for a warped product pseudoslant submanifold of the form \(M_{\theta}\times_{f}M_{\perp}\)
In this section, we obtain an inequality of warped product pseudoslant subamnifolds such that the structure vector field Î¾ is tangent to the slant submanifold \(M_{\theta}\). Taking Î¾ tangent to \(\mathcal {D}^{\theta}\), then we use the last frame.
Theorem 4.2
Let \(M=M_{\theta}\times_{f}M_{\perp}\) be a mdimensional mixed totally geodesic warped product pseudoslant submanifold of a \({2n+1}\)dimensional nearly Kenmotsu manifold MÌƒ such that \(\xi\in\Gamma(TM_{\theta})\), where \(M_{\perp}\) is an antiinvariant submanifold of dimension \(d_{2}=\alpha\) and \(M_{\theta}\) is a properslant submanifold of dimension \(d_{1}=2\beta+1\) of MÌƒ. Then:

(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 MÌƒ, respectively.
Proof
We start by the definition of the second fundamental form
Since M is a mixed totally geodesic, we get
Thus by (2.15), we obtain
The above equation can be expressed in the components of \(\phi\mathcal {D}^{\perp}\), \(F\mathcal{D}^{\theta}\), and Î½ as
We shall leave all the terms, except the second term, and we get
Thus using the adapted frame for \(F\mathcal{D}^{\theta}\), we derive
Using Lemma 3.4 for a mixed totally geodesic warped product submanifold and the fact that \(\eta(e_{j})=0\), \(1\leq j\leq d_{1}1\) for an orthonormal frame, we arrive at
Hence by hypothesis, we obtain
By the property of trigonometric functions, we arrive at
Then by adding and subtracting the same terms \(\xi\ln f\) in the above equation, we get
\(\xi\ln f=1\) [13] for a warped product submanifold of a nearly Kenmotsu manifold. Thus the above equation gives
If the equality holds, from the terms left in (4.6) and (4.7), we obtain the following conditions from the first and third terms:
and
where \(\mathcal{D}=\mathcal{D}^{\theta}\oplus\xi\), this means that \(M_{\theta}\) is a totally geodesic in MÌƒ and \(h(\mathcal {D}^{\perp}, \mathcal{D}^{\perp})\subseteq F\mathcal{D}^{\theta}\). Now from Lemma 3.4 for a mixed totally geodesic submanifold we find
for any \(Z, W\in\Gamma(TM_{\perp})\) and \(X\in\Gamma(TM_{\theta})\). The above equations imply that \(M_{\perp}\) is totally umbilical in MÌƒ, so the equality holds. This completes the proof of the theorem.â€ƒâ–¡
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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 RG27814AFR and RG27013AFR.
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Ali, A., Othman, W.A.M. & Ozel, C. Some inequalities for warped product pseudoslant submanifolds of nearly Kenmotsu manifolds. J Inequal Appl 2015, 291 (2015). https://doi.org/10.1186/s1366001508025
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DOI: https://doi.org/10.1186/s1366001508025