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An inequality for contact CRwarped product submanifolds of nearly cosymplectic manifolds
Journal of Inequalities and Applications volume 2012, Article number: 304 (2012)
Abstract
Recently, Chen (Monatshefte Math. 133:177195, 2001) established general sharp inequalities for CRwarped products in a Kaehler manifold. Afterward, Mihai obtained (Geom. Dedic. 109:165173, 2004) the same inequalities for contact CRwarped product submanifolds of Sasakian space forms and derived some applications. In this paper, we obtain an inequality for the length of the second fundamental form of the warped product submanifold of a nearly cosymplectic manifold in terms of the warping function. The equality case is also discussed.
MSC:53C40, 53C42, 53B25.
1 Introduction
An almost contact metric structure (\varphi ,\xi ,\eta ,g) satisfying ({\overline{\mathrm{\nabla}}}_{X}\varphi )X=0 is called a nearly cosymplectic structure. If we consider {S}^{5} as a totally geodesic hypersurface of {S}^{6}, then it is known that {S}^{5} has a noncosymplectic nearly cosymplectic structure. Almost contact manifolds with Killing structure tensors were defined in [1] as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also [2]). Later on, Blair and Showers [3] studied nearly cosymplectic structure (\varphi ,\xi ,\eta ,g) on a manifold \overline{M} with η closed from the topological viewpoint.
On the other hand, Chen [4] has introduced the notion of CRwarped product submanifolds in a Kaehler manifold. He has established a sharp relationship between the squared norm of the second fundamental form and the warping function. Later on, Mihai [5] studied contact CRwarped products and obtained the same inequality for contact CRwarped product submanifolds isometrically immersed in Sasakian space forms. Motivated by the studies of these authors, many articles dealing with the existence or nonexistence of warped products in different settings have appeared; one of them is [3]. In this paper, we obtain an inequality for the length of the second fundamental form in terms of the warping function for contact CRwarped product submanifolds in a more general setting, i.e., nearly cosymplectic manifold.
2 Preliminaries
A (2m+1)dimensional smooth manifold \overline{M} is said to have an almost contact structure if on \overline{M} there exist a tensor field ϕ of type (1,1), a vector field ξ and a 1form η satisfying [6]
There always exists a Riemannian metric g on \overline{M} satisfying the following compatibility condition:
where X and Y are vector fields on \overline{M} [6].
An almost contact structure (\varphi ,\xi ,\eta ) is said to be normal if the almost complex structure J on the product manifold \overline{M}\times \mathbb{R} given by
where f is a smooth function on \overline{M}\times \mathbb{R}, has no torsion, i.e., J is integrable, the condition for normality in terms of ϕ, ξ and η is [\varphi ,\varphi ]+2d\eta \otimes \xi =0 on \overline{M}, where [\varphi ,\varphi ] is the Nijenhuis tensor of ϕ. Finally, the fundamental 2form Φ is defined by \mathrm{\Phi}(X,Y)=g(X,\varphi Y).
An almost contact metric structure (\varphi ,\xi ,\eta ,g) is said to be cosymplectic if it is normal and both Φ and η are closed [6]. The structure is said to be nearly cosymplectic if ϕ is Killing, i.e., if
for any X, Y tangent to \overline{M}, where \overline{\mathrm{\nabla}} is the Riemannian connection of the metric g on \overline{M}. Equation (2.3) is equivalent to ({\overline{\mathrm{\nabla}}}_{X}\varphi )X=0 for each vector field X tangent to \overline{M}. The structure is said to be closely cosymplectic if ϕ is Killing and η is closed. It is well known that an almost contact metric manifold is cosymplectic if and only if \overline{\mathrm{\nabla}}\varphi vanishes identically, i.e., ({\overline{\mathrm{\nabla}}}_{X}\varphi )Y=0 and {\overline{\mathrm{\nabla}}}_{X}\xi =0.
Proposition 2.1 [6]
On a nearly cosymplectic manifold, the vector field ξ is Killing.
From the above proposition, we have g({\overline{\mathrm{\nabla}}}_{X}\xi ,X)=0 for any vector field X tangent to \overline{M}, where \overline{M} is a nearly cosymplectic manifold.
Let M be a submanifold of an almost contact metric manifold \overline{M} with induced metric g, and let ∇ and {\mathrm{\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 \mathrm{\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 \mathrm{\Gamma}(TM) and N\in \mathrm{\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 \overline{M}. They are related as
where g denotes the Riemannian metric on \overline{M} as well as induced on M.
For any X\in \mathrm{\Gamma}(TM), we write
where TX is the tangential component and FX is the normal component of ϕX.
A submanifold M tangent to the structure vector field ξ is said to be invariant (resp. antiinvariant) if \varphi ({T}_{x}M)\subset {T}_{x}M, \mathrm{\forall}x\in M (resp. \varphi ({T}_{x}M)\subset {T}_{x}^{\perp}M, \mathrm{\forall}x\in M).
A submanifold M tangent to the structure vector field ξ of an almost contact metric manifold \overline{M} is called a contact CRsubmanifold (or semiinvariant submanifold) if there exists a pair of orthogonal differentiable distributions \mathcal{D} and {\mathcal{D}}^{\perp} on M such that

(i)
TM=\mathcal{D}\oplus {\mathcal{D}}^{\perp}\oplus \u3008\xi \u3009, where \u3008\xi \u3009 is the onedimensional distribution spanned by ξ;

(ii)
\mathcal{D} is invariant under ϕ, i.e., \varphi ({\mathcal{D}}_{x})\subseteq {\mathcal{D}}_{x}, \mathrm{\forall}x\in M;

(iii)
{\mathcal{D}}^{\perp} is antiinvariant under ϕ, i.e., \varphi ({\mathcal{D}}_{x}^{\perp})\subset {T}_{x}^{\perp}M, \mathrm{\forall}x\in M.
A contact CRsubmanifold is invariant if {\mathcal{D}}^{\perp}=\{0\} and antiinvariant if \mathcal{D}=\{0\}, respectively. It is called a proper contact CRsubmanifold if neither \mathcal{D}=\{0\} nor {\mathcal{D}}^{\perp}=\{0\}. Moreover, if μ is the ϕinvariant subspace in the normal bundle {T}^{\perp}M, then in the case of a contact CRsubmanifold, the normal bundle {T}^{\perp}M can be decomposed as
Bishop and O’Neill [7] introduced the notion of warped product manifolds. They defined these manifolds as follows. Let ({N}_{1},{g}_{1}) and ({N}_{2},{g}_{2}) be two Riemannian manifolds and f>0 be a differentiable function on {N}_{1}. Consider the product manifold {N}_{1}\times {N}_{2} with its projections {\pi}_{1}:{N}_{1}\times {N}_{2}\to {N}_{1} and {\pi}_{2}:{N}_{1}\times {N}_{2}\to {N}_{2}. Then the warped product of {N}_{1} and {N}_{2} denoted by M={N}_{1}{\times}_{f}{N}_{2} is a Riemannian manifold {N}_{1}\times {N}_{2} equipped with the Riemannian structure such that
for each X,Y\in \mathrm{\Gamma}(TM) and ⋆ is the symbol for the tangent map. Thus, we have
The function f is called the warping function of the warped product [7]. A warped product {N}_{1}{\times}_{f}{N}_{2} is said to be trivial if the warping function f is constant.
We recall the following general result on warped product manifolds for later use.
Lemma 2.1 Let M={N}_{1}{\times}_{f}{N}_{2} be a warped product manifold with the warping function f, then

(i)
{\mathrm{\nabla}}_{X}Y\in \mathrm{\Gamma}(T{N}_{1}) is the lift of {\mathrm{\nabla}}_{X}Y on {N}_{1},

(ii)
{\mathrm{\nabla}}_{X}Z={\mathrm{\nabla}}_{Z}X=(Xlnf)Z,

(iii)
{\mathrm{\nabla}}_{Z}W={\mathrm{\nabla}}_{Z}^{{N}_{2}}Wg(Z,W)\mathrm{\nabla}lnf
for each X,Y\in \mathrm{\Gamma}(T{N}_{1}) and Z,W\in \mathrm{\Gamma}(T{N}_{2}), where \mathrm{\nabla}lnf is the gradient of the function lnf and ∇ and {\mathrm{\nabla}}^{{N}_{2}} denote the LeviCivita connections on M and {N}_{2}, respectively.
3 Contact CRwarped product submanifolds
In this section, we consider the warped product submanifolds M={N}_{1}{\times}_{f}{N}_{2} of a nearly cosymplectic manifold \overline{M}, where {N}_{1} and {N}_{2} are Riemannian submanifolds of \overline{M}. In the above product, if we assume {N}_{1}={N}_{T} and {N}_{2}={N}_{\perp}, then the warped product of {N}_{1} and {N}_{2} becomes a contact CRwarped product. In this section, we discuss the contact CRwarped products and obtain an inequality for the squared norm of the second fundamental form. For the general case of warped product submanifolds of a nearly cosymplectic manifold, we have the following result.
Theorem 3.1 [8]
A warped product submanifold M={N}_{1}{\times}_{f}{N}_{2} of a nearly cosymplectic manifold \overline{M} is a usual Riemannian product if the structure vector field ξ is tangent to {N}_{2}, where {N}_{1} and {N}_{2} are the Riemannian submanifolds of \overline{M}.
If we consider \xi \in \mathrm{\Gamma}(T{N}_{1}), then for any X\in \mathrm{\Gamma}(T{N}_{2}), we have
Taking the inner product with X\in \mathrm{\Gamma}(T{N}_{2}), then by Proposition 2.1 and Lemma 2.1(ii), we obtain that (\xi lnf){\parallel X\parallel}^{2}=0. This means that either dim{N}_{2}=0, which is not possible for warped products, or
Now, we consider the warped product contact CRsubmanifolds of the types M={N}_{\perp}{\times}_{f}{N}_{T} and M={N}_{T}{\times}_{f}{N}_{\perp} of a nearly cosymplectic manifold \overline{M}. In [8], the present author has proved that the warped product contact CRsubmanifolds of the first type are usual Riemannian products of {N}_{\perp} and {N}_{T}, where {N}_{\perp} and {N}_{T} are antiinvariant and invariant submanifolds of \overline{M}, respectively. In the following, we consider the contact CRwarped product submanifolds M={N}_{T}{\times}_{f}{N}_{\perp} and obtain a general inequality. First, we have the following preparatory result for later use.
Lemma 3.1 [8]
Let M={N}_{T}{\times}_{f}{N}_{\perp} be a contact CRwarped product submanifold of a nearly cosymplectic manifold \overline{M}. If X,Y\in \mathrm{\Gamma}(T{N}_{T}) and Z,W\in \mathrm{\Gamma}(T{N}_{\perp}), then

(i)
g(h(X,Y),FZ)=0,

(ii)
g(h(\varphi X,Z),FZ)=(Xlnf){\parallel Z\parallel}^{2}.
If we replace X by ϕX in (ii) of Lemma 3.1, then we get
For a Riemannian manifold of dimension m and a smooth function f on M, we recall

(i)
∇f, the gradient of f, is defined by
g(\mathrm{\nabla}f,X)=X(f),\phantom{\rule{1em}{0ex}}\mathrm{\forall}X\in \mathrm{\Gamma}(TM).(3.3) 
(ii)
Δf, the Laplacian of f, is defined by
\mathrm{\Delta}f=\sum _{i=1}^{m}\{({\mathrm{\nabla}}_{{e}_{i}}{e}_{i})f{e}_{i}{e}_{i}(f)\}=div\mathrm{\nabla}f,(3.4)
where ∇ is the LeviCivita connection on M and \{{e}_{1},\dots ,{e}_{m}\} is an orthonormal frame on M.
As a consequence, we have
Now, we prove the main result of this section using the above results.
Theorem 3.2 Let M={N}_{T}{\times}_{f}{N}_{\perp} be a contact CRwarped product submanifold of a nearly cosymplectic manifold \overline{M}. Then we have

(i)
The length of the second fundamental form of M satisfies the inequality
{\parallel h\parallel}^{2}\ge 2q{\parallel \mathrm{\nabla}lnf\parallel}^{2},(3.6)
where q is the dimension of {N}_{\perp} and \mathrm{\nabla}lnf is the gradient of lnf.

(ii)
If the equality sign of (3.6) holds identically, then {N}_{T} is a totally geodesic submanifold and {N}_{\perp} is a totally umbilical submanifold of \overline{M}. Moreover, M is a minimal submanifold of \overline{M}.
Proof Let \overline{M} be a (2m+1)dimensional nearly cosymplectic manifold and M={N}_{T}{\times}_{f}{N}_{\perp} be an ndimensional contact CRwarped product submanifolds of \overline{M}. Let us consider the dim{N}_{T}=2p+1 and dim{N}_{\perp}=q, then n=2p+1+q. Let \{{e}_{1},\dots ,{e}_{p},\varphi {e}_{1}={e}_{p+1},\dots ,\varphi {e}_{p}={e}_{2p},{e}_{2p+1}=\xi \} and \{{e}_{(2p+1)+1},\dots ,{e}_{n}\} be the local orthonormal frames on {N}_{T} and {N}_{\perp}, respectively. Then the orthonormal frames in the normal bundle {T}^{\perp}M of F{\mathcal{D}}^{\perp} and μ are \{F{e}_{(2p+1)+1},\dots ,F{e}_{n}\} and \{{e}_{n+q+1},\dots ,{e}_{2m+1}\}, respectively. Then the length of the second fundamental form h is defined as
For the assumed frames, the above equation can be written as
The first term on the righthand side of the above equality is the F{\mathcal{D}}^{\perp}component and the second term is the μcomponent. Here, we equate the F{\mathcal{D}}^{\perp}component, then we have
The above equation can be written for the given frame of F{\mathcal{D}}^{\perp} as
Let us decompose the above equation in terms of the components of h(\mathcal{D},\mathcal{D}), h(\mathcal{D},{\mathcal{D}}^{\perp}) and h({\mathcal{D}}^{\perp},{\mathcal{D}}^{\perp}), then we have
Using Lemma 3.1(i), the first term of (3.10) is identically zero and we shall compute the next term and leave the third term
As j,l=2p+2,\dots ,n, then we can write the above equation for one summation, and using (3.2), we obtain
Using the fact that ξ is tangent to {N}_{T} and \xi lnf=0, the above equation can be written for the given frame of the distribution \mathcal{D} as
Then from (3.5), the above expression will be
which proves the inequality (3.6). Let us denote by {h}^{\perp}, the second fundamental form of {N}_{\perp} in M, then by (2.4), we have
for any X\in \mathrm{\Gamma}(T{N}_{T}) and Z,W\in \mathrm{\Gamma}(T{N}_{\perp}). Thus, on using (3.3), we obtain
or equivalently,
Suppose the equality case holds in (3.6), then from (3.8) and (3.10), we obtain
As {N}_{T} is a totally geodesic submanifold in M (by Lemma 2.1(i)), using this fact with the first part of (3.14), we get {N}_{T} is totally geodesic in \overline{M}. On the other hand, the second condition of (3.14) with (3.13) implies that {N}_{\perp} is totally umbilical in \overline{M}. Moreover, from (3.14), we get M is a minimal submanifold of \overline{M}. This proves the theorem completely. □
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SU carried out the whole research and drafted the manuscript. KAK has given the idea of this problem and checked the calculations. All authors read and approved the final manuscript.
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Uddin, S., Khan, K.A. An inequality for contact CRwarped product submanifolds of nearly cosymplectic manifolds. J Inequal Appl 2012, 304 (2012). https://doi.org/10.1186/1029242X2012304
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DOI: https://doi.org/10.1186/1029242X2012304
Keywords
 warped product
 contact CRsubmanifold
 contact CRwarped product
 nearly cosymplectic manifold