An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds
© Uddin and Khan; licensee Springer 2012
Received: 3 July 2012
Accepted: 26 November 2012
Published: 18 December 2012
Recently, Chen (Monatshefte Math. 133:177-195, 2001) established general sharp inequalities for CR-warped products in a Kaehler manifold. Afterward, Mihai obtained (Geom. Dedic. 109:165-173, 2004) the same inequalities for contact CR-warped 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.
An almost contact metric structure satisfying is called a nearly cosymplectic structure. If we consider as a totally geodesic hypersurface of , then it is known that has a non-cosymplectic nearly cosymplectic structure. Almost contact manifolds with Killing structure tensors were defined in  as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also ). Later on, Blair and Showers  studied nearly cosymplectic structure on a manifold with η closed from the topological viewpoint.
On the other hand, Chen  has introduced the notion of CR-warped 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  studied contact CR-warped products and obtained the same inequality for contact CR-warped product submanifolds isometrically immersed in Sasakian space forms. Motivated by the studies of these authors, many articles dealing with the existence or non-existence of warped products in different settings have appeared; one of them is . In this paper, we obtain an inequality for the length of the second fundamental form in terms of the warping function for contact CR-warped product submanifolds in a more general setting, i.e., nearly cosymplectic manifold.
where X and Y are vector fields on .
where f is a smooth function on , has no torsion, i.e., J is integrable, the condition for normality in terms of ϕ, ξ and η is on , where is the Nijenhuis tensor of ϕ. Finally, the fundamental 2-form Φ is defined by .
for any X, Y tangent to , where is the Riemannian connection of the metric g on . Equation (2.3) is equivalent to for each vector field X tangent to . 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 vanishes identically, i.e., and .
Proposition 2.1 
On a nearly cosymplectic manifold, the vector field ξ is Killing.
From the above proposition, we have for any vector field X tangent to , where is a nearly cosymplectic manifold.
where g denotes the Riemannian metric on as well as induced on M.
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. anti-invariant) if , (resp. , ).
, where is the one-dimensional distribution spanned by ξ;
is invariant under ϕ, i.e., , ;
is anti-invariant under ϕ, i.e., , .
The function f is called the warping function of the warped product . A warped product 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.
is the lift of on ,
for each and , where is the gradient of the function lnf and ∇ and denote the Levi-Civita connections on M and , respectively.
3 Contact CR-warped product submanifolds
In this section, we consider the warped product submanifolds of a nearly cosymplectic manifold , where and are Riemannian submanifolds of . In the above product, if we assume and , then the warped product of and becomes a contact CR-warped product. In this section, we discuss the contact CR-warped 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 
A warped product submanifold of a nearly cosymplectic manifold is a usual Riemannian product if the structure vector field ξ is tangent to , where and are the Riemannian submanifolds of .
Now, we consider the warped product contact CR-submanifolds of the types and of a nearly cosymplectic manifold . In , the present author has proved that the warped product contact CR-submanifolds of the first type are usual Riemannian products of and , where and are anti-invariant and invariant submanifolds of , respectively. In the following, we consider the contact CR-warped product submanifolds and obtain a general inequality. First, we have the following preparatory result for later use.
Lemma 3.1 
- (i)∇f, the gradient of f, is defined by(3.3)
- (ii)Δf, the Laplacian of f, is defined by(3.4)
where ∇ is the Levi-Civita connection on M and is an orthonormal frame on M.
Now, we prove the main result of this section using the above results.
- (i)The length of the second fundamental form of M satisfies the inequality(3.6)
If the equality sign of (3.6) holds identically, then is a totally geodesic submanifold and is a totally umbilical submanifold of . Moreover, M is a minimal submanifold of .
As is a totally geodesic submanifold in M (by Lemma 2.1(i)), using this fact with the first part of (3.14), we get is totally geodesic in . On the other hand, the second condition of (3.14) with (3.13) implies that is totally umbilical in . Moreover, from (3.14), we get M is a minimal submanifold of . This proves the theorem completely. □
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