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.
Keywords:warped product; contact CR-submanifold; contact CR-warped product; nearly cosymplectic manifold
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.
A -dimensional smooth manifold is said to have an almost contact structure if on there exist a tensor field ϕ of type , a vector field ξ and a 1-form η satisfying 
There always exists a Riemannian metric g on satisfying the following compatibility condition:
where X and Y are vector fields on .
An almost contact structure is said to be normal if the almost complex structure J on the product manifold given by
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 .
An almost contact metric structure is said to be cosymplectic if it is normal and both Φ and η are closed . The structure is said to be nearly cosymplectic if ϕ is Killing, i.e., if
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 .
On a nearly cosymplectic manifold, the vector fieldξis Killing.
From the above proposition, we have for any vector field X tangent to , where is a nearly cosymplectic manifold.
Let M be a submanifold of an almost contact metric manifold with induced metric g, and let ∇ and be the induced connections on the tangent bundle TM and the normal bundle of M, respectively. Denote by the algebra of smooth functions on M and by the -module of smooth sections of TM over M. Then the Gauss and Weingarten formulas are given by
for each and , where h and are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively, for the immersion of M into . They are related as
where g denotes the Riemannian metric on as well as induced on M.
For any , 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. anti-invariant) if , (resp. , ).
A submanifold M tangent to the structure vector field ξ of an almost contact metric manifold is called a contact CR-submanifold (or semi-invariant submanifold) if there exists a pair of orthogonal differentiable distributions and on M such that
(i) , where is the one-dimensional distribution spanned by ξ;
(ii) is invariant under ϕ, i.e., , ;
(iii) is anti-invariant under ϕ, i.e., , .
A contact CR-submanifold is invariant if and anti-invariant if , respectively. It is called a proper contact CR-submanifold if neither nor . Moreover, if μ is the ϕ-invariant subspace in the normal bundle , then in the case of a contact CR-submanifold, the normal bundle can be decomposed as
Bishop and O’Neill  introduced the notion of warped product manifolds. They defined these manifolds as follows. Let and be two Riemannian manifolds and be a differentiable function on . Consider the product manifold with its projections and . Then the warped product of and denoted by is a Riemannian manifold equipped with the Riemannian structure such that
for each and ⋆ is the symbol for the tangent map. Thus, we have
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.
Lemma 2.1Let be a warped product manifold with the warping functionf, then
(i) is the lift of on ,
for each and , where is the gradient of the function lnfand ∇ and denote the Levi-Civita connections onMand , 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.
A warped product submanifold of a nearly cosymplectic manifold is a usual Riemannian product if the structure vector fieldξis tangent to , where and are the Riemannian submanifolds of .
If we consider , then for any , we have
Taking the inner product with , then by Proposition 2.1 and Lemma 2.1(ii), we obtain that . This means that either , which is not possible for warped products, or
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.
Let be a contact CR-warped product submanifold of a nearly cosymplectic manifold . If and , then
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
(ii) Δf, the Laplacian of f, is defined by
where ∇ is the Levi-Civita connection on M and 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 3.2Let be a contact CR-warped product submanifold of a nearly cosymplectic manifold . Then we have
(i) The length of the second fundamental form ofMsatisfies the inequality
whereqis the dimension of and is the gradient of lnf.
(ii) If the equality sign of (3.6) holds identically, then is a totally geodesic submanifold and is a totally umbilical submanifold of . Moreover, Mis a minimal submanifold of .
Proof Let be a -dimensional nearly cosymplectic manifold and be an n-dimensional contact CR-warped product submanifolds of . Let us consider the and , then . Let and be the local orthonormal frames on and , respectively. Then the orthonormal frames in the normal bundle of and μ are and , 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 right-hand side of the above equality is the -component and the second term is the μ-component. Here, we equate the -component, then we have
The above equation can be written for the given frame of as
Let us decompose the above equation in terms of the components of , and , 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 , then we can write the above equation for one summation, and using (3.2), we obtain
Using the fact that ξ is tangent to and , the above equation can be written for the given frame of the distribution as
Then from (3.5), the above expression will be
which proves the inequality (3.6). Let us denote by , the second fundamental form of in M, then by (2.4), we have
for any and . Thus, on using (3.3), we obtain
Suppose the equality case holds in (3.6), then from (3.8) and (3.10), we obtain
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. □
The authors declare that they have no competing interests.
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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Chen, BY: Geometry of warped product CR-submanifolds in Kaehler manifold. Monatshefte Math.. 133, 177–195 (2001). Publisher Full Text
Mihai, I: Contact CR-warped product submanifolds in Sasakian space forms. Geom. Dedic.. 109, 165–173 (2004). Publisher Full Text