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.
Keywords:
warped product; contact CRsubmanifold; contact CRwarped product; nearly cosymplectic manifold1 Introduction
An almost contact metric structure
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
There always exists a Riemannian metric g on
where X and Y are vector fields on
An almost contact structure
where f is a smooth function on
An almost contact metric structure
for any X, Y tangent to
Proposition 2.1[6]
On a nearly cosymplectic manifold, the vector fieldξis Killing.
From the above proposition, we have
Let M be a submanifold of an almost contact metric manifold
for each
where g denotes the Riemannian metric on
For any
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
A submanifold M tangent to the structure vector field ξ of an almost contact metric manifold
(i)
(ii)
(iii)
A contact CRsubmanifold is invariant if
Bishop and O’Neill [7] introduced the notion of warped product manifolds. They defined these manifolds as
follows. Let
for each
The function f is called the warping function of the warped product [7]. A warped product
We recall the following general result on warped product manifolds for later use.
Lemma 2.1Let
(i)
(ii)
(iii)
for each
3 Contact CRwarped product submanifolds
In this section, we consider the warped product submanifolds
Theorem 3.1[8]
A warped product submanifold
If we consider
Taking the inner product with
Now, we consider the warped product contact CRsubmanifolds of the types
Lemma 3.1[8]
Let
(i)
(ii)
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 LeviCivita connection on M and
As a consequence, we have
Now, we prove the main result of this section using the above results.
Theorem 3.2Let
(i) The length of the second fundamental form ofMsatisfies the inequality
whereqis the dimension of
(ii) If the equality sign of (3.6) holds identically, then
Proof Let
For the assumed frames, the above equation can be written as
The first term on the righthand side of the above equality is the
The above equation can be written for the given frame of
Let us decompose the above equation in terms of the components of
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
Using the fact that ξ is tangent to
Then from (3.5), the above expression will be
which proves the inequality (3.6). Let us denote by
for any
or equivalently,
Suppose the equality case holds in (3.6), then from (3.8) and (3.10), we obtain
As
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
References

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