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Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds
We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the type is a usual Riemannian product of and , where and are anti-invariant and invariant submanifolds of a nearly cosymplectic manifold , respectively. Thus we consider the warped product of the type and obtain a characterization for such type of warped product.
The notion of warped product manifolds was introduced by Bishop and O'Neill in 1969 as a natural generalization of the Riemannian product manifolds. Later on, the geometrical aspect of these manifolds has been studied by many researchers (cf., [1–3]). Recently, Chen  (see also ) studied warped product CR-submanifolds and showed that there exists no warped product CR-submanifolds of the form such that is a totally real submanifold and is a holomorphic submanifold of a Kaehler manifold . Therefore he considered warped product CR-submanifold in the form which is called CR-warped product, where and are holomorphic and totally real submanifolds of a Kaehler manifold . Motivated by Chen's papers, many geometers studied CR-warped product submanifolds in almost complex as well as contact setting (see [3, 5, 6]).
Almost contact manifolds with Killing structure tensors were defined in  as nearly cosymplectic manifolds, and it was shown that normal nearly cosymplectic manifolds are cosymplectic (see also ). Later on, Blair and Showers  studied nearly cosymplectic structure on a manifold with closed from the topological viewpoint.
In this paper, we have generalized the results of Chen'  in this more general setting of nearly cosymplectic manifolds and have shown that the warped product in the form is simply Riemannian product of and where is an anti-invariant submanifold and is an invariant submanifold of a nearly cosymplectic manifold . Thus we consider the warped product submanifold of the type by reversing the two factors and and simply will be called warped product semi-invariant submanifold. Thus, we derive the integrability of the involved distributions in the warped product and obtain a characterization result.
A dimensional manifold is said to have an almost contact structure if there exist on a tensor field of type , a vector field , and a 1-form satisfying  There always exists a Riemannian metric on an almost contact manifold satisfying the following compatibility condition: where and are vector fields on .
An almost contact structure is said to be normal if the almost complex structure on the product manifold given by where is a function on , has no torsion, that is, is integrable, and the condition for normality in terms of and is on , where is the Nijenhuis tensor of . Finally the fundamental 2-form is defined by .
An almost contact metric structure is said to be cosymplectic, if it is normal and both and are closed . The structure is said to be nearly cosymplectic if is Killing, that is, if for any , where is the tangent bundle of and denotes the Riemannian connection of the metric . Equation (2.4) is equivalent to , for each . The structure is said to be closely cosymplectic if is Killing and is closed. It is well known that an almost contact metric manifold is cosymplectic if and only if vanishes identically, that is, and .
Proposition 2.1 (see ). On a nearly cosymplectic manifold, the vector field is Killing.
From the above proposition we have , for any vector field tangent to , where is a nearly cosymplectic manifold.
Let be submanifold of an almost contact metric manifold with induced metric , and if and are the induced connections on the tangent bundle and the normal bundle of , respectively, then, Gauss and Weingarten formulae are given by for each and , where and are the second fundamental form and the shape operator (corresponding to the normal vector field ), respectively, for the immersion of into . They are related as where denotes the Riemannian metric on as well as being induced on .
For any , we write where is the tangential component and is the normal component of .
Similarly for any , we write where is the tangential component and is the normal component of . The covariant derivatives of the tensor fields and are defined as for all .
Let be a Riemannian manifold isometrically immersed in an almost contact metric manifold . then for every there exists a maximal invariant subspace denoted by of the tangent space of . If the dimension of is the same for all values of , then gives an invariant distribution on .
A submanifold of an almost contact metric manifold is called semi-invariant submanifold if there exists on a differentiable invariant distribution whose orthogonal complementary distribution is anti-invariant, that is, (i), (ii), (iii)
for any , where denotes the orthogonal space of in . A semi-invariant submanifold is called anti-invariant if and invariant if , respectively, for any . It is called the proper semi-invariant submanifold if neither nor , for every .
Let be a semi-invariant submanifold of an almost contact metric manifold . Then, is a subspace of . Then for every , there exists an invariant subspace of such that
A semi-invariant submanifold of an almost contact metric manifold is called Riemannian product if the invariant distribution and anti-invariant distribution are totally geodesic distributions in .
Let and be two Riemannian manifolds, and let be a positive differentiable function on . The warped product of and is the product manifold , where where is called the warping function of the warped product. The warped product is said to be trivial or simply Riemannian product if the warping function is constant. This means that the Riemannian product is a special case of warped product.
We recall the following general results obtained by Bishop and O'Neill  for warped product manifolds.
Lemma 2.2. Let be a warped product manifold with the warping function . Then (i), for each , (ii), for each and , (iii), where and denote the Levi-Civita connections on and , respectively.
In the above lemma is the gradient of the function defined by , for each . From the Lemma 2.2, we have that on a warped product manifold (i) is totally geodesic in ; (ii) is totally umbilical in .
It is straightforward to verify the following properties of and , which we enlist here for later use: (i) , (ii) ,(i) , (ii) ,
for all .
3. Warped Product Semi-Invariant Submanifolds
Throughout the section we consider the submanifold of a nearly cosymplectic manifold such that the structure vector field is tangent to . First, we prove that the warped product is trivial when is tangent to , where and are Riemannian submanifolds of a nearly cosymplectic manifold . Thus, we consider the warped product , when is tangent to the submanifold . We have the following nonexistence theorem.
Theorem 3.1. A warped product submanifold of a nearly cosymplectic manifold is a usual Riemannian product if the structure vector field is tangent to , where and are the Riemannian submanifolds of .
Proof. For any and tangent to , we have Using the fact that is Killing on a nearly cosymplectic manifold (see Proposition 2.1) and Lemma 2.2(ii), we get Equating the tangential component of (3.2), we obtain , for all , that is, is constant function on . Thus, is Riemannian product. This proves the theorem.
Now, the other case of warped product when , where and are the Riemannian submanifolds of . For any , we have By Proposition 2.1, and Lemma 2.2(ii), we obtain Thus, we consider the warped product semi-invariant submanifolds of a nearly cosymplectic manifold of the types: (i), (ii),
where and are invariant and anti-invariant submanifolds of , respectively. In the following theorem we prove that the warped product semi-invariant submanifold of the type (i) is CR-product.
Theorem 3.2. The warped product semi-invariant submanifold of a nearly cosymplectic manifold is a usual Riemannian product of and , where and are anti-invariant and invariant submanifolds of , respectively.
Proof. When , then by Theorem 3.1, is a Riemannian product. Thus, we consider . For any and , we have From the structure equation of nearly cosymplectic, the second term of right hand side vanishes identically. Thus from (2.2), we derive Then from (2.5), Lemma 2.2(ii), and Proposition 2.1, we obtain Interchanging by in (3.7) and using the fact that , we obtain It follows from (3.7) and (3.8) that , for all . Also, from (3.4) we have . Thus, the warping function is constant. This completes the proof of the theorem.
From the above theorem we have seen that the warped product of the type is a usual Riemannian product of an anti-invariant submanifold and an invariant submanifold of a nearly cosymplectic manifold . Since both and are totally geodesic in , then is CR-product. Now, we study the warped product semi-invariant submanifold of a nearly cosymplectic manifold .
Theorem 3.3. Let be a warped product semi-invariant submanifold of a nearly cosymplectic manifold . Then the invariant distribution and the anti-invariant distribution are always integrable.
Proof. For any , we have Using (2.11), we obtain Then by (2.16), we derive Thus from (2.17)(ii), we get Now, for any , we have Using the covariant derivative property of , we obtain Then by (2.5) and (2.14), we get Since is totally geodesic in (see Lemma 2.2(i)), then using (2.8) and (2.9), we obtain Equating the normal components of (3.16), we get Similarly, we obtain Then from (3.17) and (3.18), we arrive at Hence, using (2.17)(ii), we get Thus, it follows from (3.12) and (3.20) that , for all . This proves the integrability of . Now, for the integrability of , we consider any and , and we have Using Lemma 2.2(ii), we obtain Thus from (3.22), we conclude that , for each . Hence, the theorem is proved completely.
Lemma 3.4. Let be a warped product submanifold of a nearly cosymplectic manifold . If and , then (i), (ii), (iii).
Proof. For a warped product manifold , we have that is totally geodesic in ; then by (2.10), , for any , and therefore from (2.15), we get The left-hand side of (3.23) is skew symmetric in and whereas the right hand side is symmetric in and , which proves (i). Now, from (2.10) and (2.15), we have for any and . Using Lemma 2.2(ii), the first term of right-hand side is zero. Thus, taking the product with , we obtain Then by (2.2) and (2.7), we get which proves the first equality of (ii). Again, from (2.10) and (2.15), we have Thus using Lemma 2.2(ii), we derive Taking inner product with and using (2.2), we obtain Then from (2.17)(i), we get This is the second equality of (ii). Now, from (3.24) and (3.28), we have Left-hand side and the first term of right-hand side are zero on using (2.17)(i) and Lemma 2.2(i), respectively. Thus the above equation takes the form Taking the product with and on using (2.2) and (2.7), we get Interchanging by and using (2.1), we obtain Thus by (3.4)(i), the above equation reduces to This proves the lemma completely.
Theorem 3.5. A proper semi-invariant submanifold of a nearly cosymplectic manifold is locally a semi-invariant warped product if and only if the shape operator of satisfies for some function on satisfying for each .
Proof. If is a warped product semi-invariant submanifold, then by Lemma 3.4 (iii), we obtain (3.36). In this case .
Conversely, suppose is a semi-invariant submanifold of a nearly cosymplectic manifold satisfying (3.36). Then Now, from (2.5) and the property of covariant derivative of , we have Then from (2.5), (2.14), and (3.37), the above equation takes the form Using (2.10) and (2.15), we obtain Thus by (2.2), the above equation reduces to Hence using (2.7) and (3.36), we get which implies , that is, is integrable and its leaves are totally geodesic in . Now, for any and , we have Then, using (2.6) and (2.14), we obtain Thus from (2.7) and the property , we arrive at Again using (2.7) and (2.17)(i), we get On the other hand, from (2.10) and (2.15), we have Taking the product with and using (3.36), we obtain The first term of right-hand side of above equation is zero using the fact that , for any . Again using (2.7), we get Thus from (3.36), we derive Then from (3.36), (3.46), and (3.50), we obtain Let be a leaf of , and let be the second fundamental form of the immersion of into . Then for any , we have Hence, from (3.51) and (3.52), we conclude that This means that integral manifold of is totally umbilical in . Since the anti-invariant distribution of a semi-invariant submanifold is always integrable (Theorem 3.3) and for each , which implies that the integral manifold of is an extrinsic sphere in ; that is, it is totally umbilical and its mean curvature vector field is nonzero and parallel along . Hence by virtue of results obtained in , is locally a warped product , where and denote the integral manifolds of the distributions and , respectively and is the warping function. Thus the theorem is proved.
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