Research Article | Open Access
Falleh R. Al-Solamy, Meraj Ali Khan, "Warped Product Submanifolds of Riemannian Product Manifolds", Abstract and Applied Analysis, vol. 2012, Article ID 724898, 12 pages, 2012. https://doi.org/10.1155/2012/724898
Warped Product Submanifolds of Riemannian Product Manifolds
We study warped product of the type and , where , , and are proper slant, invariant, and anti-invariant submanifolds, respectively, and we prove some basic results and finally obtain some inequalities for squared norm of second fundamental form.
Bishop and O’Neil  introduced the notion of warped product manifolds that occur naturally; for example, surface of revolution is a warped product manifold. With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on warped product manifolds . CR-warped product was introduced by Chen ; he studied warped product CR-submanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product of the form ; therefore he considered warped product CR-submanifolds of type and established a relationship between the warping function and the squared norm of second fundamental form . In  Atçeken studied semi-slant warped product of Riemannian product manifolds. In fact they proved that there exists no warped product if spheric submanifold of warped product submanifold is proper slant submanifold. On the other hand they proved the existence of warped product of the type and via some examples. In this continuation we have studied the warped product submanifolds in which proper slant submanifolds are totally geodesic; that is, we study the warped product of the types and and called them semi-slant warped product and hemi-slant warped product submanifolds, respectively.
Let and be the Riemannian manifolds with dimensions and , respectively, and let be Riemannian product manifold of and . We denote projection mapping of onto and by and , respectively. Then we have , and , where denotes the differential.
Riemannian metric of the Riemannian product manifold is defined by for any . If we set , then , , and satisfies the condition for any ; thus defines an almost Riemannian product structure on . We denote Levi-Civita connection on by ; then the covariant derivative of is defined as for any . We say that is parallel with respect to the connection if we have . Here from , we know that is parallel; that is, is Riemannian product structure.
Let be a Riemannian product manifold with Riemannian product structure and an immersed submanifold of ; we also denote by the induced metric tensor on as well as on . If is the Levi-Civita connection on , then the Gauss and Weingarten formulas are given, respectively, as for any and , where is the connection on and is the connection in the normal bundle, is the second fundamental form of , and is the shape operator of . The second fundamental form and the shape operator are related by For any , we can write where and are the tangential and normal components of , respectively, and for , where and are the tangential and normal components of , respectively, and the submanifold is said to be invariant if is identically zero. On the other hand is said to be an anti-invariant submanifold if is identically zero.
Let be an immersed submanifold of a Riemannian product manifold , for each nonzero vector tangent to at a point , and we denote by the angle between and . The angle is called the slant angle of immersion.
Let be an immersed submanifold of a Riemannian product manifold . is said to be slant submanifold of Riemannian product manifold if the slant angle is constant which is independent of choice of and .
Invariant and anti-invariant submanifolds are particular cases of slant submanifolds with angles and , respectively. A slant submanifold which is neither invariant nor anti-invariant is called proper slant submanifold. The following characterization of slant submanifolds of Riemannian product manifolds is proved by Atçeken .
Theorem 2.1. Let be an immersed submanifold of a Riemannian product manifold . Then is a slant submanifold if and only if there exists a constant such that .
Moreover, if is the slant angle of , then it satisfies .
Hence, for a slant submanifold we have the following relations which are consequences of the above theorem: for any .
Papaghuic  introduced a class of submanifolds in almost Hermitian manifolds called semi-slant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al.  initiated the study of contact version of semi-slant submanifolds and also gave the notion of Bi-slant submanifolds. A step forward Carriazo  defined and studied Bi-slant submanifolds and simultaneously gave the notion of anti-slant submanifolds; after that V. A. Khan and M. A. Khan  have studied anti-slant submanifolds with the name pseudo-slant submanifolds. Recently, Sahin  renamed these submanifolds and studied these submanifolds with the name hemi-slant submanifolds for their warped product.
Definition 2.2. A submanifold of a Riemannian product manifold is said to be semi-slant submanifold if there exist two orthogonal complementary distributions and such that is invariant and is slant distribution with slant angle .
It is straight forward to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds with and , respectively.
If is invariant subspace under of the normal bundle , then in the case of semi-slant submanifold, the normal bundle can be decomposed as
A semi-slant submanifold is called a semi-slant product if the distributions and are parallel on . In this case is foliated by the leaves of these distributions.
Definition 2.3. A submanifold of a Riemannian product manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions and such that is totally real and is slant distribution with slant angle .
It is easy to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds with and , respectively. The normal bundle can be decomposed as follows:
As and are orthogonal distributions on , then it is easy to see that the distributions and are mutually perpendicular. In fact, the decomposition (2.18) is an orthogonal direct decomposition. A hemi-slant submanifold is called a hemi-slant product if the distributions and are parallel on .
As a generalization of product manifold and in particular of semi-slant product submanifolds (hemi-slant product submanifolds) one can consider warped product of manifolds which are defined as.
Definition 2.4. Let and be two Riemannian manifolds with Riemannian metric and , respectively, and a positive differentiable function on . The warped product of and is the Riemannian manifold , where
For a warped product manifold , we denote by and the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words, is obtained by the tangent vectors of via the horizontal lift and is obtained by the tangent vectors of via vertical lift. In case of semi-slant warped product submanifolds and are replaced by and , respectively.
The warped product manifold is denoted by . If is the tangent vector field to at , then
Bishop and O’Neill  proved the following.
Theorem 2.5. Let be warped product manifolds. If and , then(i), (ii), (iii). is the gradient of and is defined as for all .
Corollary 2.6. On a warped product manifold , the following statements hold: (i) is totally geodesic in ;(ii) is totally umbilical in .
Throughout, we denote by , , and invariant, anti-invariant, and slant submanifolds, respectively, of a Riemannian product manifold .
3. Semi-Slant Warped Product Submanifolds
In this section we will consider the warped product of the type .
For the warped product of the type by Theorem 2.5 we have for any and .
Lemma 3.1. Let be a semi-slant warped product submanifold of a Riemannian product manifold; then(i),
for any and .
Proof. For any , ; then from (2.13)
Taking inner product with we have
This is part (i) of the lemma.
Now for any , from (2.13) and (2.9), Taking inner product with , the above equation yields Using (3.1), the above equation gives In particular This proves part (ii) of the lemma. Now we have the following corollary.
Corollary 3.2. For the warped product of the type following statements are equivalent: (i), (ii) or the warping function is constant; that is, there does not exist warped product.
Now we have the following characterization for semi-slant warped product submanifolds.
Theorem 3.3. A semi-slant submanifold of Riemannian product manifolds with integrable invariant distribution and the slant distribution is locally a semi-slant warped product if and only if and there exist a -function on with for all such that for all and .
Proof. If is a semi-slant warped product of the type , then for any and from (2.9), (2.13), and (3.1), we have
Taking inner product with , the above equation gives
By part (i) of Lemma 3.1, we also have
From (3.12) and (3.13) we have the following equation:
Conversely, let be a semi-slant submanifold of satisfying the hypothesis of the theorem; then for any and we have
This mean .
From (2.14), we have Comparing components of and , we get It is evident from the above equation that ; this means for any and hence is totally geodesic. Further, let be a leaf of and a second fundamental form of the immersion in ; then for any and or Using the hypothesis, we get Finally, the above equation yields That is, is totally umbilical and as , for all , this means that mean curvature vector of is parallel; that is, the leaves of are extrinsic spheres in . Hence by virtue of result of  which says that if the tangent bundle of a Riemannian manifold splits into an orthogonal sum of nontrivial vector subbundles such that is spherical and its orthogonal complement is auto parallel, then the manifold is locally isometric to a warped product , we can say is locally semi-slant warped product submanifold , where warping function .
Let us denote by and the tangent bundles on and , respectively, and let and be local orthonormal frames of vector fields on and , respectively, with and being real dimensions: Now, on a semi-slant warped product submanifold of a Riemannian product manifold, we prove the following.
Theorem 3.4. Let be a semi-slant warped product submanifold of a Riemannian product manifold with and invariant and slant submanifolds, respectively, of . Then the squared norm of the second fundamental form satisfies
Proof. For , in view of decomposition (2.17), we may write for each , where and with where for each . For any and , by (3.25) we have Now, using (3.26), (3.7), and (2.16), the above equation takes the form Now summing over , and again using (3.7) and (3.26), we have or By similar calculation, from (3.25), (3.26), (3.3), and (2.16) it is easy to see that The result follows from (3.22), (3.30), and (3.31).
4. Hemi-Slant Warped Product Submanifolds
In this section we will study the warped product of the type . For warped product of type from Theorem 2.5 we have for any and .
Now we have the following lemma.
Lemma 4.1. Let be a hemi-slant warped product submanifold of a Riemannian product manifold; then(i), (ii), (iii), for any and .
Proof. For any , ; then from (2.13) we have
or equivalently the above equation gives
which proves part (i).
From (2.9), (2.13), we have Taking inner product with the above equation is reduced to Using (2.9), (2.13), and (4.1), we derive Taking inner product with and using (4.5), Now we have the following corollary.
Corollary 4.2. For the warped product of the type following statements are equivalent: (i), (ii)the warping function is constant; that is, there does not exist warped product.
Proof. As is totally umbilical, then from (4.7) and from Theorem 2.5
Since , hence from the previous equation it is easy to see that statements (i) and (ii) are equivalent.
Let us denote by and the tangent bundles on and , respectively, and let and be local orthonormal frames of vector fields on and , respectively, with and being their real dimensions; then Now, on a hemi-slant warped product submanifold of a Riemannian product manifold, we prove the following inequality.
Theorem 4.3. Let be a hemi-slant warped product submanifold of a Riemannian product manifold with and anti-invariant and slant submanifolds, respectively, of . Then the squared norm of the second fundamental form satisfies
Proof. In view of decomposition (2.18), the second fundamental form can be decomposed as follows:
for each , where , , and with
for each .
Now making use of (2.18), (4.12), (4.13), and (4.7) we have for any and .
Again using (4.12) and (4.13), the previous equation gives Summing over and , we have Similarly, for any and by (4.12), (4.13), and (4.5) it is easy to see that The result follows from (4.9), (4.16), and (4.17).
The work is supported by Deanship of Scientific Research, University of Tabuk, Saudi Arabia.
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Copyright © 2012 Falleh R. Al-Solamy and Meraj Ali Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.