Research Article | Open Access
Falleh R. Al-Solamy, Meraj Ali Khan, "Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold", Mathematical Problems in Engineering, vol. 2012, Article ID 708191, 10 pages, 2012. https://doi.org/10.1155/2012/708191
Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold
We study semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of second fundamental form.
In  Tanno classified the connected almost contact metric manifold whose automorphism group has maximum dimension, there are three classes:(a)homogeneous normal contact Riemannian manifolds with constant holomorphic sectional curvature if the sectional curvature of the plane section contains , say ;(b)global Riemannian product of a line or a circle and Kaehlerian manifold with constant holomorphic sectional curvature, ;(c)a warped product space , if .
Manifolds of class (a) are characterized by some tensorial equations, it has a Sasakian structure and manifolds of class (b) are characterized by some tensor equations called Cosymplectic manifolds. Kenmotsu  obtained some tensorial equations to Characterize manifolds of class (c), these manifolds are called Kenmotsu manifolds.
The notion of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghiuc  after that cabrerizo et al.  defined and studied semi-slant submanifolds in the setting of almost contact manifolds.
Bishop and O’Neill  introduced the notion of warped product manifolds. These manifolds are generalization of Riemannian product manifolds and occur naturally. Recently, many important physical applications of warped product manifolds have been discovered, giving motivation to study of these spaces with differential geometric point of view. For instance, it has been accomplished that warped product manifolds provide an excellent setting to model space time near black hole or bodies with large gravitational fields (c.f., [5–7]). Due to wide applications of these manifolds in physics as well as engineering this becomes a fascinating and interesting topic for research, and many articles are available in literature (c.f., [3, 8, 9]).
Recently, Atçeken  proved that the warped product submanifolds of type and of a Kenmotsu manifold do not exist where the manifolds and (resp. are proper slant and invariant (resp., anti-invariant) submanifolds of Kenmotsu manifold , respectively. After that Siraj-Uddin et al.  investigated warped product of the types and and obtained some interesting results. In this continuation we obtain a characterization and an inequality for squared norm of second fundamental form.
A dimensional manifold is said to have an almost contact structure if there exist on a tensor field of type , a vector field , and 1-form satisfying the following properties: There always exists a Riemannian metric on an almost contact manifold satisfying the following conditions: where are vector fields on .
An almost contact metric structure is said to be Kenmotsu manifold, if it satisfies the following tensorial equation : for any , where is the tangent bundle of and denotes the Riemannian connection of the metric . Moreover, for a Kenmotsu manifold
Let be a submanifold of an almost contact metric manifold with induced metric and if and are the induced connection 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, respectively, for the immersion of into and they are related as where denotes the Riemannian metric on as well as 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 field and are defined as From (2.3), (2.5), (2.7) and (2.8) we have
Definition 2.1 (see ). A submanifold of an almost contact metric manifold is said to be slant submanifold if for any and is constant. The constant angle is then called slant angle of in . If the submanifold is invariant submanifold, if then it is anti-invariant submanifold, if , then it is proper slant submanifold.
For slant submanifolds of contact manifolds Cabrerizo et al.  proved the following lemma.
Lemma 2.2. Let be a submanifold of an almost contact manifold , such that then is slant submanifold if and only if there exists a constant such that
Thus, one has the following consequences of above formulae:
A submanifold of is said to be semi-slant submanifold of an almost contact manifold if there exist two orthogonal complementary distributions and on such that(i), (ii)the distribution is invariant that is, ,(iii)the distribution is slant 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.
As a generalization of the product manifolds and in particular of a semi-slant product submanifold, one can consider warped product of manifolds which are defined as
Definition 2.3. Let and be two Riemannian manifolds with Riemannian metric and , respectively, and be 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 2.4. Let be warped product manifolds. If , and , then(i),
is the gradient of and is defined as for all .
Corollary 2.5. On a warped product manifold , the following statements hold:(i) is totally geodesic in ,(ii) is totally umbilical in .
Throughout, one denotes by and an invariant and a slant submanifold, respectively, of an almost contact metric manifold .
Khan et al.  proved the following corollary.
Corollary 2.6. Let be a Kenmotsu manifold and and be any Riemannian submanifolds of , then there do not exist a warped product submanifold of such that is tangential to .
Thus, one assumes that the structure vector field is tangential to of a warped product submanifold of .
In this paper we will consider the warped product of the type and . The warped product of the type is called warped product semi-slant submanifolds; this type of warped product is studied by Atçeken , they proved that the warped product does not exist. The warped product of the type is called semi-slant warped product; these submanifolds were studied by Siraj-Uddin et al.  and they proved the following Lemma
Lemma 2.7. Let be warped product semi-slant submanifold of a Kenmotsu manifold such that is tangent to , where and are invariant and proper slant submanifolds of . then(i),
for any and .
Replacing by in part (ii) of above lemma one has
3. Semi-Slant Warped Product Submanifolds
Throughout this section we will study the warped product of the type , for these submanifolds by Theorem 2.4 we have for any and .
Lemma 3.1. Let be a semi-slant warped product submanifolds of a Kenmotsu manifold , then for any and .
Now we have the following Characterization.
Theorem 3.2. A semi-slant submanifold of a Kenmotsu manifold with integrable invariant distribution and integrable slant distribution is locally a semi-slant warped product if and only if and there exists a function on with , for all and .
Proof. From (2.10) and (3.1) we have
from (3.5) and (3.6), we get
taking inner product with , we have
From Lemma 3.1 and (3.8) we get the desired result.
Conversely, let be a semi-slant submanifold of satisfying the hypothesis of the theorem, then for any and that means . Then from (2.11) Since , then we have , that is, . Hence, each leaf of is totally geodesic in .
Further, suppose be a leaf of and be second fundamental form of the immersion of in , then for any and , we have using (2.7) and (2.5), the above equation yields applying (3.4), we get Replacing by , the above equation gives From above equation it is easy to derive that is, is totally umbilical and as , for all is defined on , this mean 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 autoparallel, then the manifold is locally isometric to a warped product , we can say is locally semi-slant warped product submanifold , where the 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 dimension. Since for all , then the second fundamental form can be written as Now, on a semi-slant warped product submanifold of a Kenmotsu manifold, we prove the following.
Theorem 3.3. Let be a semi-slant warped product submanifold of a Kenmotsu manifold with and invariant and slant submanifolds, respectively, of . If for all , then the squared norm of the second fundamental form satisfies where is the gradient of and is the dimension .
Proof. In view of the decomposition (2.15), we may write
for each , where and with
for each . In view of above formulae we have
Now using (2.14) and (2.19)
In view of (3.20) and (2.19), we get
Summing over and the above equation yields
Since we have choose the orthonormal frame of vector fields on as , then the second term in the right-hand side of (3.24) is written as
From part (i) of Lemma 2.7, the first two terms of above equation can be written as
In account of to hypothesis the above expression is greater than equal to the following term:
Using above inequality into (3.24), we have
The inequality (3.17) follows from (3.16) and (3.28).
The equality holds if , , is orthogonal to and for all and and .
In this paper we study nontrivial warped product submanifolds of a Kenmotsu manifold and in this study there emerge natural problems of finding the estimates of the squared norm of second fundamental form and to find the relation between shape operator and warping function. This study predict the geometric behavior of underlying warped product submanifolds. Further, as it is known that the warping function of a warped product manifold is a solution of some partial differential equations (c.f., ) and most of physical phenomenon is described by partial differential equations. We hope that our study may find applications in physics as well as in engineering.
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.