Hemi-Slant Warped Product Submanifolds of Nearly Kaehler Manifolds

Abstract

Hemi-slant warped product submanifolds of nearly Kaehler manifolds are studied and some interesting results are obtained. Moreover, an inequality is established for squared norm of second fundamental form and equality case is also discussed. The results obtained are also true if ambient manifold is replaced by a Kaehler manifold. These results generalize several known results in the literature.

1. Introduction

In [1] Bishop and O’Neill introduced the notion of warped product manifolds as a natural generalization of Riemannian product manifolds. For instance, a surface of revolution is a warped product manifold. So far as its applications are concerned, it has been shown that warped product manifolds provide excellent setting to model space time near black holes or bodies with large gravitational forces (see [1, 2]). Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research and many articles are available in literature (see [1, 3–5]). Chen [6] initiated the study of warped product submanifolds by showing that there do not exist warped product CR-submanifolds of the type , and he considered warped product CR-submanifolds of the types and established a relationship between the warping function and the squared norm of the second fundamental form. Extending the study of Chen, Sahin [7] proved that there exist no semislant warped product submanifolds in a Kaehler manifold. In [8], V. A. Khan and K. A. Khan studied generic warped product submanifolds of nearly Kaehler manifolds and obtained an inequality for squared norm of second fundamental form in terms of warping function. Recently, Sahin [4] investigated hemi-slant warped product for Kaehler manifolds and obtained an inequality for squared norm of second fundamental form for mixed totally geodesic submanifolds. In view of the interesting geometric characteristic of nearly Kaehler manifolds and the nonexistence of CR-product submanifolds in [9], it will be significant to explore hemi-slant warped product submanifolds of a nearly Kaehler manifold. In this continuation we have achieved success in extending the results of Sahin [4] and Chen [6] to the setting of nearly Kaehler manifolds.

2. Preliminaries

Let be a nearly Kaehler manifold with an almost complex structure and Hermitian metric and a Levi-Civita connection such that for all vector fields and on . Six-dimensional sphere is a classic example of a nearly Kaehler non-Kaehler manifold. It has an almost complex structure defined by the vector cross product in the space of purely imaginary Cayley numbers which satisfies the condition . Let be the Cayley division algebra generated by , over and the subspace of consisting of all purely imaginary Cayley numbers. We may identify with a -dimensional Euclidean space with the canonical inner product . The automorphism group of is the compact simple Lie group and furthermore the inner product is invariant under the action of and hence, the group may be considered as a subgroup of . A vector cross product for vectors in is defined by Then the multiplication table for is given by Considering as , the almost complex structure on is defined by where and . The almost complex structure defined by the above equation together with the induced metric on from on gives rise to a nearly Kaehler structure [10].

Let be a submanifold of . Then the induced Riemannian metric on is denoted by the same symbol and the induced connection on is denoted by the symbol . If and denote the tangent bundle on and , respectively, and , the normal bundle on , then the Gauss and Weingarten formulae are given by for , and where denotes the connection on the normal bundle . and are the second fundamental form and the shape operator of immersions of into . Corresponding to the normal vector field they are related as The mean curvature vector of is given by where is the dimension of and is a local orthonormal frame of vector fields on . The squared norm of the second fundamental form is defined as A submanifold of is said to be a totally geodesic submanifold if for each , , and totally umbilical submanifold if .

For and we write where and are tangential components of and , respectively, and and are the normal components of and .

The covariant differentiation of the tensors , , , and is defined, respectively, as Furthermore, for any , , the tangential and normal parts of are denoted by and ; that is, On using (6)–(13), we may obtain that Similarly, for , denoting by and , respectively, the tangential and normal parts of , we find that On a submanifold of a nearly Kaehler manifold by (2) and (16) for any , .

Let be an almost Hermitian manifold with an almost complex structure and Hermitian metric and let be a submanifold of . Submanifold is said to be CR-submanifold if there exist two orthogonal complementary distributions and such that is holomorphic distribution, that is, and is totally real distribution, that is, .

An immersed submanifold of an almost Hermitian manifold is said to be slant submanifold if the Wirtinger angle between and , and (cf. [11]). Holomorphic and totally real submanifolds are slant submanifolds with wirtinger angle 0 and . A submanifold is called proper slant if it is neither holomorphic nor totally real. More generally, a distribution on is called a slant distribution if the angle between and has the same value for each and and .

If is a slant submanifold of an almost Hermitian manifold , then we have (cf. [11]) where is the wirtinger angle of in . Hence we have for any , .

Papaghiuc [12] introduced a class of submanifolds in almost Hermitian manifolds called the semislant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al. [13] initiated the study of contact version of semislant submanifold and also bislant submanifolds. As a step forward, Carriazo [14] defined and studied bislant submanifolds and simultaneously gave the notion of antislant submanifolds in almost Hermitian manifolds; after that V. A. Khan and M. A. Khan [15] studied antislant submanifolds with the name pseudo-slant submanifolds in the setting of Sasakian manifolds.

Recently, Sahin [4] renamed pseudo-slant submanifolds as hemi-slant submanifolds and studied hemi-slant submanifolds for their warped product.

Definition 1. A submanifold of an almost Hermitian manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions and such that is totally real, that is, and is slant distribution with slant angle .

It is straight forward to see that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with and , respectively.

If is the invariant subspace of the normal bundle then in the case of hemi-slant submanifold, 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 (24) is an orthogonal direct decomposition.

A hemi-slant submanifold is called a hemi-slant product if the distributions and are parallel on . In this case is foliated by the leaves of these distributions. In particular if is CR-submanifold with parallel distribution then it is called CR-product. In general, if and are Riemannian manifolds with Riemannian metrics and , respectively, then the product manifold , is a Riemannian manifold with Riemannian metric defined as where are the projection maps of onto and , respectively, and are their differentials.

As a generalization of the product manifold and in particular of a hemi-slant product submanifold, one can consider warped product of manifolds which are defined in the following.

Definition 2. Let and be two Riemannian manifolds with Riemannian metrics 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-invariant 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 [1] proved the following.

Theorem 3. Let be warped product manifolds. If , and , then(i),(ii),(iii). is the gradient of and is defined as for all .

Corollary 4. On a warped product manifold , the following statements hold: (i) is totally geodesic in (ii) is totally umbilical in .

In what follows, and will denote a totally real and slant submanifold, respectively, of an almost Hermitian manifold .

A warped product manifold is said to be trivial if its warping function is constant. More generally, a trivial warped product manifold is a Riemannian product , where is the manifold with the Riemannian metric which is homothetic to the original metric of . For example, a trivial CR-warped product is CR-product.

Sahin [4] extended the study of warped product hemi-slant submanifolds and hemi-slant warped product of Kaehler manifolds introducing warped product submanifolds as and , where is the slant angle.

3. Hemi-Slant Warped Product Submanifolds

In [5] Uddin and Chi investigated warped product pseudo-slant (hemi-slant) submanifolds of nearly Kaehler manifolds and they only showed that there do not exist warped products of the form in nearly Kaehler manifolds, where is totally real submanifold and is slant submanifold. In this section we study the warped products of the types .

Let be a hemi-slant warped product of a nearly Kaehler manifold . Then by Theorem 3, for any , .

Now by formula (12) and Theorem 3, for each , . Now we will investigate some interesting results of the second fundamental form.

Proposition 5. On a hemi-slant warped product submanifold of a nearly Kaehler manifold , one has(i),(ii), for any , and .

Proof. As is totally geodesic in then and therefore by formula (17), or Similarly, we have Adding above two equations and using (20)(a), we get part (i).
Now by formula (17) and (20)(a), and by (29) the above equation gives Taking inner product of (35) with , we get which proves part (ii).

Theorem 6. For a hemi-slant warped product submanifold of a nearly Kaehler manifold the warping function satisfies the following relation: for any and .

Proof. If is a hemi-slant warped product submanifold of a nearly Kaehler manifold then for each and , and thus by (17), On the other hand Now using (29), the above equation takes the form Adding (38) and (40) and using (20)(a), taking inner product with , and using the fact that is totally umbilical, one gets the following equation: By replacing by the required result follows.

Remark 7. In [4] Sahin proved that hemi-slant warped products of the type do not exist in the setting of Kaehler manifolds. Therefore, in the following Corollary we discuss the warped products of the type .

Corollary 8. For a hemi-slant warped product submanifold of a Kaehler manifold the warping function satisfies the following relation: for any and .

Proof. Since is a hemi-slant warped product submanifold of a Kaehler manifold, then by tensorial equation of Kaehler manifold, it is easy to see that , for any and , and using this fact in (40) and taking inner product with , we get the required result.

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 we calculate the inequality for the squared norm of second fundamental form in the following theorem.

Theorem 9. Let be a hemi-slant warped product of a nearly Kaehler manifold with and slant and totally real submanifolds, respectively, of . If , then(i)the second fundamental form satisfies the following inequality: (ii)equality holds if , , is normal to and is normal to and where and ( and ),where is the gradient of and is dimension of .

Proof. In view of (24) the second fundamental form can be decomposed as follows: for each , where , and with where, where, for each and .
Now, making use of (37) with assumption and formulae (48) and (49), we obtain Summing over and and (37) and assumption with formula (50) the above equation gives Let us consider the orthonormal frame of vector fields on as , and the second term in the right hand side of last equation on using (50) can be written as On applying (37), the first part of above expression reduced to Taking account of the above equation into (52), we obtain The inequality follows from (44) and (55).
To discuss the equality case we will explore the expression as follows.
Making use of (47), (48), and (23) and summing over and we find Let us choose the orthonormal frame of vectors fields on as . Then the right hand side of the above equation with the help of (48) can be written as From (44), (50), and expression (53), it is clear that equality holds if , is normal to and is normal to and , , , where and .

Remark 10. Since (37) is the key result of the paper which helps to get the inequality in Theorem 9 and moreover (37) is also true for the Kaehler manifolds, hence the results in Theorem 9 are also true for hemi-slant warped product submanifolds of a Kaehler manifold.

Now we compile some results of [16] and give the following example of a warped product submanifold in .

Example 11. Let be the canonical basis of the Cayley division algebra on over and the subspace of generated by the purely imaginary Cayley numbers . Then is a unit sphere admitting a nearly Kaehler structure as has been specified earlier. Now suppose that is a unit 2-sphere. For a real triple with and , define a mapping of to as follows: where and . Then gives rise to an isometric immersion from warped product Riemannian manifold into . Moreover, induced warped product metric on is given by where and are natural projections and is the Riemannian metric on and from (27) it is evident that warping function is as follows:

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are highly thankful to anonymous referees for their valuable suggestions and comments which have improved the paper.