Pointwise Hemislant Submanifolds in a Complex Space Form

Abstract

In this paper, pointwise hemislant submanifolds were introduced in a Kahler manifold. The integrability conditions for the distributions which are involved in the definition of a pointwise hemislant submanifold were investigated. In addition, the necessary and sufficient conditions were given for a pointwise hemislant submanifold to be a pointwise hemislant product.

1. Introduction

The concept of pointwise slant submanifolds appeared under the name of quasi-slant submanifolds by Etayo [1] as a generalization of slant submanifolds introduced by Chen [2, 3]. Then, in [4], Chen and Garay studied pointwise slant submanifolds of almost Hermitian manifolds and proved many interesting results. Later on, pointwise slant submanifolds were investigated on Riemannian manifolds equipped with various structures [5–9].

On the other hand, the notation of hemislant submanifolds was first defined by Carrizo et al. [10, 11], and they named them pseudoslant submanifolds. After that, in [12], Sahin studied hemislant submanifolds and their warped products of Kahler manifolds. Al-Solamy et al. [13] defined the totally umbilical hemislant submanifolds in Kahler manifolds and derived several results. Recently, hemislant submanifolds were studied by different authors in many ambient spaces (see [14–17]). Furthermore, the notation of a quasi-hemislant submanifold was studied by Prasad et al. in [18].

In the present paper, the purpose is to study the geometry of pointwise hemislant submanifolds of a Kahler manifold. Some basic formulas and definitions are recalled in Section 2, which are useful to the next section. Section 3 defines the pointwise hemislant submanifold of a Kahler manifold and gives some basic results on such submanifolds. The integrability condition of the distributions on the pointwise hemislant submanifold of a Kahler manifold is constructed. Following the procedure, the necessary and sufficient conditions are given for a pointwise hemislant submanifold to be a pointwise hemislant product.

2. Preliminaries

Let be an almost Hermitian manifold with structure where is a (1, 1) tensor field and is a Riemannian metric on satisfying the following properties:for all vector fields on . If, in addition to the above relations,holds, then is said to be Kahler manifold, where is the Levi-Civita connection of . The covariant derivative of the complex structure is given by

Let be an isometrical immersed submanifold of with the induced metric . Let and be the differential vector fields set tangent and normal to , respectively. Then, Gauss and Weingarten formulas are, respectively, given byandfor all and , where and are the induced connections on and , respectively, is the second fundamental form of , and is the shape operator of the second fundamental form, which is related by

For any orthonormal frame of , the mean curvature vector is given bywhere . The submanifold is totally geodesic in if and minimal if . If for all , then is totally umbilical. For any and ,andwhere and are the tangential components and and are the normal components of and , respectively. A submanifold of an almost Hermitian manifold is said to be holomorphic (resp. totally real) if (resp. [19]. The covariant derivatives of the tangential and normal components of and are defined by

For any , we have . Also, by using (2), (9), and (10), we have , for any . That is, and are skew-symmetric tensor fields. Furthermore, the relation between the tensor fields and is given byfor any and .

3. Pointwise Hemislant Submanifolds of a Kahler Manifold

In this section, a brief introduction of pointwise hemislant submanifolds of a Kahler manifold is given. We shall obtain some results.

Chen defined slant and pointwise slant submanifolds as follows.

For a nonzero vector , , the angle between and is called the Wirtinger angle of . A submanifold is said to be slant if the Wirtinger angle is constant on ; i.e., it is independent of the choice of and [2, 3]. In this case, is called the slant angle of .

A submanifold is said to be pointwise slant if the Wirtinger angle can be regarded as a function on , which is known as the slant function in [4]. A pointwise slant submanifold with a slant function is simply called a pointwise slant submanifold. Clearly, a pointwise slant submanifold is a slant submanifold if its slant function is a constant function on .

Holomorphic and totally real submanifolds are slant submanifolds with slant angles 0 and , respectively. A slant submanifold is called proper slant if it is neither holomorphic nor totally real.

We recall the following basic result from [4] for pointwise slant submanifolds of an almost Hermitian manifold.

Theorem 1. Let be a submanifold of an almost Hermitian manifold. Then, is pointwise slant if and only iffor some real-valued function defined on .

Following relations are straightforward consequence of equation (16):for all . Clearly, we also have

In [20], Uddin and Stankovic defined a pointwise hemislant submanifold as follows.

Definition 2. A submanifold of a Kahler manifold is said to be a pointwise hemislant submanifold if there exist two orthogonal complementary distributions and such that:(i)The tangent space admits the orthogonal direct decomposition (ii)The distribution is pointwise slant with a slant function (iii)The distribution is a totally real, i.e., If the dimensions of the distributions and are denoted by and , respectively, then the following cases are obtained:(i)If , then is totally real submanifold(ii)If , then is a pointwise slant submanifold(iii)If and , then is a holomorphic submanifold(iv)If is constant on , then is a hemislant submanifold with a slant angle (v)If and is not constant, then is a proper pointwise hemislant submanifoldWe mention the following example of pointwise hemislant submanifolds in the Euclidean space.

Example 1. Let be the Euclidean 6-space with the cartesian coordinates , and the almost comlex structure is defined byand the standard Euclidean metric is on . Consider a submanifold of given by the immersion as follows:for nonvanishing real valued functions on . Then, the tangent bundle of is spanned by the following tangent vectors:Then,Clearly, is an orthogonal to ; hence, is a totally real distribution, and is a proper pointwise slant distribution with a slant function . Thus, is proper pointwise hemislant submanifold of .

Lemma 3. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, .

Proof. For any and , by (9), we haveBut, from (2), we haveThus,which means that .
From the above lemma, the normal bundle can be decomposed aswhere is the invariant distribution of under .

Lemma 4. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, we have

Proof. The proof is direct, and it can be obtained by using (1), (2), (9), and (16).

Lemma 5. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, we haveandfor all .

Proof. In a Kahler manifold, we have thatwhich gives thatFrom (5) and (9), we obtainAgain, by (5), (6), (9), and (10), we can writeComparing the tangential and normal parts with using (11) and (12), we get the required results. Hence, the lemma is proved completely.

By a similar argument, we have the following Lemma.

Lemma 6. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, we haveandfor all and .

Lemma 7. Let be a pointwise hemislant submanifold of a Kahler manifold . Then,for all .

Proof. For any and , using (3), (5), (6), and (7), we haveTherefore,

It follows from (28) thatfor any .

Theorem 8. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the covariant derivation of the endomorphism is skew-symmetric, i.e.,for any .

Proof. For any , using (7), (15), and (21), we getThis proves our assertion.

Theorem 9. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, we havefor any and .

Proof. For any and , using (7), (22), and (23), we obtainwhich verifies our assertion.

Theorem 10. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the tensor is parallel if and only iffor any .

Proof. For any , using (7), (15), and (21), we haveHence, the result is obtained.

Theorem 11. Let be a pointwise hemislant submanifold in a Kahler manifold . Then, the tensor is parallel if and only iffor any and .

Proof. For any and , using (7) and (22), we haveThe proof is completed.

Theorem 12. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the covariant derivation of the endomorphism is skew-symmetric, that is,for any and .

Proof. For any and , using (7), (15), and (24), we getwhich is the required result.

Theorem 13. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the tensor is parallel if and only iffor all .

Proof. Let and , using (7), (15), and (24), we getThis proves our assertion.

Theorem 14. Let be a pointwise hemislant submanifold of a Kahler manifold . Then,for any .

Proof. For any and , using (3), (5), (6), (9), and (20), we haveThus, the result is concluded.

Theorem 15. Let be a proper pointwise hemislant submanifold of a Kahler manifold . If the tensor is parallel, then, is a totally geodesic submanifold of .

Proof. Suppose that is parallel, then making use of (9) and (24), we havefor all and . Applying to the above relation with using (1) and (10), we findTaking the inner product with and then using (7), (15), (19), and (92), we obtainwhich implies thatAs is a proper pointwise hemislant submanifold, we obtain , which means that is a totally geodesic submanifold of .

Definition 16. A pointwise hemislant submanifold of a Kahler manifold is said to be -geodesic (resp. -geodesic) if , for any (resp. , for any ), and is called a mixed geodesic submanifold if , for any and .

Theorem 17. Let be a proper pointwise hemislant submanifold of a Kahler manifold . If the tensor is parallel, then, is a mixed geodesic submanifold of .

Proof. If is parallel, then from Theorem 9 and (28) with (22), we obtainfor any and . Also, we can writePutting with using (16), we findSince is a proper pointwise hemislant submanifold, we conclude . That is, is a mixed geodesic submanifold of .

Theorem 18. Let be a pointwise hemislant submanifold of a Kahler manifold . If the tensor is parallel, then is either -geodesic or a totally real submanifold of .

Proof. Suppose that is parallel, then if we put in (23), we obtainfor any . This tells us that is either totally real or . Again by (35), we derivewhich implies thatfor any . That is, is a -geodesic or a totally real submanifold of .

Theorem 19. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the totally real distribution is always integrable, and its maximal integral submanifold is totally real submanifold of .

Proof. For any and , by (2), (3), (5), (6), and (9), we haveNow, from (7), (12), (19), (28), (30), and (40), we deriveThus, by (7), (18), and (20), we can writewhich implies thatTherefore, for any , which means that the totally real distribution is always integrable, and its maximal integral submanifold is a totally real submanifold of . Hence, the proof is completed.

Corollary 20. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, we havefor any .

Proof. Since the ambient manifold is Kahler, for any , we havewhich can be written asNow, using (5) and (9), we findHence, by (6), (9), and (10), we deduceIf we take the tangential components of the above equation, we obtainSimilarly, we getThus, by (26), (35), and (36), we derive

Theorem 21. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the slant distribution is integrable if and only iffor any .

Proof. Let and , by using (2) and (3), we haveThen, by (5), (6), and (9), we obtainSince , we deduce the result.

Theorem 22. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the pointwise slant distribution is integrable if and only iffor any and .

Proof. For any and , by using (2), (3), and (9), we haveThus, by (6), (12), and (22), we findHence,which is the required result.

Theorem 23. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, the slant distribution is minimal if and only if the normal bundle is parallel andfor any and .

Proof. For any and , we haveUsing (2), (6), and (20), we deriveHence, we conclude that is minimal if and only if the normal bundle is parallel and

Definition 24. A pointwise hemislant submanifold of a Kahler manifold is said to be pointwise hemislant product if the distributions and are totally geodesic in .

Theorem 25. Let be a pointwise hemislant submanifold of a Kahler manifold . Then, is a pointwise hemislant product if and only if the second fundamental form of satisfies the following condition:for any and .

Proof. For all and , we haveNow, by (6), (9), (12), and (28), we haveThus, using (7), (18), and (22), we findwhich gives thatBy similar argument, we obtainHence,So, from (92) and (94), we conclude that and are totally geodesic if and only if (74) is satisfied.

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Conflicts of Interest

The authors declare that they have no conflicts of interest.