Warped Product Submanifolds of LP-Sasakian Manifolds

Abstract

We study of warped product submanifolds, especially warped product hemi-slant submanifolds of LP-Sasakian manifolds. We obtain the results on the nonexistance or existence of warped product hemi-slant submanifolds and give some examples of LP-Sasakian manifolds. The existence of warped product hemi-slant submanifolds of an LP-Sasakian manifold is also ensured by an interesting example.

1. Introduction

The notion of warped product manifolds was introduced by Bishop and O'Neill [1], and later it was studied by many mathematicians and physicists. These manifolds are generalization of Riemannian product manifolds. The existence or nonexistence of warped product manifolds plays some important role in differential geometry as well as in physics.

On the analogy of Sasakian manifolds, in 1989, Matsumoto [2] introduced the notion of LP-Sasakian manifolds. The same notion is also introduced by Mihai and Roşca [3] and obtained many interesting results. Later on, LP-Sasakian manifolds are also studied by several authors.

The notion of slant submanifolds in a complex manifold was introduced and studied by Chen [4], which is a natural generalization of both invariant and anti-invariant submanifolds. Chen [4] also found examples of slant submanifolds of complex Euclidean spaces and . Then, Lotta [5] has defined and studied the slant immersions of a Riemannian manifold into an almost contact metric manifold and proved some properties of such immersions. Also, Cabrerizo et al. [6] studied slant immersions of K-contact and Sasakian manifolds.

In 1994, Papaghuic [7] introduced the notion of semi-slant submanifolds of almost Hermitian manifolds. Then, Cabrerizo et al. [8] defined and investigated semi-slant submanifolds of Sasakian manifolds. The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bi-slant submanifolds and he called them anti-slant submanifolds [9]. Recently, these submanifolds were studied by Sahin for their warped products of Kähler manifolds [10]. Recently, Uddin [11] studied warped product CR-submanifolds of LP-Sasakian manifolds.

The purpose of the present paper is to study the warped product hemi-slant submanifolds of LP-Sasakian manifolds. The paper is organized as follows. Section 2 is concerned with some preliminaries. Section 3 deals with the study of warped and doubly warped product submanifolds of LP-Sasakian manifolds. In Section 4, we define hemi-slant submanifolds of LP-contact manifolds and investigate their warped products. Section 5 consists some examples of LP-Sasakian manifolds and their warped products.

2. Preliminaries

An -dimensional smooth manifold is said to be an LP-Sasakian manifold [3] if it admits a tensor field , a unit timelike contravariant vector field , an 1-form , and Lorentzian metric , which satisfy where denotes the operator of covariant differentiation with respect to the Lorentzian metric . It can be easily seen that, in an LP-Sasakian manifold, the following relations hold: Again, we put for any vector fields , tangent to . The tensor field is a symmetric (0,2) tensor field [2]. Also, since the vector field is closed in an LP-Sasakian manifold, we have [2] for any vector fields and tangent to .

Let be a submanifold of an LP-Sasakian manifold with induced metric and let and be the induced connections on the tangent bundle and the normal bundle of , respectively. Then, the Gauss and Weingarten formulae are given by for all , and , where and are second fundamental form and the shape operator (corresponding to the normal vector field ), respectively, for the immersion of into . The second fundamental form and the shape operator are related by [12] for any and

For any , we may write where is the tangential component and is the normal component of .

Also, for any , we have where and are the tangential and normal components of , respectively. The covariant derivatives of the tensor fields and are defined as for any .

Throughout the paper, we consider to be tangent to . The submanifold is said to be invariant if is identically zero, that is, for any . On the other hand, is said to anti-invariant if is identically zero, that is, for any .

Furthermore, for a submanifold tangent to the structure vector field , there is another class of submanifolds which is called a slant submanifold. For each nonzero vector tangent to at , the angle , between and is called the slant angle or wirtinger angle. If the slant angle is constant then the submanifold is called aslant submanifold. Invariant and anti-invariant submanifolds are particular classes of slant submanifolds with slant angle and , respectively. A slant submanifold is said to be proper slant if the slant angle lies strictly between and , that is, [6].

Theorem 2.1 (see [13]). Let be a submanifold of a Lorentzian almost paracontact manifold such that is tangent to . Then, is slant submanifold if and only if there exists a constant such that

Furthermore, if is the slant angle of , then . Also from (2.14), we have for any , tangent to .

The study of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghuic [7], which was extended to almost contact manifold by Cabrerizo et al. [8]. The submanifold is called semi-slant submanifold of if there exist an orthogonal direct decomposition of as where is an invariant distribution, that is, and is slant with slant angle . The orthogonal complement of in the normal bundle is an invariant subbundle of and is denoted by . Thus, we have for a semi-slant submanifold For an LP-contact manifold this study is extended by Yüksel et al. [13].

3. Warped and Doubly Warped Products

The notion of warped product manifolds was introduced by Bishop and O’Neill [1]. They defined the warped product manifolds as follows.

Definition 3.1. Let and be two semi-Riemannian manifolds and be a positive differentiable function on . Then, the warped product of and is a manifold, denoted by , where A warped product manifold is said to be trivial if the warping function is constant.
More explicitely, if the vector fields and are tangent to at , then where are the canonical projections of onto and , respectively, and * stands for the derivative map.
Let be a warped product manifold, which means that and are totally geodesic and totally umbilical submanifolds of , respectively.

For the warped product manifolds, we have the following result for later use [1].

Proposition 3.2. Let be a warped product manifold. Then, (I) is  the  lift  of on ,(II), (III), for any and , where and denote the Levi-Civita connections on and , respectively.

Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Ünal [14]. A doubly warped product manifold of and , denoted as is endowed with a metric defined as where and are positive differentiable functions on and , respectively.

In this case formula (II) of Proposition 3.2 is generalized as for each in and in [15].

One has the following theorem for doubly warped product submanifolds of an LP-Sasakian manifold [11].

Theorem 3.3. Let be a doubly warped product submanifold of an LP-Sasakian manifold where and are submanifolds of . Then, is constant and is anti-invariant if the structure vector field is tangent to , and is constant and is anti-invariant if is tangent to .

The following corollaries are immediate consequences of the above theorem.

Corollary 3.4. There does not exist a proper doubly warped product submanifold in LP-Sasakian manifolds.

Corollary 3.5. There does not exist a warped product submanifold of an LP-Sasakian manifold such that is tangent to .

From the above theorem and Corollary 3.5, we have only the remaining case is to study the warped product submanifold with structure vector field is tangent to .

4. Warped Product Hemi-Slant Submanifolds

In this section, first we define hemi-slant submanifolds of an LP-contact manifold and then we will discuss their warped products.

Definition 4.1. A submanifold of an LP-contact manifold is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions and satisfying: (i), (ii) is a slant distribution with slant angle ,(iii) is anti-invariant, that is, .If is -invariant subspace of the normal bundle , then in case of hemi-slant submanifold, the normal bundle can be decomposed as
Now, we discuss the warped product hemi-slant submanifolds of an LP-Sasakian manifold . If be a warped product hemi-slant submanifold of an LP-Sasakian manifold and and are slant and anti-invariant submanifolds of an LP-Sasakian manifold , respectively then their warped product hemi-slant submanifolds may be given by one of the following forms: (i)(ii). In the following theorem, we start with the case (i).

Theorem 4.2. There does not exist a proper warped product hemi-slant submanifold of an LP-Sasakian manifold such that is tangent to , where and are anti-invariant and proper slant submanifolds of , respectively.

Proof. Let be a proper warped product hemi-slant submanifold of an LP-Sasakian manifold such that is tangent to . Then, for any and , we have By virtue of (2.3) and (2.7)–(2.11), it follows from (4.2) that Using Proposition 3.2(II) in (4.3) and then equating the tangential components, we get Taking the inner product with in (4.4) and using the fact that and are mutually orthogonal vector fields, then we have Using (2.9) and (2.15), we get Replacing by in (4.6) and using (2.14), we obtain Adding (4.6) and (4.7), we get Since is proper slant and is nonnull, (4.8) yields , which shows that is constant and consequently the theorem is proved.

The second case is dealt with the following theorem.

Theorem 4.3. Let be a warped product hemi-slant submanifold of an LP-Sasakian manifold such that is a proper slant submanifold tangent to and is an anti-invariant submanifold of . Then, lies in the invariant normal subbundle , for each and .

Proof. Consider be a warped product hemi-slant submanifold of an LP-Sasakian manifold such that is a proper slant submanifold tangent to and is an anti-invariant submanifold of . Then, for any and , we have Using (2.7) and (2.8), we obtain By virtue of (2.10), (2.11) and Proposition 3.2(II), it follows from (4.10) that Equating the normal components, we obtain Taking the inner product of with , for any in (4.13), we get Also for any and , we have Taking the inner product for any in (4.14) and using (2.1) and (2.2), we derive By virtue of (4.13), the above equation yields Similarly, if any , then from (2.13), we obtain Since the product of tangential component with normal is zero and is a proper slant submanifold, we may conclude from (4.17) that From (4.16) and (4.18), it follows that and hence the proof is complete.

5. Examples on LP-Sasakian Manifolds

Example 5.1. We consider a 3-dimensional manifold , where are the standard coordinates in . Let be a linearly independent global frame on given by where is a nonzero constant such that . Let be the Lorentzian metric defined by , , . Let be the 1-form defined by for any . Let be the (1,1) tensor field defined by , , and . Then, using the linearity of and we have , , and for any . Thus for defines a Lorentzian paracontact structure on .
Let be the Levi-Civita connection with respect to the Lorentzian metric . Then, we have Using Koszul formula for the Lorentzian metric g, we can easily calculate From the above computations, it can be easily seen that for , is an LP-Sasakian structure on . Consequently, is an LP-Sasakian manifold.

Example 5.2 (see [16]). Let be the 5-dimensional real number space with a coordinate system . Define the structure becomes an LP-Sasakian structure in .

Example 5.3. Consider a 4-dimensional submanifold of with the cordinate system and the structure is defined as Hence, the structure is an LP-contact structure on . Now, for any and nonzero and , we define the submanifold as follows: Then, the tangent space is spanned by the vectors: Then the distributions is a slant distribution tangent to and is an anti-invariant distribution, respectively. Let us denote by and their integral submanifolds, then the metric on is given by Hence, the submanifold is a hemi-slant-warped product submanifold of with the warping function .