Warped Product Submanifolds of LP-Sasakian Manifolds

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.


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 Ros ¸ca 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 C 2 and C 4 .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.

Preliminaries
An n-dimensional smooth manifold M is said to be an LP-Sasakian manifold 3 if it admits a 1, 1 tensor field φ, a unit timelike contravariant vector field ξ, an 1-form η, and a Lorentzian metric g, which satisfy where ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric g.It can be easily seen that, in an LP-Sasakian manifold, the following relations hold: Again, we put Ω X, Y g X, φY 2.5 for any vector fields X, Y tangent to M. The tensor field Ω X, Y is a symmetric 0,2 tensor field 2 .Also, since the vector field η is closed in an LP-Sasakian manifold, we have 2 for any vector fields X and Y tangent to M.
Let N be a submanifold of an LP-Sasakian manifold M with induced metric g and let ∇ and ∇ ⊥ be the induced connections on the tangent bundle TN and the normal bundle T ⊥ N of N, respectively.Then, the Gauss and Weingarten formulae are given by for all X, Y ∈ TN and V ∈ T ⊥ N, where h and A V are second fundamental form and the shape operator corresponding to the normal vector field V , respectively, for the immersion of N into M.The second fundamental form h and the shape operator A V are related by 12 for any X, Y ∈ TN and V ∈ T ⊥ N For any X ∈ TN, we may write where EX is the tangential component and FX is the normal component of φX.Also, for any V ∈ T ⊥ N, we have where BV and CV are the tangential and normal components of φV , respectively.The covariant derivatives of the tensor fields E and F are defined as for any X, Y ∈ TN.
Throughout the paper, we consider ξ to be tangent to N. The submanifold N is said to be invariant if F is identically zero, that is, φX ∈ TN for any X ∈ TN.On the other hand, N is said to anti-invariant if E is identically zero, that is, φX ∈ T ⊥ N for any X ∈ TN.
Furthermore, for a submanifold tangent to the structure vector field ξ, there is another class of submanifolds which is called a slant submanifold.For each nonzero vector X tangent to N at x ∈ N, the angle θ X , 0 ≤ θ X π/2 between φX and EX 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 θ 0 and θ π/2, respectively.A slant submanifold is said to be proper slant if the slant angle θ lies strictly between 0 and π/2, that is, 0 < θ < π/2 6 .Theorem 2.1 see 13 .Let N be a submanifold of a Lorentzian almost paracontact manifold M such that ξ is tangent to N.Then, N is slant submanifold if and only if there exists a constant λ ∈ 0, 1 such that Furthermore, if θ is the slant angle of N, then λ cos 2 θ.Also from 2.14 , we have for any X, Y tangent to N.
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 N is called semi-slant submanifold of M if there exist an orthogonal direct decomposition of TN as where D 1 is an invariant distribution, that is, φ D 1 D 1 and D 2 is slant with slant angle θ / 0. The orthogonal complement of FD 2 in the normal bundle T ⊥ N is an invariant subbundle of T ⊥ N and is denoted by μ.Thus, we have for a semi-slant submanifold

2.18
For an LP-contact manifold this study is extended by Y üksel et al. 13 .

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 N 1 , g 1 and N 2 , g 2 be two semi-Riemannian manifolds and f be a positive differentiable function on N 1 .Then, the warped product of N 1 and N 2 is a manifold, denoted by A warped product manifold N 1 × f N 2 is said to be trivial if the warping function f is constant.More explicitely, if the vector fields X and Y are tangent to where π i i 1, 2 are the canonical projections of N 1 × N 2 onto N 1 and N 2 , respectively, and * stands for the derivative map.
Let N N 1 × f N 2 be a warped product manifold, which means that N 1 and N 2 are totally geodesic and totally umbilical submanifolds of N, respectively.
For the warped product manifolds, we have the following result for later use 1 .
Proposition 3.2.Let N N 1 × f N 2 be a warped product manifold.Then, for any X, Y ∈ TN 1 and U, V ∈ TN 2 , where ∇ and ∇ denote the Levi-Civita connections on N and N 2 , respectively.
Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Ünal 14 .A doubly warped product manifold of N 1 and N 2 , denoted as f 2 N 1 × f 1 N 2 is endowed with a metric g defined as where f 1 and f 2 are positive differentiable functions on N 1 and N 2 , respectively.In this case formula II of Proposition 3.2 is generalized as for each X in TN 1 and Z in TN 2 15 .One has the following theorem for doubly warped product submanifolds of an LP-Sasakian manifold 11 .
a doubly warped product submanifold of an LP-Sasakian manifold M where N 1 and N 2 are submanifolds of M.Then, f 2 is constant and N 2 is anti-invariant if the structure vector field ξ is tangent to N 1 , and f 1 is constant and 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
From the above theorem and Corollary 3.5, we have only the remaining case is to study the warped product submanifold N 1 × f N 2 with structure vector field ξ is tangent to N 1 .
Taking the inner product with EX in 4.4 and using the fact that X and EX are mutually orthogonal vector fields, then we have g A FU X, EX U ln f g EX, EX g Bh X, U , EX 0. 4.5 Using 2.9 and 2.15 , we get Replacing X by EX in 4.6 and using 2.14 , we obtain Since N θ is proper slant and X is nonnull, 4.8 yields U ln f 0, which shows that f is constant and consequently the theorem is proved.
The second case is dealt with the following theorem.
Theorem 4.3.Let N N θ × f N ⊥ be a warped product hemi-slant submanifold of an LP-Sasakian manifold M such that N θ is a proper slant submanifold tangent to ξ and N ⊥ is an anti-invariant submanifold of M.Then, ∇ X F U lies in the invariant normal subbundle μ, for each X ∈ TN θ and U ∈ TN ⊥ .
Proof.Consider N N θ × f N ⊥ be a warped product hemi-slant submanifold of an LP-Sasakian manifold M such that N θ is a proper slant submanifold tangent to ξ and N ⊥ is an anti-invariant submanifold of M.Then, for any X ∈ TN θ and U ∈ TN ⊥ , we have ∇ X φU φ∇ X U. 4.9 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

4.11
Equating the normal components, we obtain Taking the inner product of with FW 1 , for any W 1 ∈ TN ⊥ in 4.13 , we get

4.13
Also for any X ∈ TN θ and U ∈ TN ⊥ , we have Taking the inner product FW 1 for any W 1 ∈ TN ⊥ in 4.14 and using 2.1 and 2.2 , we derive By virtue of 4.13 , the above equation yields Similarly, if any W 2 ∈ TN θ , then from 2.13 , we obtain g ∇ X F U, φW 2 g ∇ ⊥ X FU, φW 2 − g F∇ X U, φW 2 .

4.17
Since the product of tangential component with normal is zero and N θ is a proper slant submanifold, we may conclude from 4.17 that g ∇ X F U, φW 2 0 for any X, W 2 ∈ TN θ , U ∈ TN ⊥ .

4.18
From 4.16 and 4.18 , it follows that ∇ X F U ∈ μ and hence the proof is complete.

Examples on LP-Sasakian Manifolds
Example 5.1.We consider a 3-dimensional manifold M { x, y, z ∈ R 3 : z > 0}, where x, y, z are the standard coordinates in R 3 .Let {E 1 , E 2 , E 3 } be a linearly independent global frame on M given by where a is a nonzero constant such that a / 1.Let g be the Lorentzian metric defined by Let η be the 1-form defined by η U g U, E 3 for any U ∈ TM.Let θ be the 1,1 tensor field defined by ηE 1 −E 1 , φE 2 −E 2 , and φE 3 0.Then, using the linearity of φ and g we have η E 3 −1, φ 2 U U η U E 3 , and g φU, φW g U, W η U η W for any U, W ∈ TM.Thus for E 3 ξ, φ, ξ, η, g defines a Lorentzian paracontact structure on M.
Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g.Then, we have Using Koszul formula for the Lorentzian metric g, we can easily calculate

5.6
Then, the tangent space TN is spanned by the vectors:

5.7
Then the distributions D θ span{e 1 , e 2 , e 4 } is a slant distribution tangent to ξ e 4 and D ⊥ span{e 3 } is an anti-invariant distribution, respectively.Let us denote by N θ and N ⊥ their integral submanifolds, then the metric g on N is given by g 2 du 2 dv 2 u 2 v 2 dα 2 .

5.8
Hence, the submanifold N N θ × f N ⊥ is a hemi-slant-warped product submanifold of R 7 with the warping function f u 2 v 2 .
Example 5.3.Consider a 4-dimensional submanifold N of R 7 with the cordinate system x 1 , x 2 , . . ., x 6 , t and the structure is defined as Example 5.2 see 16 .Let R 5 be the 5-dimensional real number space with a coordinate system x, y, z, t, s .Define j η ⊗ η.