WARPED PRODUCT PSEUDO-SLANT SUBMANIFOLDS OF QUASI-SASAKIAN MANIFOLDS

The object of the present paper is to study warped product and doubly warped product pseudo-slant submanifolds of quasi-Sasakian manifolds. We derive an example of proper pseudo-slant submanifold of almost contact metric manifold. Also, we study the non-existence of warped product and doubly warped product pseudoslant submanifolds of quasi-Sasakian manifolds.


Introduction
The idea of warped product manifolds was given by Bishop and Neill [1] and these manifolds were studied by many authors.The warped product manifolds are generalization of Riemannian manifolds.The warped product manifolds play important role in differential geometry as well as in theory of relativity.
Carriazo [2] introduced the concept of pseudo-slant submanifolds in almost Hermitian manifolds and the contact version of pseudo-slant submanifolds has been defined and studied by Khan and Khan in [3].Atceken and Hui [4] studied slant and pseudo slant submanifold of (LCS) n −manifolds where as pseudo slant submanifold of trans-Sasakian manifolds were studied by Khan and Chahal [5] .
In 1967, Blair [6] introduced the notion of quasi-Sasakian manifold to unify Sasakian and cosymplectic manifolds.Again in 1977, Kanemaki [7] defined quasi-Sasakian manifolds.Motivated by the studies, the object of the present paper is to study warped product pseudo-slant submanifolds of quasi-Sasakian manifolds.This paper is organized as follows.Section 2 is concerned with preliminaries and we derive an example of proper pseudo-slant submanifold of contact metric manifold.In section 3, we study warped product pseudo-slant submanifolds of quasi-Sasakian manifolds.Section 4 we study doubly warped product pseudo-slant submanifolds of quasi-Sasakian manifolds.

Preliminaries
A (2n + 1) −dimensional Riemannian manifold M, g is called an almost contact metric manifold if the following results hold [6] : (1) φ is called the fundamental two form of the manifold.If the characteristic vector field ξ is a killing vector, then the contact manifold is called a K-contact manifold.A contact metric This was first introduced by Blair [6].There are many types of quasi-Sasakian structure ranging from the cosymplectic case, dη = 0 (rank η = 1), to the Sasakian case, η ∧ (dη) n = 0 (rank . The 1−form η has rank r = 2p if (dη) p = 0 and η ∧ (dη) p = 0 and has rank r = 2p + 1 if (dη) p = 0 and η ∧ (dη) p = 0. We also say that r is the rank of the quasi-Sasakian structure.Blair [6] also proved that there are no quasi-Sasakian manifold of even rank.In order to study the properties of quasi-Sasakian manifolds Blair [6] proved some theorems regarding Kaehlerian manifolds and the existence of quasi-Sasakian manifolds.The fundamental vector field ξ of a quasi-Sasakian structure is a killing vector field that is £ ξ g = 0.
Let M be a submanifold immersed in M with induced metric g.Also let ∇ and ∇ ⊥ are induced Levi-Civita connections on the tangent bundle T M and T ⊥ M of M respectively.Then the Gauss and Weingarten formulae are given by ( 10) and ( 11) for all X,Y ∈ T M and V ∈ T ⊥ M, h and A V are second fundamental form and Weingarten map associated with V as For any X ∈ T x M, we write where T X ∈ T x M and NX ∈ T ⊥ x M. Similarly, for V ∈ T ⊥ x M, we have where tV is the tangential component and nV is the normal component of φV.From (4) and (13), we have where x ∈ M, and X,Y ∈ T x M.
Definition 2.1.A submanifold M of an almost contact metric manifold M is said to be a slant submanifold if for each x ∈ M and X ∈ T x M , linearly independent to ξ , the angle between 2 then submanifold is anti-invariant submanifold if θ lies strictly between 0 and π 2 , i.e 0 < θ < π 2 then it is called proper slant submanifold.
Definition 2.2.A submanifold M is called pseudo-slant submanifold of an almost contact metric manifold M, if there exist two orthogonal distribution D ⊥ and D θ on M such that From the above definition, it is obvious that if θ = 0 and θ = π 2 , then the pseudo-slant submanifold becomes semi-invariant submanifold and anti-invariant submanifold respectively.
On the other hand if we denote the dimensions of D θ and D ⊥ by d 1 and d 2 respectively then we have the following cases.
A pseudo slant submanifold is called proper if d 1 , d 2 = 0, θ = 0 and θ = π 2 .Now, we derive an example of proper pseudo-slant submanifold of almost contact metric manifold.
Example.Let (R 9 , φ , ξ , η, g) be an almost contact metric manifold with cartesian coordinates (x 1 , y 1 , x 2 , y 2 , x 3 , y 3 , x 4, y 4, t) and the almost contact structure where ξ = ∂ ∂t , η = dt and g is the standard Euclidean metric on R 9 .Consider a submanifold M of R 9 defined by χ(x, y, u, v,t) = (x + y, x − y, x, y, x cos u, x sin u, x cos v, x sin v,t), such that u, v (u = v) are non vanishing real valued functions on M. Then the tangent space T M is spanned by the following vector fields

Now we have
Then D ⊥ = Span{e 3 , e 4 } is an anti-nvariant distribution and D θ = Span{e 1 , e 2 } is a slant distribution with slant angle θ = cos −1 2 √ 15 .Thus M is 5−dimensional proper pseudo-slant submanifold of R 9 .[1] defined the warped product submanifold as follows: Definition 2.3.Let (N 1 , g 1 ) and (N 2 , g 2 ) be two Riemannian manifolds with Riemannian meteic g 1 and g 2 respectively and f be a positive definite smooth function on N 1 .The warped product of N 1 and N 2 is the Riemannian manifold

Bishop and Neill
If the vector field U is tangent to where π i (i = 1, 2) are the cononical projections of N 1 × N 2 onto N 1 and N 2 respectively and * stands for the derivative map.
For warped product manifold we have [8] Lemma 2.1.
where ∇ and ∇ denote the Levi-Civita connection on N 1 and N 2 respectively.
Doubly warped product manifolds were introduced as a generalization of warped product manifold by Unal [9] .
Definition 2.4.A doubly warped product (M, g) is a product manifold of the form 3. warped product pseudo-slant submanifolds of quasi-Sasakain manifolds From equations (9),( 10) and (21) , we get ξ In f = 0, for all Z ∈ N ⊥ .Which shows f is constant on M threrfore in this case warped product does not exist.
Corollary 3.2.A warped product pseudo-slant submanifold of (1) Sasakian (2) cosymplectic manifold M of type M = N θ × f N ⊥ , where N θ is a proper slant submanifold tangent to ξ and N ⊥ is an anti-invariant submanifold of M, does not exist.

Doubly warped product pseudo-slant submanifolds of quasi-Sasakain manifolds
In this section, we will study doubly warped product pseudo-slant submanifolds of quasi-Sasakain manifolds.Let M = f N 1 × b N 2 be a doubly warped product pseudo-slant submanifold of quasi-Sasakain manifolds.If N θ and N ⊥ are proper slant submanifolds and anti-invariant submanifolds of quasi-Sasakain manifolds M then their doubly warped product pseudo-slant submanifolds may be given by one of the following: A doubly warped product pseudo-slant submanifold of a quasi-Sasakain manifold M of type M = f N ⊥ × b N θ , where N ⊥ is an anti-invariant submanifold and N θ is a proper slant submanifold of M such that ξ is tangent to N θ , does not exist.
Proof.Let M = f N ⊥ + b N θ be doubly warped product pseudo-slant submanifold of quasi-Sasakain manifold M, ξ is tangent to N θ then, for any Z ∈ N ⊥ , we get From equations (9), ( 10) and (22) , we get Application.The warped product manifolds are used in the theory of relativity as well as in physics.

∞) are smooth maps and g 1 ,
g 2 are the metric on Riemannian manifolds B and F respectively.If either b = 1 or f = 1, then we obtain a single warped product.If both b = 1 and f = 1, then we have a warped product.If neither b nor f is constant, then we have a non trivial doubly warped product.If both f and b are constant.So, there does not exist doubly warped product pseudo-slant submanifold of M of type M = f N ⊥ + b N θ , with ξ tangent to N θ .Corollary 4.1.A doubly warped product pseudo-slant submanifold of (1) Sasakian (2) cosymplectic manifold M of type M = f N ⊥ × b N θ , where N ⊥ is an anti-invariant submanifold and N θ is a proper slant submanifold of M such that ξ is tangent to N θ , does not exist.Theorem 4.2.A doubly warped product pseudo-slant submanifold of a quasi-Sasakain manifold M of type M = f N θ + b N ⊥ , where N θ is a proper slant submanifold of M tangent to ξ and N ⊥ is an anti-invariant submanifold, does not exist.Proof.Let M = f N θ + b N ⊥ be pseudo-slant doubly warped product submanifold of quasi-Sasakain manifold, ξ is tangential to N θ .For any X ∈ N θ and Z ∈ N ⊥ , from equations (9),(10) and (22) , we get (25) Z (In f ) = 0, (26) ξ (Inb) = 0. Therefore both f and b are constant.Hence there does not exist doubly warped product pseudoslant submanifold of M of type M = f N θ + b N ⊥ in M, with ξ tangent to N θ .Corollary 4.2.A doubly warped product pseudo-slant submanifold of (1) Sasakian (2) cosymplectic manifold M of type M = f N θ + b N ⊥ in M, where N θ is a proper slant submanifold of M tangent to ξ and N ⊥ is an anti-invariant submanifold, does not exist.