On pseudo-slant submanifolds of trans-Sasakian manifolds

The object of the present paper is to study pseudo-slant submanifolds of trans-Sasakian manifolds. Integrability conditions of the distributions on these submanifolds are worked out. Some interesting results regarding such manifolds have also been deduced. An example of a pseudo-slant submanifold of a trans-Sasakian manifold is given.


INTRODUCTION
In 1990, Chen [6] introduced the concept of slant immersions as a generalization of both holomorphic and totally real immersions.Many authors have studied slant immersions in Hermitian manifolds.Lotta [9], introduced the notion of slant immersions in contact manifolds.In papers [3,4], slant submanifolds of K-contact and Sasakian manifolds have been characterized by Cabrerizo et al.Recently, Carriazo [5] defined and studied bi-slant immersions in almost Hermitian manifolds and simultaneously gave the notion of pseudo-slant submanifolds in almost Hermitian manifolds.The contact version of pseudo-slant submanifolds has been defined and studied by V. A. Khan and M. A. Khan [8].Slant submanifolds of trans-Sasakian manifolds have been studied by Gupta et al. [7].In an analogous way we would like to extend the notion of pseudo-slant submanifolds in trans-Sasakian manifolds.The present paper is organized as follows.
Preliminaries are given in Section 2. In Section 3, we define pseudo-slant submanifolds of trans-Sasakian manifolds.Section 4 deals with the integrability conditions of the distributions of such submanifolds and some other geometric results.Section 5 contains an example of a pseudo-slant submanifold of a trans-Sasakian manifold.

PRELIMINARIES
Let M be a (2n + 1)-dimensional C ∞ -differentiable manifold endowed with the almost contact metric structure (φ , ξ , η, g), where φ is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form and g a Riemannian metric on M, all these tensor fields satisfying [1] φ 2 (X) = −X + η(X)ξ , η(ξ ) = 1, g(X, ξ ) = η(X), (2.1) φ ξ = 0, ηφ = 0, g(X, φY for any X,Y ∈ T M.Here T M is the standard notation for the tangent bundle of M. The two-form Φ denotes the fundamental two-form and is given by g(X, φY ) = Φ(X,Y ).The manifold is said to be contact if Φ = dη.If ξ is a Killing vector field with respect to g, the contact metric structure is called a K-contact structure.It is known that a contact metric manifold is K-contact if and only if ∇X ξ = −φ X, where ∇ denotes the Levi-Civita connection on M. The almost contact structure M is said to be normal if [φ , φ ] + 2dη ⊗ ξ = 0, where [φ , φ ] is the Nijenhuis tensor of φ .A Sasakian manifold is a normal contact metric manifold.Every Sasakian manifold is K-contact.A three-dimensional K-contact manifold is Sasakian.An almost contact metric manifold is Sasakian if and only if (2.4) Moreover, on a Sasakian manifold ∇X ξ = −φ X, ( for any X ∈ T M and ξ is the structure vector field. An almost contact metric structure (φ , ξ , η, g) on M is called a trans-Sasakian structure of type (α, β ) if it satisfies for certain functions α and β on M, where ∇ means the covariant differentiation with respect to g.In particular, it is normal and generalizes cosymplectic, α-Sasakian, and β -Kenmotsu manifolds.If β = 0, then the structure is called α-Sasakian.If α = 0, then the structure is called β -Kenmotsu.If both α and β are zero, then the manifold reduces to a cosymplectic manifold [2].If α and β are not simultaneously zero, then we shall call a trans-Sasakian manifold a proper trans-Sasakian manifold.Again, it is known that a trans-Sasakian manifold of dimension ≥ 5 is either α-Sasakian or β -Kenmotsu or cosymplectic [10].We know that a trans-Sasakian structure satisfies for any X ∈ T M and ξ is the structure vector field.
Let M be a submanifold immersed in a (2n + 1)-dimensional contact metric manifold M; we denote by the same symbol g the induced metric on M. T M is the tangent bundle of the manifold M and T ⊥ M is the set of vector fields normal to M. Then the Gauss and Weingarten formula is given by for any X,Y ∈ T M and N ∈ T M, where ∇ ⊥ is the connection in the normal bundle.The second fundamental form h and A N are related by (2.12) The submanifold M is invariant if N is identically zero.On the other hand, M is anti-invariant if T is identically zero.From (2.3) and (2.11), we have for any X,Y ∈ T M.
From now on, we put Q = T 2 .We define for any X,Y ∈ T M. In view of (2.8), (2.11), and (2.7) it follows that (2.18) 3. PSEUDO-SLANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS Definition 3.1.We say that M is a pseudo-slant submanifold of a trans-Sasakian manifold M if there exist two orthogonal distributions D 1 and D 2 on M such that [8] (i) T M admits the orthogonal direct decomposition From the above definition it is clear that if θ = 0, then the pseudo-slant submanifold is a semi-invariant submanifold.On the other hand, if we denote the dimension of D i by d i , for i = 1, 2, then we find the following cases: for all X,Y ∈ D 1 .
Proof.In view of (2.10), By virtue of (2.8), (4.2) reduces to Combining (4.3) and (4.4), we obtain Hence, from (4.5) we obtain, with the help of (2.6), The above equation yields Therefore, the above equation reduces to Remark 4.1.As particular cases the above result holds for α-Sasakian, β -Kenmotsu, and cosymplectic manifolds.For the Sasakian case the above result has been proved in [8].
Theorem 4.2.Let M be a pseudo-slant submanifold of a trans-Sasakian manifold M. Then the distribution Since X ∈ D 1 and Z ∈ D 2 , where D 1 and D 2 are two orthogonal distributions and D 1 is anti-invariant, in view of (2.7), (4.8) we obtain from (4.9) In view of (2.6), In virtue of (2.11) and (4.11), equation (4.10) yields As particular cases the above result holds for α-Sasakian, β -Kenmotsu, and cosymplectic manifolds.For the Sasakian case the above result has been proved in [8].
Theorem 4.3.Let M be a pseudo-slant submanifold of a trans-Sasakian manifold M. Then for any In view of (4.8) we have from above The above equation gives the following: Corollary 4.1.In a proper trans-Sasakian manifold and α-Sasakian manifold the distribution D 1 ⊕ D 2 is not integrable.
In other words, we have the following Proof.For any X ∈ T M, let where P i , i = 1, 2 are projection maps on the distribution D i .From (4.15) it follows that In view of (2.6) and (2.9) and keeping in mind that g(U,V ) = 0 for U ∈ D 1 and V ∈ D 2 , we obtain from (4.16) For X,Y ∈ D 1 , we get η(X) = η(Y ) = 0. Hence Theorem 4.1 and the above equation yield g The immediate consequence of the above theorem is the following: Corollary 4.4.On a pseudo-slant submanifold M of a trans-Sasakian manifold M, the distribution D 1 ⊕ < ξ > is integrable.
For a Sasakian manifold the above result was proved by V. A. Khan and M. A. Khan [8].
Theorem 4.5.Let M be a pseudo-slant submanifold of a trans-Sasakian manifold M. Then the slant distribution D 2 is not integrable.
The above theorem produces Corollary 4.5.In an α-Sasakian manifold the slant distribution D 2 is not integrable.
Theorem 4.6.Let M be a submanifold of an almost contact metric manifold M, such that ξ ∈ T M. Then M is a pseudo-slant submanifold if and only if there exists a constant λ ∈ (0, 1] such that Furthermore, in this case λ = cos 2 θ , where θ denotes the slant angle of D.
Theorem 4.7.Let M be a pseudo-slant submanifold of a trans-Sasakian manifold M. Then ∇Q = 0 if and only if M is an anti-invariant submanifold.
Proof.If we consider the distribution D 2 ⊕ < ξ >, then from Theorem 4.6 we can write Denote by θ the slant angle of M.Then, replacing X by ∇ X Y, we get from (4.19) Equation (4.19) also gives for any X ∈ T M. Putting the value of ∇ X ξ in (4.21), we obtain Combining (4.20) and (4.23), we find for any X,Y ∈ D 2 ⊕ < ξ >.Here, we note that Hence ∇Q = 0 if and only if θ = π 2 holds in D 2 ⊕ < ξ >.Again, D 1 is anti-invariant by definition.Thus, the theorem follows.
As a consequence of Theorem 4.7 we immediately obtain Corollary 4.6.In a pseudo-slant submanifold of an α-Sasakian manifold ∇Q = 0 if and only if the submanifold is anti-invariant.Corollary 4.7.In a pseudo-slant submanifold of a β -Kenmotsu manifold ∇Q = 0 if and only if the submanifold is anti-invariant.
But for a cosymplectic manifold we have Corollary 4.8.In a submanifold of a cosymplectic manifold ∇Q is always zero, whether the submanifold is anti-invariant or not.Proof.Follows from [9].
Proof.Statements (a) and (b) follow from (4.24) and Theorem 4.8.Conversely, suppose that (a) and (b) hold.Let λ 1 (x) be the eigenvalue of Q |D 2 at each point x of M and Y ∈ D 2 be a unit eigenvector associated with λ 1 , that is, QY = λ 1 Y.Then from (b) we get for any X ∈ T M. Since both ∇ X Y and Q(∇ X Y ) are perpendicular to Y, we conclude that λ 1 is a constant on M.
Now we want to prove that M is pseudo-slant.To fulfill our purpose, in view of Theorem 4.6, it is sufficient to show that there exists a constant µ such that where λ 1 is the eigenvalue of . This shows that φY ∈ T ⊥ M. Therefore TY = 0. Thus, by the use of Theorem 4.6, the above theorem is proved.Since Q |D 2 = λ 1 I, we have QX = λ 1 X and so λ 1 X = λ 1 (X − η(X)ξ ).By taking µ = −λ 1 , we get M is pseudo-slant.
For an α-Sasakian manifold the above theorem yields the following: for any X,Y ∈ D 2 ⊕ < ξ >.Moreover, if θ is the slant angle of M, then λ = cos 2 θ .
In the β -Kenmotsu case Theorem 4.9 takes the following form: for any X,Y ∈ D 2 ⊕ < ξ >.Moreover, if θ is the slant angle of M, then λ = cos 2 θ .
For the cosymplectic case we obtain the following: for any X,Y ∈ D 2 ⊕ < ξ >.

.31)
Comparing tangential and normal parts, we have

.33)
As a consequence of the above theorem we obtain the following: Corollary 4.12.In a pseudo-slant submanifold of an α-Sasakian manifold (4.34) Corollary 4.13.In a pseudo-slant submanifold of a β -Kenmotsu manifold
In this case, the distribution D 1 is anti-invariant while the distribution D 2 is slant.Hence the submanifold under consideration is pseudo-slant.

CONCLUSION
Trans-Sasakian manifolds generalize both α-Sasakian and β -Kenmotsu manifolds.Pseudo-slant submanifolds mainly extend the notion of semi-invariant submanifolds.The integrability of the distributions of the tangent bundle of a submanifold determines the nature of the submanifold.In the present paper we consider the direct orthogonal decomposition of the tangent bundle T M of the pseudo-slant submanifold M of a trans-Sasakian manifold M as T M = D 1 ⊕ D 2 ⊕ < ξ >, where D 1 is the anti-invariant distribution and D 2 is the slant distribution.We mainly show that the distributions D 1 ⊕ < ξ > and D 1 are integrable but the distribution D 2 is not integrable.A necessary and sufficient condition for a pseudo-slant submanifold to be anti-invariant is obtained.An example of a pseudo-slant submanifold of a trans-Sasakian manifold is constructed.
Submanifold theory has an important role in many branches of applied mathematics.The results obtained in this paper can be used in many problems of dynamical system and critical point theory.

Corollary 4 . 2 .
In a β -Kenmotsu manifold the distribution D 1 ⊕ D 2 is integrable.Again, in a similar manner we have Corollary 4.3.In a cosymplectic manifold the distribution D 1 ⊕ D 2 is integrable.Theorem 4.4.Let M be a pseudo-slant submanifold of a trans-Sasakian manifold M. Then the anti-invariant distribution D 1 is integrable.

Theorem 4 . 8 .
Let M be a submanifold of an almost contact metric manifold M with a slant angle θ .Then, at each point x ∈ M, Q |D has only one eigenvalue λ 1 = cos 2 θ , for the slant distribution D of M.

Theorem 4 . 9 .
Let M be a submanifold of a trans-Sasakian manifold M with T M = D 1 ⊕ D 2 ⊕ < ξ >.Then M is pseudo-slant if and only if (a) the endomorphism Q |D 2 has only one eigenvalue at each point of M, (b) there exists a function λ :

Corollary 4 . 9 .
Let M be a submanifold of an α-Sasakian manifold M with T M = D 1 ⊕ D 2 ⊕ < ξ >.Then M is pseudo-slant if and only if (a) the endomorphism Q |D 2 has only one eigenvalue at each point of M, (b) there exists a function λ : M → [0, 1] such that

Corollary 4 . 10 .
Let M be a submanifold of a β -Kenmotsu manifold M with T M = D 1 ⊕ D 2 ⊕ < ξ >.Then M is pseudo-slant if and only if (a) the endomorphism Q |D 2 has only one eigenvalue at each point of M, (b) there exists a function λ : M → [0, 1] such that