Slant and hemislant submanifolds of a 3−dimensional indefinite trans-Sasakian manifold

Abstract. In this paper we would like to establish some of the properties of slant and hemislant submanifolds of an indefinite trans-Sasakian manifold. We have four sections in this paper. Section (1) is introductory. In Section (2)we recall some necessary details of an indefinite trans-Sasakian manifold. In Section (3) we have obtained some interesting properties on a totally umbilical slant submanifolds of an indefinite trans-Sasakian manifold. Finally, in Section (4), some results on integrability conditions of the distributions of hemislant submanifolds of an indefinite trans-Sasakian manifold have been obtained.


Introduction
The study of slant submanifolds in complex spaces was initiated by B.Y.Chen as a natural generalization of both holomorphic and totally real submanifolds in ( [1], [2]). After him, A.Lotta in 1996 extended the notion to the setting of almost contact metric manifolds [3]. Further modifications regarding semislant submanifolds were introduced by N.Papaghiuc [4]. These submanifolds are a generalized version of CR-submanifolds. J.L.Cabrerizo et.al. ([5], [6]) extended the study of semislant submanifolds of Kaehler manifold to the setting of Sasakian manifolds. In particular, totally umbilical proper slant submanifold of a Kaehler manifold has also been discussed in [7]. Recently, Khan et.al. [8] carried some investigation on these submanifolds in the setting of Lorentzian paracontact manifolds. The idea of hemislant submanifold was introduced by Carriazo as a particular class of bislant submanifolds, and he called them antislant submanifolds in [9]. Later on, in 2011 Siraj Uddin et.al. studied totally umbilical proper slant and hemislant submanifolds of an LP-cosymplectic manifold. In the present note, our aim is to extend the study of slant and hemislant submanifolds of an indefinite trans-Sasakian manifold.

Preliminaries
LetM be an (2n + 1)-dimensional indefinite almost contact metric manifold with indefinite almost contact metric structure (ϕ, ξ, η,g), where ϕ is a tensor of type (1, 1) having rank 2n, ξ is a vector field, η is a 1-form andg is Riemannian metric, satisfying following properties : for all vector fields X, Y onM . It is easy to see that g(ξ, ξ) = ϵ = ±1. An indefinite almost contact metric structure (ϕ, ξ, η,g) is called an indefinite trans-Sasakian structure if for functions α and β onM of type (α, β), where∇ is the Levi-Civita connection onM . On indefinite trans-Sasakian manifold we have, for any X ∈ TM where TM is the Lie algebra of vector fields onM .
for any X, Y ∈ T M , N ∈ T ⊥ M , h is the second fundamental form and A N is the Weingarten map associated with N via For any X ∈ Γ(T M ) we can write, where P X is the tangential component and F X is the normal component of ϕX. Similarly for any N ∈ Γ(T ⊥ M ) we can put On the other hand, M is said to be anti-invariant if P is identically zero, i.e., ϕX ∈ Γ(T ⊥ M ) for any X ∈ Γ(T M ). A submanifold M of an indefinite trans-Sasakian manifoldM is called totally umbilical if for any X, Y ∈ Γ(T M ). The mean curvature vector H is denoted by where the orthogonal complementary distribution D of < ξ > is known as the slant distribution on M . If µ is ϕ-invariant of the normal bundle T ⊥ M , then Defining the endomorphism P : T M −→ T M , whose square, P 2 will be denoted by Q. Then the tensor fields on M of type (1, 1) determined by these endomorphism will be denoted by the same letters, respectively P and Q.
We are already having the following result for a slant submanifold.
We can easily draw the following consequences for a 3−dimensional indefinite trans-Sasakian mani-foldM from [6] ( for any X, Y tangent to M . Proof : From (2.14) we have Using (2.6) and (3.4) we can write Using (2.9) we can have From (2.11) we get

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By using equation (2.4), (2.9), (2.14) and Gauss and Weingarten formulae we calculate Equating the normal components we get On the other hand from (3.5) we infer for any X ∈ Γ(T M ). Taking the covariant derivative of the above equation w.r.t P X, we obtain Using the property of metric connection and using (2.12) and (2.13) we have (3.15) g(∇ ⊥ P X F X, F X) = sin 2 θg(∇ P X X, X).
Now taking the inner product in (3.12) with F X, for any X ∈ Γ(T M ), then After using (2.6), (3.5) and (3.15) and having some brief calculation we derive Since∇ is the metric connection then the above equation can be written as AsM is a 3−dimensional indefinite trans-Sasakian manifold, then using the fact that we can easily conclude from , if H ∈ Γ(µ) then, tan θ = 0 =tan(nπ). Since θ ∈ [0, π/2] hence θ = 0.

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Therefore M is invariant i.e. M is not a proper slant manifold.
Using equation (2.6), (2.7), (2.9), (2.10) and (3.25) we calculate Taking inner product with F Y , for any Y ∈ Γ(T M ) we have Since C∇ ⊥ X H ∈ Γ(µ), then by (3.5) the above equation takes the form Using (2.7), (2.8) and (2.14) and having some brief calculations we obtain The above equation can be written as Again using the fact that H ∈ Γ(µ) then by (2.7) we have From (3.24) and (3.31) we get After having some calculations we infer Putting Y = ∇ X Y we have

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By taking the covariant derivative of (3.34) w.r.t X ∈ T M we get Again using (3.35) and (3.36) we obtain Using (2.5) in (3.37) we can easily observe that if θ = π/2 then (∇ X Q)Y = 0 i.e. Q is parallel and thus assertion is proved.

Hemislant submanifold
We assume that M is a hemislant submanifold of a 3−dimensional indefinite trans-Sasakian manifold such that the structure vector field ξ is tangent to M . At first, we define a hemislant submanifold and then we derive integrability condition of the involved distributions D 1 and D 2 .

Definition(4.1) :
A submanifold M of a 3−dimensional indefinite trans-Sasakian manifold is said to be a hemislant submanifold if there exist two orthogonal complementary distributions D 1 and D 2 satisfying the following properties : Further if µ is ϕ-invariant subspace of the normal bundle T ⊥ M , then for hemislant submanifold, the normal bundle T ⊥ M can be decomposed as, Now, our task is to obtain the integrability conditions of the involved distributions.

Proposition(4.1) :
Let M be a hemislant submanifold of a 3−dimensional indefinite trans-Sasakian manifoldM , then the anti-invariant distribution D 2 is integrable iff for any Z, W ∈ D 2 .
Proof : For any Z, W ∈ D 2 , we know After using (2.7) and (2.11) we get

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From the fact thatM is a 3−dimensional indefinite trans-Sasakian manifold we infer Therefore we have As D 2 is an anti-invariant distribution, the tangential part of above equation should be identically zero, hence we obtain the required result.
Proof : For any Z, W ∈ D 1 ⊕ < ξ >, we have SinceM is a 3−dimensional indefinite trans-Sasakian manifold and using (2.11) we get Again using (2.4) and (2.9) we obtain Taking inner product with F X for any X ∈ D 2 and using Gauss and Weingarten formulae we calculate Since ξ is tangential to D 1 , we obtain the required integrability condition.
Taking vectors Z, W ∈ D 2 , then from Gauss and Weingarten formulae we have Using (2.4), (2.9) and (2.10) we obtain Equating the tangential components we calculate  Taking the inner product with V ∈ D 2 , we get   As M is totally umbilical, we infer x(u, v, t) = 2(ucosθ, usinθ, v, 0, t).