A Note on Quasi bi-slant submanifolds of cosymplectic manifolds

The main purpose of the present paper is to define and study the notion of quasi bi-slant submanifolds of almost contact metric manifolds. We mainly concerned with quasi bi-slant submanifolds of cosymplectic manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant and quasi hemi-slant submanifolds. First, we give non-trivial examples in order to demostrate the method presented in this paper is effective and investigate the geometry of distributions. Moreover, We study these types of submanifolds with parallel canonical structures.


Introduction
Study of submanifolds theory has shown an increasing development in image processing, computer design, economic modeling as well as in mathematical physics and in mechanics. In this manner, B-Y. Chen [6] initiated the notion of slant submanifold as a generalization of both holomorphic (invariant) and totally real submanifold (anti-invariant) of an almost Hermitian manifold. Inspried by B-Y. Chen's paper, many geometers have studied this notion in the different kind of structures: (see [22], [23]). Many consequent results on slant submanifolds are collected in his book [5]. After this notion, as a generalization of semi-slant submanifold which was defined by N. Papaghiuc [19] (see also [8]). A. Carriazo [3] and [4] introduced the notion of bi-slant submanifold under the name anti-slant submanifold. However, B. Şahin called these submanifolds hemi-slant submanifolds in [21]. (See also [9] and [10], [20], [24]).
Furthermore, the submanifolds of a cosymplectic manifold have been studied by many geometers: See [11], [12], [13], [14], [15], [16], [18]. Taking into account of the above studies, we are motivated to fill a gap in the literature by giving the notion of quasi bi-slant submanifolds in which the tangent bundle consist of one invariant and two slant distributions and the Reeb vector field. In this paper, as a generalization of slant, semi-slant, hemi-slant, bi-slant and quasi hemi-slant submanifolds, we introduce quasi bi-slant submanifolds and investigate the geometry of distributions in detail.
The paper is organized as follows: In section 2, we recall basic formulas and definitions for a cosymplectic manifold and their submanifolds. In section 3, we introduce the notion of quasi bi-slant submanifolds, giving an non-tirivial example and obtain some basic results for the next sections. In section 4, we give some necessary and sufficient conditions for the geometry of distributions. Finally, we study these types of submanifolds with parallel canonical structures.

Preliminaries
In this section, we give the definition of cosymplectic manifold and some background on submanifolds theory.
A (2m + 1)-dimensional C ∞ -manifold M said to have an almost contact structure if there exist on M a tensor field ϕ of type (1,1), a vector field ξ and 1-form η satisfying: There always exists a Riemannian metric g on an almost contact manifold M satisfying the following conditions where X, Y are vector fields on M. An almost contact structure (ϕ, ξ, η) is said to be normal if the almost complex structure J on the product manifold M × R is given by where f is a C ∞ -function on M × R has no torsion i.e., J is integrable. The condition for normality in terms of ϕ, ξ and η is [ϕ, ϕ] + 2dη ⊗ ξ = 0 on M, where [ϕ, ϕ] is the Nijenhuis tensor of ϕ. Finally, the fundamental two-form Φ is defined Φ(X, ϕY ) = g(X, ϕY ). An almost contact metric structure (ϕ, ξ, η, g) is said to be cosymplectic, if it is normal and both Φ and η are closed ( [1], [2], [16]), and the structure equation of a cosymplectic manifold is given by for any X, Y tangent to M, where ∇ denotes the Riemannian connection of the metric g on M. Moreover, for cosymplectic manifold Example. ( [17]) R 2n+1 with Cartesian coordinates (x i , y i , z)(i = 1, ..., n) and its usual contact form here ξ is the characteristic vector field and its Riemannian metric g and tensor field ϕ are given by This gives a cosymplectic manifold on R 2n+1 . The vector fields e i = ∂ ∂yi , e n+i = ∂ ∂xi , ξ form a ϕ-basis for the cosymplectic structure. On the other hand, it can be shown that R 2n+1 (ϕ, ξ, η, g) is a cosymplectic manifold.
Let M be a Riemannian manifold isometrically immersed inM and induced Riemannian metric on M is denoted by the same symbol g throughout this paper. Let A and h denote the shape operator and second fundamental form, respectively, of immersion of M intoM . The Gauss and Weingarten formulas of M intoM are given by [6] for any vector fields X, Y ∈ Γ(T M ) and V ∈ Γ(T ⊥ M ).
If h(X, Y ) = 0 for all X, Y ∈ Γ(T M ), then M is said to be totally geodesic.

Quasi bi-slant submanifolds of cosmyplectic manifolds
In this section, we define the concept of quasi bi-slant submanifolds of cosymplectic manifolds, giving a non-trivial exmaple and obtain some related results for later use. Taking the dimension of distributions D, D 1 and D 2 are m 1 , m 2 and m 3 , respectively. One can easily see the following cases: • If m 1 = 0 and m 2 = m 3 = 0, then M is a invariant submanifold.
Remark 3.2. In this paper, we assume that M is proper quasi bi-slant submanifold of a cosymplectic manifoldM . Now, we present an example of proper quasi bi-slant submanifold in R 11 .
Let M be a quasi bi-slant submanifold of a cosymplectic manifoldM . Then, for any X ∈ Γ(T M ), we have where P, Q and R denotes the projections on the distributions D, D 1 and D 2 , recpectively.
where T X and F X are tangential and normal components on M. Making now use of (3.1) and (3.2), we get immediately here since ϕD = D, we have F PX = 0. Thus we get and where BZ ∈ Γ(T M ) and CZ ∈ Γ(T ⊥ M ). Taking into account of the condition (iii) in Definition (3.1), (3.2) and (3.6), we obtain the followings: With the help of (3.2) and (3.6), we obtain the following Lemma.
We need the following lemma for later use.
Lemma 3.6. Let M be a quasi bi-slant submanifold of a cosymplectic manifold M , then for any Z 1 , Z 2 ∈ Γ(T M ), we have the following and Proof. SinceM is a cosmyplectic manifold, we have that which implies that∇ Z1 ϕZ 2 − ϕ∇ Z1 Z 2 = 0. By using (2.5) and (3.2), we get Taking into account of (2.5), (2.6), (3.2) and (3.6), we obtain Comparing the tangential and normal components, we have the required results.
In a similar way, we have: Lemma 3.7. Let M be a quasi bi-slant submanifold of a cosymplectic manifold M , then we have the following and for any Z 1 ∈ Γ(T M ) and W 1 ∈ Γ(T ⊥ M ).

Integrability and Totally geodesic foliations
In this section we give some necessary and sufficient condition for the integrability of the distributions. First, we have the following theorem: for any X, Y ∈ Γ(D) and Z ∈ Γ(D 1 ⊕ D 2 ).

Proof. The distribution D is integrable on
Now, using (2.4), (3.2) and F Y = 0 for any Y ∈ Γ(D), we have Taking into account of (2.5) and (3.3) in the above equation, we get

Now again taking into account the equation (3.2), we obtain
which completes the proof.
For the slant distrubition D 1 , we have: Taking into account the equation lemma (3.5) (i) in the above equation, we get

Now, using (2.6) and (3.3), we obtain
In a similar way, we obtain the following case for the slant distribution D 2 .
Lemma 5.1. Let M be a quasi bi-slant submanifold of a cosymplectic manifold M . Then for any Z 1 , Z 2 ∈ Γ(T M ) and W 1 ∈ Γ(T M ) ⊥ we obtain First, we have the following theorem: Proof. For any X, Y ∈ Γ(D), from (5.5), we have here we have used A F Y X = 0 since F Y = 0 for any Y ∈ Γ(D). Thus, our assertion comes from (5.9). Proof. Assume that F is parallel. Now, from (5.6), we have Now, taking inner product with V ∈ Γ(T M ) ⊥ in the above equation and using (2.5), we obtain which gives the assertion. Proof. By using (2.5), (5.6) and (5.7), we get for any Z 1 , Z 2 ∈ Γ(T M ) and W 1 ∈ Γ(T M ) ⊥ . This proves our assertion.
Finally, we mention another non-trivial example of quasi bi-slant submanifold of a cosymplectic manifold.
By direct calculations, we obtain the distribution D = span{e 1 , e 2 } is an invariant distribution, the distribution D 1 = span{e 3 , e 4 } is a slant distribution with slant angle θ 1 = π 4 and the distribution D 2 = span{e 5 , e 6 } is also a slant distribution with slant angle θ 2 = α, 0 < α < π 2 . Thus M is a 7−dimensional proper quasi bi-slant submanifold of R 11 with its usual almost contact metric structure.