A Classification of a Totally Umbilical Slant Submanifold of Cosymplectic Manifolds

and Applied Analysis 3 For any X ∈ Γ TM , we write


Introduction
The study of slant submanifolds in complex spaces was initiated by Chen as a natural generalization of both holomorphic and totally real submanifolds 1, 2 . Since then, many research papers have appeared concerning the existence of these submanifolds as well as on the geometry of the existent slant submanifolds in different known spaces cf. 3, 4 . The slant submanifolds of an almost contact metric manifold were defined and studied by Lotta 4 . Later on, these submanifolds were studied by Cabrerizo et al. in the setting of Sasakian manifolds 3 .
Recently, Şahin proved that a totally umbilical proper slant submanifold of a Kaehler manifold is totally geodesic 5 . Our aim in the present paper is to investigate slant submanifolds in contact manifolds. Thus, we study slant submanifolds of a cosymplectic manifold. We have shown that a totally umbilical slant submanifold M of a cosymplectic manifold M is either an anti-invariant submanifold or the dim M 1 or the mean curvature vector H ∈ Γ μ , and then we have obtained an interesting result for a totally umbilical proper slant submanifold of a cosymplectic manifold.

Preliminaries
Let M be a 2n 1 -dimensional manifold with 1, 1 tensor field φ satisfying 6 : where I is the identity transformation, ξ a vector field, and η a 1-form on M satisfying φξ η • φ 0 and η ξ 1. Then M is said to have an almost contact structure. There always exists a Riemannian metric g on M such that for all vector fields X, Y , on M. From 2.2 , it is easy to observe that g φX, Y g X, φY 0.

2.3
The fundamental 2-form Φ is defined as: Φ X, Y g X, φY . If φ, φ dη ⊗ ξ 0, then the almost contact structure is said to be normal, where φ, φ X, Y φ 2 X, Y φX, φY − φ φX, Y − φ X, φY . If Φ dη, the almost contact structure is a contact structure. A normal almost contact structure such that Φ is closed and dη 0 is called cosymplectic structure. It is well known 7 that the cosymplectic structure is characterized by for all vector fields X, Y , on M, where ∇ is the Levi-Civita connection of g. From the formula ∇ X φ 0, it follows that ∇ X ξ 0. Let M be submanifold of an almost contact metric manifold M with induced metric g and let ∇ and ∇ ⊥ be the induced connections on the tangent bundle TM and the normal bundle T ⊥ M of M, respectively. Denote by F M the algebra of smooth functions on M and by Γ TM the F M -module of smooth sections of a vector bundle TM over M, then Gauss and Weingarten formulae are given by for each X, Y ∈ Γ TM and N ∈ Γ T ⊥ M , where h and A N are the second fundamental form and the shape operator corresponding to the normal vector field N , respectively for the immersion of M into M. They are related as where g denotes the Riemannian metric on M as well as the one induced on M 8 .
We can see that μ is an invariant subbundle with respect to φ. Furthermore, the covariant derivatives of the tensor fields P and F are defined as for any X, Y ∈ Γ TM . A submanifold M is said to be invariant if F is identically zero, that is, φX ∈ Γ TM for any X ∈ Γ TM . On the other hand, M is said to be anti-invariant if P is identically zero, that is, φX ∈ Γ T ⊥ M , for any X ∈ Γ TM . A

Slant Submanifolds
Throughout the section, we assume that M is a slant submanifold of a cosymplectic manifold M. We always consider such submanifold tangent to the structure vector field ξ. For each nonzero vector X tangent to M at x, we denote by 0 ≤ θ X ≤ π/2, the angle between φX and T x M, known as the Wirtinger angle of X. If the Wirtinger angle θ X is constant, that is, independent of the choice of x ∈ M and X ∈ T x M − {ξ}, then M is said to be a slant submanifold 4 . In this case the constant angle θ is called slant angle of the slant submanifold. Obviously if θ 0, M is invariant and if θ π/2, M is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant submanifold. If M is a slant submanifold of an almost contact metric manifold, then the tangent bundle TM is decomposed as where ξ denotes the distribution spanned by the structure vector field ξ and D is the complementary distribution of ξ in TM, known as the slant distribution.
We recall the following result for a slant submanifold.
The following relations are straightforward consequence of 3.2 : for any X, Y tangent to M. Now, we prove the following. ii M is a 1-dimensional submanifold; iii

3.9
Equating the normal components, we get On the other hand, from 3.4 , we have g FX, FX sin 2 θ g X, X − η X η X , 3.11 for any X ∈ Γ TM . Taking the covariant derivative of the above equation with respect to PX, we obtain 2g ∇ PX FX, FX 2sin 2 θg ∇ PX X, X − 2sin 2 θη X g ∇ PX X, ξ − 2sin 2 θη X g X, ∇ PX ξ .

3.12
Using the property of metric connection ∇, the last two terms of the right-hand side are cancelling each other, thus we have Since ∇ is the metric connection, then the above equation can be written as sin 2 θη X g X, ∇ PX ξ cos 2 θ X 2 − η 2 X g H, FX .

3.19
As M is cosymplectic thus using the fact that ∇ PX ξ 0, the left hand side of the above equation vanishes identically, then cos 2 θ X 2 − η 2 X g H, FX 0.

3.20
Thus from 3.20 , it follows that either θ π/2 or X ξ or H ∈ Γ μ , where μ is the invariant normal subbundle orthogonal to FTM. This completes the proof. Proof. As M is cosymplectic, then we have for any U, V ∈ Γ T M . Using this fact and formulae 2.5 and 2.8 we obtain that ∇ X PY ∇ X FY P ∇ X Y F∇ X Y φh X, Y , 3.22 for any X, Y ∈ Γ TM . Then from 2.5 , 2.6 and 2.12 , we get ∇ X PY h X, P Y − A FY X ∇ ⊥ X FY P ∇ X Y F∇ X Y g X, Y φH. 3.23