Totally Umbilical Proper Slant and Hemislant Submanifolds of an LP-Cosymplectic Manifold

In the present note, we study slant and hemislant submanifolds of an LP-cosymplectic manifold which are totally umbilical. We prove that every totally umbilical proper slant submanifold M of an LP-cosymplectic manifold M is either totally geodesic or if M is not totally geodesic in M then we derive a formula for slant angle of M. Also, we obtain the integrability conditions of the distributions of a hemi-slant submanifold, and then we give a result on its classification.


Introduction
A manifold M with Lorentzian paracontact metric structure φ, ξ, η, g satisfying ∇ X φ Y 0 is called an LP-cosymplectic manifold, where ∇ is the Levi-Civita connection corresponding to the Lorentzian metric g on M. The study of slant submanifolds was initiated by Chen 1 .Since then, many research papers have appeared in this field.Slant submanifolds are the natural generalization of both holomorphic and totally real submanifolds.Lotta 2 defined and studied these submanifolds in contact geometry.Later on, Cabrerizo et al. studied slant, semi-slant, and bislant submanifolds in contact geometry 3, 4 .In particular, totally umbilical proper slant submanifold of a Kaehler manifold has also been studied in 5 .Recently, Khan et al. 6 studied these submanifolds in the setting of Lorentzian paracontact manifolds.
The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bislant submanifolds, and he called them antislant submanifolds 7 .Recently, these submanifolds are studied by Sahin for their warped products 8 .In this paper, we study slant and hemi-slant submanifolds of an LP-cosymplectic manifold.We prove that a totally umbilical proper slant submanifold M is either totally geodesic in M or if it is not totally geodesic, then the slant angle θ tan −1 g X, Y /η X η Y .Also, we define hemislant submanifolds of an LP-contact manifold.After we find integrability conditions of the distributions, we investigate a classification of totally umbilical hemi-slant submanifolds of an LP-cosymplectic manifold.

Preliminaries
Let M be a n-dimensional paracontact manifold with the Lorentzian paracontact metric structure φ, ξ, η, g , that is, φ is a 1, 1 tensor field, ξ is a contravariant vector field, η is a 1-form, and g is a Lorentzian metric with signature −, , , . . ., on M, satisfying 9 , for all X, Y ∈ TM.
A Lorentzian paracontact metric structure on M is called a Lorentzian para-cosymplectic structure if ∇φ 0, where ∇ denotes the Levi-Civita connection with respect to g.The manifold M in this case is called a Lorentzian para-cosymplectic in brief, an LP-cosymplectic manifold 10 .From formula ∇φ 0, it follows that ∇ X ξ 0.
Let M be a submanifold of a Lorentzian almost paracontact manifold M with Lorentzian almost paracontact structure φ, ξ, η, g .Let the induced metric on M also be denoted by g, then Gauss and Weingarten formulae are given by for any X, Y in TM and N in T ⊥ M, where TM is the Lie algebra of vector field in M and T ⊥ M is the set of all vector fields normal to M. ∇ ⊥ is the connection in the normal bundle, h is the second fundamental form, and A N is the Weingarten endomorphism associated with N. It is easy to see that For any X ∈ TM, we write where PX is the tangential component and FX is the normal component of φX.Similarly for N ∈ T ⊥ M, we write where BN is the tangential component and CN is the normal component of φN.
The covariant derivatives of the tensor fields φ, P , and F are defined as Moreover, for an LP-cosymplectic manifold, one has where H is the mean curvature vector.Furthermore, if h X, Y 0 for all X, Y ∈ TM, then M is said to be totally geodesic, and if H 0, then M is minimal in M.
A submanifold M of a paracontact manifold M is said to be a slant submanifold if for any x ∈ M and X ∈ T x M− ξ , the angle between φX and T x M is constant.The constant angle θ ∈ 0, π/2 is then called slant angle of M. The tangent bundle TM of M is decomposed as where the orthogonal complementary distribution D of ξ is known as the slant distribution on M. If μ is φ-invariant subspace of the normal bundle T ⊥ M, then Khan et al. 6 proved the following theorem for a slant submanifold M of a Lorentzian paracontact manifold M with slant angle θ.Theorem 2.1.Let M be a submanifold of an LP -contact manifold M such that ξ ∈ TM, then M is slant submanifold if and only if there exists a constant λ ∈ 0, 1 such that
Thus, one has the following consequences of formula 2.16 : for any X, Y ∈ TM.

Totally Umbilical Proper Slant Submanifold
In this section, we consider M as a totally umbilical proper slant submanifold of an LPcosymplectic manifold M. Such submanifolds we always consider tangent to the structure vector field ξ.
Taking the product with ξ and using 2.9 , we obtain g ∇ X PY, ξ g A FY X, ξ g Bh X, Y , ξ .

3.2
Using 2.5 and the fact that M is totally umbilical, the above equation takes the form −g PY, ∇ X ξ g H, FY η X g X, Y g BH, ξ .

3.3
Then, from the characteristic equation of LP-cosymplectic manifold, we obtain 0 g H, FY η X .

3.4
Thus, from 3.4 , it follows that either H ∈ μ or M is trivial.Now, for an LP-cosymplectic manifold, one has, from 2.8 , for any X, Y ∈ TM.From 2.3 and 2.6 , we obtain Again using 2.3 , 2.4 , and 2.6 , we get As M is totally umbilical, then Taking the inner product with φH and using the fact that H ∈ μ, we obtain Then from 2.2 and 2.13 , we get

3.10
Again, using 2.2 and the fact that H ∈ μ, then φH is also lies in μ; thus, we obtain

3.12
Now, for any X ∈ TM, one has

3.13
Using the fact that as M is an LP-cosymplectic manifold, we obtain ∇ X φH φ∇ X H.

3.15
Taking the product in 3.15 with FY for any Y ∈ TM and using the fact C∇ ⊥ X H ∈ μ, the above equation gives g ∇ ⊥ X φH, FY −g FA H X, FY .

3.16
Using 2.18 , we obtain g ∇ X FY, φH sin 2 θ g A H X, Y η A H X η Y , 3.17 then, from 2.5 and 2.13 , we get

3.18
Thus, from 3.12 and 3.18 , we derive Hence, 3.19 gives either H 0 or if H / 0, then the slant angle of M is θ tan −1 g X, Y /η X η Y .This proves the theorem completely.

Hemislant Submanifolds
In the following section, we assume that M is a hemi-slant submanifold of an LPcosymplectic manifold M such that the structure vector field ξ tangent to M. First, we define a hemi-slant submanifold, and then we obtain the integrability conditions of the involved distributions D 1 and D 2 in the definition of a hemi-slant submanifold M of an LPcosymplectic manifold M. Definition 4.1.A submanifold M of an LP-contact manifold M is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions D 1 and D 2 satisfying If μ is φ-invariant subspace of the normal bundle T ⊥ M, then in case of hemi-slant submanifold, the normal bundle T ⊥ M can be decomposed as 4.1 In the following, we obtain the integrability conditions of involved distributions in the definition of hemi-slant submanifold.
Taking the product in 4.10 with FZ, for any Z ∈ D 2 , we obtain Thus, the assertion follows from 4.11 after using 2.2 and the fact that ξ is tangential to D 1 .Now, we consider M as a totally umbilical hemi-slant submanifold of an LPcosymplectic manifold M. For any X, Y ∈ TM, one has ∇ X φY φ∇ X Y.

4.12
Using this fact, if we take for any Z, W ∈ D 2 , then from 2.3 and 2.4 , the above equation takes the form Thus, on using 2.6 and 2.7 , we obtain −A FW Z ∇ ⊥ Z FW P ∇ Z W F∇ Z W Bh Z, W Ch Z, W .

4.14
Equating the tangential components, we get

4.15
Taking the product with V ∈ D 2 , we obtain g P ∇ Z W, V −g A FW Z, V − g Bh Z, W , V .

4.16
Using 2.2 , 2.5 , and the fact that PW 0, for any W ∈ D 2 , thus, the above equation takes the form 0 g h Z, V , FW g Bh Z, W , V .

4.17
As M is totally umbilical, we derive 0 g Z, V g H, FW g Z, W g BH, V .

4.18
Thus, 4.18 has a solution if either Z W V ξ, that is, dim D 2 1 or H ∈ μ or D 2 {0}.Hence, we state the following theorem.
Theorem 4.4.Let M be a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold M, then at least one of the following statements is true: i the dimension of anti-invariant distribution is one, that is, dim D 2 1, ii the mean curvature vector H ∈ μ, iii M is proper slant submanifold of M.
As D 2 is an anti-invariant distribution, then the tangential part of 4.5 should be identically zero; hence, we obtain the required result.