Nonexistence of PR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}\mathcal {R}$$\end{document}-semi-slant warped product submanifolds in paracosymplectic manifolds

In the present paper, we prove that there does not exist any PR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}\mathcal {R}$$\end{document}-semi-slant warped product submanifolds in paracosymplectic manifolds. In addition, by presenting a non-trivial example we find that there is no proper PR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}\mathcal {R}$$\end{document}-semi-slant warped product submanifold other than PR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}\mathcal {R}$$\end{document}-semi-invariant warped products.

present paper, we investigate the existence or nonexistence of PR-semi-slant warped product submanifolds in paracosymplectic manifolds.
The organization of the article is as follows. In Sect. 2, we recall some basic information about paracontact manifolds and their submanifolds. Moreover we review some known facts about warped product submanifolds and give some preparatory results for the existence or nonexistence of warped product submanifolds in a paracosymplectic manifold. In Sect. 3, we first define PR-semi-slant submanifolds and then derive necessary and sufficient conditions for the distributions equipped with the definition of such submanifold to be involutive and totally geodesic. Section 4 deals with the nonexistence of the non-trivial PR-semi-slant warped product submanifolds of the forms M T × f M λ and M λ × f M T in a paracosymplectic manifold M such that the characteristic vector field ξ is tangent to first factor or second factor in both cases, where M T and M λ are invariant submanifold and proper slant submanifold of M. Finally, in Sect. 5, we present a non-trivial example of PR-semi-invariant warped product submanifold which can be viewed as the non-trivial PR-semi-slant warped product submanifold with an improper slant coefficient, i.e., λ = 0.

Preliminaries
Let M be an odd-dimensional smooth manifold. An almost paracontact structure on M is a triplet (φ, ξ, η) [22], such that φ is a tensor field of (1, 1)-type, and ξ is a vector field and, η is a 1-form satisfying the following conditions: where I is the identity transformation and the tensor field φ induces on 2m-dimensional horizontal distribution D = ker(η) an almost paracomplex structure J ; that is, J 2 = I and the eigen subbundles D ± corresponding to the eigenvalues ±1 of J , respectively, have equal dimension m; hence D = D + ⊕ D − . The direct consequence of Eq. (2.1) is that the structure endomorphism φ has rank 2m, φξ = 0 and η • φ = 0. If a manifold M with (φ, ξ, η)-structure admits a pseudo-Riemannian metric g of signature (m + 1, m) such that then M is said to have an almost paracontact metric structure (φ, ξ, η, g) and the manifold M equipped with (φ, ξ, η, g)-structure is called an almost paracontact metric manifold, here g is known as compatible metric (see [29,43]). With respect to g, η is metrically dual to the unitary vector field ξ , i.e., η = g(·, ξ). With the consequences of Eqs. (2.1) and (2.2) we deduce that φ is a g-skew-symmetric operator, that is, for any X, Y ∈ (T M); (T M) denotes the section of the tangent bundle (T M) of M. The fundamental 2-form = g(·, φ·) is a non-degenerate on the horizontal distribution D and η ∧ m = 0. If = dη, then η is a paracontact 1-form and the almost paracontact metric manifold M(φ, ξ, η, g) is called a paracontact metric manifold. Let us consider that M is an isometrically immersed submanifold of a paracosymplectic manifold M in the sense of B. O'Neill [27]. Let g denote the induced metric on M such that g = g| M [18], (T M ⊥ ) indicates the set of all vector fields normal to M and (T M) the sections of tangent bundle T M of M. Then the Gauss-Weingarten formulas are given, respectively, by for any X, Y ∈ (T M) and ζ ∈ (T M ⊥ ), where ∇ is the induced connection, ∇ ⊥ is the normal connection on the normal bundle (T M ⊥ ), h is the second fundamental form, and the shape operator A ζ associated with the normal section ζ is given in [11] by Now, for any X ∈ (T M) and ζ ∈ (T M ⊥ ), we write where t X (resp., n X) is tangential (resp., normal) part of φ X and t ζ (resp., n ζ ) is tangential (resp., normal) part of φζ . Then the submanifold M is said to be invariant if n is identically zero and anti-invariant if t is identically zero. From Eqs. (2.3) and (2.8), we obtain that for any X, Y ∈ (T M) By virtue of Gauss formula and the fact that the structure is paracosymplectic, we can give the following result for later use: Then the manifold M = B × f F is said to be a warped product if it is equipped with the following warped metric for all X, Y ∈ (T M) and ' * ' stands for derivation map, or equivalently, The function f is called the warping function and a warped product manifold M is said to be trivial if f is constant. For the sake of simplicity, we will determine a vector field X on B with its lift X and a vector field Z on F with its lift Z on M = B × f F (see also [5,13]).

Proposition 2.3 [5]
For X, Y ∈ (T B) and Z , W ∈ (T F), we obtain for the warped product manifold where ∇ denotes the Levi-Civita connection on M and ∇ f is the gradient of f defined by g(∇ f, X ) = X f and ∇ is the connection on F.

Remark 2.4
It is also important to note that for a warped product M = B × f F; B is totally geodesic and F is totally umbilical in M [5].
Furthermore, we prove an important theorem for later use;

Theorem 2.5 Let M(φ, ξ, η, g) be a paracosymplectic manifold. Then there does not exist a non-trivial warped
Proof In view of Lemma 2.2 and Proposition 2.3, we obtain that X (ln f )ξ = 0, for any non-degenerate vector fields X ∈ (T B) and ξ ∈ (T F). This implies that f is constant function, since X, ξ are non-degenerate vector fields in M. Hence, the proof is complete.
Here, we recall the following important results from [30] for later use when ξ ∈ (T B): for any X, Y ∈ (T B) and Z , W ∈ (T F).
In [30], we have defined PR-semi-invariant submanifolds in paracontact manifold as follows: Definition 2.7 Let M be an isometrically immersed submanifold of an almost paracontact metric manifold M(φ, ξ, η, g) such that the characteristic vector field ξ ∈ (T M). Then the submanifold M is called PRsemi-invariant if it is furnished with a pair of non-degenerate orthogonal distribution (D T , D ⊥ ) which satisfies the following conditions: If the warping function f is constant then a PR-semi-invariant warped product submanifold is said to be a PR-semi-invariant product or trivial product.

PR-semi-slant submanifolds
In this section, by following [1-3,6,31], we introduce PR-semi-slant submanifolds in M which generalizes [30]. Since, submanifold M is non-degenerate submanifold so this class of submanifolds can be viewed as the generalization of submanifolds defined in [4,9,10,19,24] which includes the space-like vector fields only. Let M be a non-degenerate submanifold of an almost paracontact metric manifold M such that On the other hand, In particular, from Eqs. (3.1) and (3.2), we obtain for X = Y , that g(φ X,t X) Here we call λ a slant constant coefficient or simply slant coefficient and consequently M a slant submanifold. Conversely, assume that M is a slant submanifold then λ |φ X | |t X| = |t X| |ϕ X | , where X is a non-light-like vector field. We obtain by the consequence of previous equation for any non-light like vector fields X, tY ) by virtue of the fact that structure is paracosymplectic and X is non-degenerate vector fields.

Remark 3.1
The slant coefficient λ is sometimes cos 2 θ or cosh 2 θ or − sinh 2 θ for all vector fields tangent to M, where θ is a slant angle [1] and [3]. Now as a consequence of the above theory we have the following definitions.
where λ is a slant coefficient and slant distribution D λ indicate the non-degenerate distribution on M [31]. (b) PR-semi-slant if it is furnished with a pair of non-degenerate orthogonal distribution (D T , D λ ) satisfies the following conditions: Here, {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 , e 8 , e 9 } is a local orthonormal frame for (T M), given by e i = ∂ ∂ x i , e j = ∂ ∂ x j and e 9 = ∂ ∂ x 9 .

Theorem 3.7 Let M be a proper PR-semi-slant submanifold of a paracosymplectic manifold M. Then the distribution (D T ⊕ {ξ }) (i) is involutive if and only if h(X, tY ) = h(t X, Y ); (ii) defines a totally geodesic foliation if and only if A n Z tY = A nt Z Y ; for any X, Y ∈ (D T ⊕ {ξ }) and Z ∈ (D λ ).
Proof For M to be a proper PR-semi-slant submanifold M of a paracosymplectic manifold, we have Then, by virtue of Eqs. (2.2), (2.5) and fact that ξ and Z are orthogonal, we achieve that Employing Eq. (2.4), (2.7), (2.8), (3.2) and Gauss-Weingarten formulas in Eq. (3.9), we obtain that (3.10) From above equation, we can write that

PR-semi-slant warped product submanifolds
In this section, we investigate the nonexistence of PR-semi-slant warped product submanifolds of the forms M T × f M λ and M λ × f M T in a paracosymplectic manifold M, whether the structure vector field ξ is tangent to first factor or second factor, where M T and M λ are invariant and proper slant submanifolds of M, respectively. As a straight forward consequence of Theorem 2.5, we conclude the following results when ξ is tangent to the fiber in each case.
for any non-degenerate vector fields X ∈ (T M T ) and Z ∈ (T M λ ). Employing Eqs. (2.8), (2.9) and Proposition (2.3) in (4.1), we compute that for all X ∈ (T M T ) and Z ∈ (T M λ ). Moreover, we can write from Eq. (4.5), that Now, by taking inner product of t Z with Eq. (4.3), we obtain  Proof Let us assume that M = M λ × f M T is a PR-semi-slant warped product submanifold of a paracosymplectic manifold M(φ, ξ, η, g) such that ξ ∈ (T M λ ). Then we can write from Proposition 2.3, Gauss formula and the Connection property for ∇, that for any X ∈ (T M T ) and Z ∈ (T M λ ). We also, from Eqs. (2.2), (2.5), (2.8), Lemma 2.2 and the fact that structure is paracosymplectic, achieve that Above equation by the use of Connection property for ∇ and Proposition 2.3 is reduced to Hence by virtue of Eqs. (4.10) and (4.12) and the orthogonality of vector fields, we obtain Interchanging X by φ X in (4.13) and using the fact that ξ ∈ (T M λ ), then using (2.2), we derive (4.14) Thus, the result follows from Eqs. (4.13) and (4.14), which proves the theorem completely.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.