Almost Paracontact 3-Submersions

In this paper, we discuss some geometric properties of Riemannian submersions whose total space is an almost paracontact manifold with 3-structure. The study is focused on the transference of structures, the geometry of the fibres and sectional curvature tensor. 2010. Mathematics Subject Classification: 53C15, 53C26, 53C55.

Deshmukh and Khan [9, p.448] have constructed a properly almost paracontact 3−structure on R 3 as follows  Note that, recently, other geometers have been interested by the problem of mixing paracontact and contact structures. This is the case of Mixed 3-Sasakian structures studied by [4] and Bi-paracontact structures studied by [5] as the paper of V. Martin-Molina [15].
An almost paracontact manifold with 3-structure is called:  between Riemannian manifolds such that π * |(Kerπ * ) ⊥ is a linear isometry [16]. The tangent bundle T (M ) of the total space M admits an orthogonal decomposition where V (M ) and H(M ) denote , respectively, the vertical and the horizontal distribution. We denote by V and H the vertical and the horizontal projections of T (M ) onto V (M ) and H(M ) respectively. For all points x ∈ M , the closed embedded submanifold F x = π −1 (x ) is called the fibre of π over x . It is known that dimF x = dimM − dimM .
A vector field X of the horizontal distribution is called a basic vector field if it is π-related to a vector field X * of the base space M . That is X * = π * X. On the base space, tensors and other operators will be specified by a prime Journal of Advances in Mathematics Vol 17 (2019) ISSN: 2347-1921 https://rajpub.com/index.php/jam ( ) while those of the fibres will be denoted by a caret() . For instance, ∇, ∇ , and∇ will designate the Levi-Civita connection of the total space, the base space and the fibres respectively.
Here, we put the definitions and some properties of the two types of submersions which will be considered.
) be an almost paracontact manifold with 3-structure and (M 4m , g , (J i ) 3 i=1 ) be an almost hyper-para-Hermitian manifold. A Riemannian submersion is called an almost paracontact 3-submersion of type I, if it satisfies π * ϕ i = J i π * , i = 1, 2, 3. Proof. Clearly the fibres are 4(m − m ) + 3-dimensional submanifolds. Let (g, ϕ i , ξ i , η i ) be the almost paracontact metric 3−structure of the total space; note by (ĝ,φ i ,ξ i ,η i ) its restriction on the fibres. The problem is to show that is an almost paracontact metric 3−structure. Indeed, let U and V be two vertical vector fields tangent to the fibres. We havê Surely the fibres areφ i −invariant. By definition, it is known that π * φi U = J i π * U. Let us denote by kerπ * the fibres; taking U ∈ kerπ * , then π * U = 0 = π * φi U which means that Now, we are going to introduce a new type of almost paracontact 3-submersions. This type is closely related to almost contact 3-submersions of type II [18].
is said to be an almost paracontact 3-submersion of type II if : J i =φ i ,we have to show that (g, J i ) is an almost para-Hermitian structure. Indeed, which achieves the proof.

Transference of Structures
The aim of this subsection is to determine the structure of the fibres and the base space according to that of the total space. (a) Let X, Y and Z be basic vector fields. It is not hard to see that π * Ω i = Φ i from which we deduce π * dΩ i = dΦ i .
(b) It is well known that the vertical distribution is invariant by ϕ i . Since η i (U ) =η i (U ) and One can easily show that dΩ i = 0 which is the defining relation of a 3-para-almost Kähler structure on the base space for i=1,2,3. Accordingly, the base space is hyper-para-almost Kähler.
The case of the structure of the fibres is treated as in the preceding Proposition 3.5. Now, we want to examine the structure of the fibres of a type II almost paracontact 3-submersion.  Proof. By absurd, suppose that the total space is a 3−para-Sasakian manifold and let us describe the structure of the fibres.
Recall from Proposition 3.4 that the fibres are almost hyper-para-Hermitian. Consider U and V two vertical vector fields tangent to the fibres. Since the total space is supposed to be 3−para-Sasakian, it is defined by Φ i = dη i and But it can be shown that dη i (U, V ) = 0 which implies that on the fibres, the fundamental local 2 − f orm Ω i (U, V ) = 0. This is to say that, the fibres have the fundamental local 2 − f orm zero for all i = 1, 2, 3. Which is surely a non sense. Thus, the total space cannot be a 3−para-Sasakian manifold.

The Geometry of the Fibres
We want to determine the classes of submersions with minimal or totally geodesic fibres. Let us recall that, on the total space of a Riemnnian submersion, B. O'Neill [16] has defined two configuration tensors, T and A of type (1,2) by setting We shall be interested with the tensor T which is a usefull tool in the study of the fibres. The tensor A is used to measure the integrability of the horizontal distribution.
Note that the study of the ϕ i -linearity of these tensors , in [17], is with helpfull. Recall that if T ϕiU V = ϕ i T U V , then the fibres are minimal and if T ≡ 0 they are totally geodesic.
Since ϕ i U is vertical and from the symmetric property of T on vertical vector fields, we deduce

It is clear that
On the other hand, it can be shown that T U ξ i = 0 and then Proof. It can be shown that the manifold under consideration satisfies the preceding Corrollary 3.10. The vanishing of T is the condition for the fibres to be totally geodesic.

Sectional Curvature Tensor
Here, we want to determine the classes of almost paracontact 3-submersions which preserve the ϕ i -holomorphic sectional curvature tensor on horizontal or vertical distribution.
Proof. From Proposition 3.11, we have T = 0 which implies that H ϕi (U ) = Hφ i (U ) . Obviously, this manifold satisfies A ϕiX Y = ϕ i A X Y which gives rise to A X ϕ i X = 0 and leads to H ϕi (X) = H J i (X * ).

Some Examples
In order to construct an example of a type I submersion, we will consider a manifold product as follows.
) be an almost hyper-para-Hermitian manifold and ) an almost paracontact manifold with 3-structure. It is clear that the manifold productM = M × M is of dimension 4(m + m) + 3 which will be taken 4p + 3 where p = m + m. We can define a 3-structure onM by settinḡ It is not hard to show that (M ,ḡ, is an almost paracontact manifold with 3-structure whereḡ is defined byḡ We can refer to [23], where the product of paracontact manifolds is treated.
) be an almost hyper-para-Hermitian manifold and (M 4m+3 , g, is an almost paracontact manifold with 3-structure, constructed as above, then it is 3-para-cosymplectic if, and only if, M is 3-para-cosymplectic and M is hyper-para-Kähler; Proof. Let us note that, in this product, the calculation gives the following relation (3.1) Suppose that (g, (ϕ i , ξ i , η i ) 3 i=1 ) is a 3−para-cosymplectic structure. It is known that in this case (∇ D η i ) = 0 from which (3.1) gives which is the defining relation of a hyper-para-Kähler structure.