# Paracomplex Paracontact Pseudo-Riemannian Submersions

Authors: S. S. Shukla; Uma Shankar Verma
Journal: Geometry
 (2014)

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## Abstract

We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifolds onto almost paracontact metric manifolds. We discuss the transference of structures on total manifolds and base manifolds and provide some examples. We also obtain the integrability condition of horizontal distribution and investigate curvature properties under such submersions.


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## Body

## 1. Introduction

The theory of Riemannian submersion was introduced by O’Neill [1, 2] and Gray [3]. It is known that the applications of such Riemannian submersion are extensively used in Kaluza-Klein theories [4, 5], Yang-Mill equations [6, 7], the theory of robotics [8], and supergravity and superstring theories [5, 9].There is detailed literature on the Riemannian submersion with suitable smooth surjective map followed by different conditions applied to total space and on the fibres of surjective map. The Riemannian submersions between almost Hermitian manifolds have been studied by Watson [10]. The Riemannian submersions between almost contact manifolds were studied by Chinea [11]. He also concluded that if M¯ is an almost Hermitian manifold with structure (J¯,g¯) and M is an almost contact metric manifold with structure (ϕ,ξ,η,g), then there does not exist a Riemannian submersion f:M¯→M which commutes with the structures on M¯ and M; that is, we cannot have the condition f*∘J¯=ϕ∘f*. Chinea also defined the Riemannian submersion between almost complex manifolds and almost contact manifolds and studied some properties and interrelations between them [12]. In [13], Gündüzalp and Sahin gave the concept of paracontact paracomplex semi-Riemannian submersion between almost paracontact metric manifolds and almost para-Hermitian manifolds submersion giving an example and studied some geometric properties of such submersions.An almost paracontact structure on a differentiable manifold was introduced by Sato [14], which is an analogue of an almost contact structure and is closely related to almost product structure. An almost contact manifold is always odd dimensional but an almost paracontact manifold could be even dimensional as well.The paracomplex geometry has been studied since the first papers by Rashevskij [15], Libermann [16], and Patterson [17] until now, from several different points of view. The subject has applications to several topics such as negatively curved manifolds, mechanics, elliptic geometry, and pseudo-Riemannian space forms. Paracomplex and paracontact geometries are topics with many analogies and also with differences with complex and contact geometries.This motivated us to study the pseudo-Riemannian submersion between pseudo-Riemannian manifolds equipped with paracomplex and paracontact structures.In this paper, we give the notion of paracomplex paracontact pseudo-Riemannian submersion between almost paracomplex manifolds and almost paracontact pseudometric manifolds giving some examples and study the geometric properties and interrelations under such submersions.The composition of the paper is as follows. In Section2, we collect some basic definitions, formulas, and results on almost paracomplex manifolds, almost paracontact pseudometric manifolds, and pseudo-Riemannian submersion. In Section 3, we define paracomplex paracontact pseudo-Riemannian submersion giving some relevant examples and investigate transference of structures on the total manifolds and base manifolds under such submersions. In Section 4, curvature relations between total manifolds, base manifolds, and fibres are studied.


## 2. Preliminaries

### 2.1. Almost Paracontact Manifolds

LetM be a (2n+1)-dimensional Riemannian manifold, ϕ a (1,1)-type tensor field, ξ a vector field, called characteristic vector field, and η a 1-form on M. Then, (ϕ,ξ,η) is called an almost paracontact structure on M if
(1)ϕ2X=X-η(X)ξ;η(ξ)=1,
and the tensor field ϕ induces an almost paracomplex structure on the distribution 𝒟=ker⁡(η) [18, 19].M is said to be an almost paracontact manifold, if it is equipped with an almost paracontact structure. Again, M is called an almost paracontact pseudometric manifold if it is endowed with a pseudo-Riemannian metric g of signature (-,-,-,…,-︸(n-times),+,+,+,…,+︸(n+1)-times)) such that
(2)g(ϕX,ϕY)=g(X,Y)-εη(X)η(Y),∀X,Y∈Γ(TM),
where ε=1 or -1 according to the characteristic vector field ξ is spacelike or timelike. It follows that
(3)g(ξ,ξ)=ε,(4)g(ξ,X)=εη(X),(5)g(X,ϕY)=g(ϕX,Y),∀X,Y∈Γ(TM).
In particular, if index(g)=1, then the manifold (M2n+1,ϕ,ξ,η,g,ε) is called a Lorentzian almost paracontact manifold.If the metricg is positive definite, then the manifold (M2n+1,ϕ,ξ,η,g) is the usual almost paracontact metric manifold [14].The fundamental 2-formΦ on M is defined by
(6)Φ(X,Y)=g(X,ϕY).
Let M2n+1 be an almost paracontact manifold with the structure (ϕ,ξ,η). An almost paracomplex structure J on M2n+1×ℝ1 is defined by
(7)J(X,fddt)=(ϕX+fξ,η(X)ddt),
where X is tangent to M2n+1, t is the coordinate on ℝ1, and f is a smooth function on M2n+1.An almost paracontact structure(ϕ,ξ,η) is said to be normal, if the Nijenhuis tensor NJ of almost paracomplex structure J defined as
(8)NJ(X,Y)=[J,J](X,Y)=[JX,JY]+J2[X,Y]-J[JX,Y]-J[X,JY],
for any vector fields X,Y∈Γ(TM), vanishes.IfX and Y are vector fields on M2n+1, then we have [18–20]
(9)NJ((X,0),(Y,0))=(ddtNϕ(X,Y)-2dη(X,Y)ξ,hhhhh{(ℒϕXη)Y-(ℒϕYη)X}ddt),(10)NJ((X,0),(0,ddt))=-((ℒξϕ)X,((ℒξη)X)ddt),
where Nϕ is Nijenhuis tensor of ϕ,ℒX is Lie derivative with respect to a vector field X, and N(1),  N(2),  N(3),  and  N(4) are defined as
(11)Nϕ(X,Y)=[ϕ,ϕ](X,Y)=[ϕX,ϕY]+ϕ2[X,Y]-ϕ[ϕX,Y]-ϕ[X,ϕY],(12)N(1)(X,Y)=Nϕ(X,Y)-2dη(X,Y)ξ,(13)N(2)(X,Y)=(ℒϕXη)Y-(ℒϕYη)X,(14)N(3)(X)=(ℒξϕ)X,(15)N(4)(X)=(ℒξη)X.
The almost paracontact structure (ϕ,ξ,η) is normal if and only if the four tensors N(1),  N(2),N(3), and  N(4) vanish.For an almost paracontact structure(ϕ,ξ,η), vanishing of N(1) implies the vanishing of N(2),  N(3), and N(4). Moreover, N(2) vanishes if and only if ξ is a killing vector field.An almost paracontact pseudometric manifold(M2n+1,ϕ,ξ,η,g,ε) is called(i)
normal, ifNϕ-2dη⊗ξ=0,(ii)
paracontact, ifΦ=dη,(iii)
K-paracontact, if M is paracontact and ξ is killing,(iv)
paracosymplectic, if∇Φ=0, which implies ∇η=0, where ∇ is the Levi-Civita connection on M,(v)
almost paracosymplectic, ifdη=0 and dΦ=0,(vi)
weakly paracosymplectic, ifM is almost paracosymplectic and [R(X,Y),ϕ]=R(X,Y)ϕ-ϕR(X,Y)=0, where R is Riemannian curvature tensor,(vii)
para-Sasakian, ifΦ=dη and M is normal,(viii)
quasi-para-Sasakian, ifdϕ=0 and M is normal.


### 2.2. Almost Paracomplex Manifolds

A(1,1)-type tensor field J on 2m-dimensional smooth manifold M is said to be an almost paracomplex structure if J2=I and (M2m,J) is called almost paracomplex manifold.An almost paracomplex manifold(M,J) is such that the two eigenbundles T+M and T-M corresponding to respective eigenvalues +1 and -1 of J have the same rank [21, 22].An almost para-Hermitian manifold(M,J,g) is a smooth manifold endowed with an almost paracomplex structure J and a pseudo-Riemannian metric g such that
(16)g(JX,JY)=-g(X,Y),∀X,Y∈Γ(TM).
Here, the metric g is neutral; that is, g has signature (m,m).The fundamental 2-form of the almost para-Hermitian manifold is defined by(17)F(X,Y)=g(X,JY).
We have the following properties [21, 22]:
(18)g(JX,Y)=-g(X,JY),(19)F(X,Y)=-F(Y,X),(20)F(JX,JY)=-F(X,Y),(21)3dF(X,Y,Z)=X(F(Y,Z))-Y(F(X,Z))+Z(F(X,Y))-F([X,Y],Z)+F([X,Z],Y)-F([Y,Z],X),(22)(∇XF)(Y,Z)=g(Y,(∇XJ)Z)=-g(Z,(∇XJ)Y),(23)3dF(X,Y,Z)=(∇XF)(Y,Z)+(∇YF)(Z,X)+(∇ZF)(X,Y),(24)theco-differential,(δF)(X)=∑i=12mεi(∇eiF)(ei,X).
An almost para-Hermitian manifold is called (i)
para-Hermitian, ifNJ=0; equivalently, (∇JXJ)JY+(∇XJ)Y=0,(ii)
para-Kähler, if, for anyX∈Γ(TM),  ∇XJ=0; that is, ∇J=0,(iii)
almost para-Kähler, ifdF=0,(iv)
nearly para-Kähler, if(∇XJ)X=0,(v)
almost semi-para-Kähler, ifδF=0,(vi)
semi-para-Kähler, ifδF=0 and NJ=0.


### 2.3. Pseudo-Riemannian Submersion

Let(M¯m,g¯) and (Mn,g) be two connected pseudo-Riemannian manifolds of indices s¯(0≤s¯≤m) and s(0≤s≤n), respectively, with s¯≥s.A pseudo-Riemannian submersion is a smooth mapf:M¯m→Mn, which is onto and satisfies the following conditions [2, 3, 23, 24].(i)
The derivative mapf*p:TpM¯→Tf(p)M is surjective at each point p∈M¯.(ii)
The fibresf-1(q) of f over q∈M are either pseudo-Riemannian submanifolds of M¯ of dimension (m-n) and index ν or the degenerate submanifolds of M¯ of dimension (m-n) and index ν with degenerate metric g¯|f-1(q) of type (0,0,0,…,0︸μ-times,-,-,-,…,-︸ν-times,+,+,+,…,+︸(m-n-μ-ν)-times)), where μ=dim⁡(𝒱p∩ℋp) and ν=s¯-s=index ofg¯|f-1(q).(iii)
f* preserves the length of horizontal vectors.We denote the vertical and horizontal projections of a vector fieldE on M¯ by Ev (or by vE) and Eh (or by hE), respectively. A horizontal vector field X¯ on M¯ is said to be basic if X¯ is f-related to a vector field X on M. Thus, every vector field X on M has a unique horizontal lift X¯ on M¯.Lemma 1 (see [1, 23]).
Iff:M¯→M is a pseudo-Riemannian submersion and X¯,  Y¯ are basic vector fields on M¯ that are f-related to the vector fields X,  Y on M, respectively, then one has the following properties:(i)
g¯(X¯,Y¯)=g(X,Y)∘f,(ii)
h[X¯,Y¯] is a vector field and h[X¯,Y¯]=[X,Y]∘f,(iii)
h(∇¯X¯Y¯) is a basic vector field f-related to ∇XY, where ∇¯ and ∇ are the Levi-Civita connections on M¯ and M, respectively,(iv)
[E,U]∈𝒱, for any vector field U∈𝒱 and for any vector field E∈Γ(TM¯).A pseudo-Riemannian submersionf:M¯→M determines tensor fields 𝒯 and 𝒜 of type (1,2) on M¯ defined by formulas [1, 2, 23]
(25)𝒯(E,F)=𝒯EF=h(∇¯vEvF)+v(∇¯vEhF),(26)𝒜(E,F)=𝒜EF=v(∇¯hEhF)+h(∇¯hEvF),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiforanyE,F∈Γ(TM¯).LetX¯,  Y¯ be horizontal vector fields and let U,  V be vertical vector fields on M¯. Then, one has
(27)𝒯UX¯=v(∇¯UX¯),𝒯UV=h(∇¯UV),(28)∇¯UX¯=𝒯UX¯+h(∇¯UX¯),(29)𝒯X¯F=0,𝒯EF=𝒯vEF,(30)∇¯UV=𝒯UV+v(∇¯UV),(31)𝒜X¯Y¯=v(∇¯X¯Y¯),𝒜X¯U=h(∇¯X¯U),(32)∇¯X¯U=𝒜X¯U+v(∇¯X¯U),(33)𝒜UF=0,𝒜EF=𝒜hEF,(34)∇¯X¯Y¯=𝒜X¯Y¯+h(∇X¯Y¯),(35)h(∇¯UX¯)=h(∇¯X¯U)=𝒜X¯U,(36)𝒜X¯Y¯=12v[X¯,Y¯],(37)𝒜X¯Y¯=-𝒜Y¯X¯,(38)𝒯UV=𝒯VU,
for all E,F∈Γ(TM¯).Moreover,𝒯UV coincides with second fundamental form of the submersion of the fibre submanifolds. The distribution ℋ is completely integrable. In view of (37) and (38), 𝒜 is alternating on the horizontal distribution and 𝒯 is symmetric on the vertical distribution.


## 2.1. Almost Paracontact Manifolds

LetM be a (2n+1)-dimensional Riemannian manifold, ϕ a (1,1)-type tensor field, ξ a vector field, called characteristic vector field, and η a 1-form on M. Then, (ϕ,ξ,η) is called an almost paracontact structure on M if
(1)ϕ2X=X-η(X)ξ;η(ξ)=1,
and the tensor field ϕ induces an almost paracomplex structure on the distribution 𝒟=ker⁡(η) [18, 19].M is said to be an almost paracontact manifold, if it is equipped with an almost paracontact structure. Again, M is called an almost paracontact pseudometric manifold if it is endowed with a pseudo-Riemannian metric g of signature (-,-,-,…,-︸(n-times),+,+,+,…,+︸(n+1)-times)) such that
(2)g(ϕX,ϕY)=g(X,Y)-εη(X)η(Y),∀X,Y∈Γ(TM),
where ε=1 or -1 according to the characteristic vector field ξ is spacelike or timelike. It follows that
(3)g(ξ,ξ)=ε,(4)g(ξ,X)=εη(X),(5)g(X,ϕY)=g(ϕX,Y),∀X,Y∈Γ(TM).
In particular, if index(g)=1, then the manifold (M2n+1,ϕ,ξ,η,g,ε) is called a Lorentzian almost paracontact manifold.If the metricg is positive definite, then the manifold (M2n+1,ϕ,ξ,η,g) is the usual almost paracontact metric manifold [14].The fundamental 2-formΦ on M is defined by
(6)Φ(X,Y)=g(X,ϕY).
Let M2n+1 be an almost paracontact manifold with the structure (ϕ,ξ,η). An almost paracomplex structure J on M2n+1×ℝ1 is defined by
(7)J(X,fddt)=(ϕX+fξ,η(X)ddt),
where X is tangent to M2n+1, t is the coordinate on ℝ1, and f is a smooth function on M2n+1.An almost paracontact structure(ϕ,ξ,η) is said to be normal, if the Nijenhuis tensor NJ of almost paracomplex structure J defined as
(8)NJ(X,Y)=[J,J](X,Y)=[JX,JY]+J2[X,Y]-J[JX,Y]-J[X,JY],
for any vector fields X,Y∈Γ(TM), vanishes.IfX and Y are vector fields on M2n+1, then we have [18–20]
(9)NJ((X,0),(Y,0))=(ddtNϕ(X,Y)-2dη(X,Y)ξ,hhhhh{(ℒϕXη)Y-(ℒϕYη)X}ddt),(10)NJ((X,0),(0,ddt))=-((ℒξϕ)X,((ℒξη)X)ddt),
where Nϕ is Nijenhuis tensor of ϕ,ℒX is Lie derivative with respect to a vector field X, and N(1),  N(2),  N(3),  and  N(4) are defined as
(11)Nϕ(X,Y)=[ϕ,ϕ](X,Y)=[ϕX,ϕY]+ϕ2[X,Y]-ϕ[ϕX,Y]-ϕ[X,ϕY],(12)N(1)(X,Y)=Nϕ(X,Y)-2dη(X,Y)ξ,(13)N(2)(X,Y)=(ℒϕXη)Y-(ℒϕYη)X,(14)N(3)(X)=(ℒξϕ)X,(15)N(4)(X)=(ℒξη)X.
The almost paracontact structure (ϕ,ξ,η) is normal if and only if the four tensors N(1),  N(2),N(3), and  N(4) vanish.For an almost paracontact structure(ϕ,ξ,η), vanishing of N(1) implies the vanishing of N(2),  N(3), and N(4). Moreover, N(2) vanishes if and only if ξ is a killing vector field.An almost paracontact pseudometric manifold(M2n+1,ϕ,ξ,η,g,ε) is called(i)
normal, ifNϕ-2dη⊗ξ=0,(ii)
paracontact, ifΦ=dη,(iii)
K-paracontact, if M is paracontact and ξ is killing,(iv)
paracosymplectic, if∇Φ=0, which implies ∇η=0, where ∇ is the Levi-Civita connection on M,(v)
almost paracosymplectic, ifdη=0 and dΦ=0,(vi)
weakly paracosymplectic, ifM is almost paracosymplectic and [R(X,Y),ϕ]=R(X,Y)ϕ-ϕR(X,Y)=0, where R is Riemannian curvature tensor,(vii)
para-Sasakian, ifΦ=dη and M is normal,(viii)
quasi-para-Sasakian, ifdϕ=0 and M is normal.


## 2.2. Almost Paracomplex Manifolds

A(1,1)-type tensor field J on 2m-dimensional smooth manifold M is said to be an almost paracomplex structure if J2=I and (M2m,J) is called almost paracomplex manifold.An almost paracomplex manifold(M,J) is such that the two eigenbundles T+M and T-M corresponding to respective eigenvalues +1 and -1 of J have the same rank [21, 22].An almost para-Hermitian manifold(M,J,g) is a smooth manifold endowed with an almost paracomplex structure J and a pseudo-Riemannian metric g such that
(16)g(JX,JY)=-g(X,Y),∀X,Y∈Γ(TM).
Here, the metric g is neutral; that is, g has signature (m,m).The fundamental 2-form of the almost para-Hermitian manifold is defined by(17)F(X,Y)=g(X,JY).
We have the following properties [21, 22]:
(18)g(JX,Y)=-g(X,JY),(19)F(X,Y)=-F(Y,X),(20)F(JX,JY)=-F(X,Y),(21)3dF(X,Y,Z)=X(F(Y,Z))-Y(F(X,Z))+Z(F(X,Y))-F([X,Y],Z)+F([X,Z],Y)-F([Y,Z],X),(22)(∇XF)(Y,Z)=g(Y,(∇XJ)Z)=-g(Z,(∇XJ)Y),(23)3dF(X,Y,Z)=(∇XF)(Y,Z)+(∇YF)(Z,X)+(∇ZF)(X,Y),(24)theco-differential,(δF)(X)=∑i=12mεi(∇eiF)(ei,X).
An almost para-Hermitian manifold is called (i)
para-Hermitian, ifNJ=0; equivalently, (∇JXJ)JY+(∇XJ)Y=0,(ii)
para-Kähler, if, for anyX∈Γ(TM),  ∇XJ=0; that is, ∇J=0,(iii)
almost para-Kähler, ifdF=0,(iv)
nearly para-Kähler, if(∇XJ)X=0,(v)
almost semi-para-Kähler, ifδF=0,(vi)
semi-para-Kähler, ifδF=0 and NJ=0.


## 2.3. Pseudo-Riemannian Submersion

Let(M¯m,g¯) and (Mn,g) be two connected pseudo-Riemannian manifolds of indices s¯(0≤s¯≤m) and s(0≤s≤n), respectively, with s¯≥s.A pseudo-Riemannian submersion is a smooth mapf:M¯m→Mn, which is onto and satisfies the following conditions [2, 3, 23, 24].(i)
The derivative mapf*p:TpM¯→Tf(p)M is surjective at each point p∈M¯.(ii)
The fibresf-1(q) of f over q∈M are either pseudo-Riemannian submanifolds of M¯ of dimension (m-n) and index ν or the degenerate submanifolds of M¯ of dimension (m-n) and index ν with degenerate metric g¯|f-1(q) of type (0,0,0,…,0︸μ-times,-,-,-,…,-︸ν-times,+,+,+,…,+︸(m-n-μ-ν)-times)), where μ=dim⁡(𝒱p∩ℋp) and ν=s¯-s=index ofg¯|f-1(q).(iii)
f* preserves the length of horizontal vectors.We denote the vertical and horizontal projections of a vector fieldE on M¯ by Ev (or by vE) and Eh (or by hE), respectively. A horizontal vector field X¯ on M¯ is said to be basic if X¯ is f-related to a vector field X on M. Thus, every vector field X on M has a unique horizontal lift X¯ on M¯.Lemma 1 (see [1, 23]).
Iff:M¯→M is a pseudo-Riemannian submersion and X¯,  Y¯ are basic vector fields on M¯ that are f-related to the vector fields X,  Y on M, respectively, then one has the following properties:(i)
g¯(X¯,Y¯)=g(X,Y)∘f,(ii)
h[X¯,Y¯] is a vector field and h[X¯,Y¯]=[X,Y]∘f,(iii)
h(∇¯X¯Y¯) is a basic vector field f-related to ∇XY, where ∇¯ and ∇ are the Levi-Civita connections on M¯ and M, respectively,(iv)
[E,U]∈𝒱, for any vector field U∈𝒱 and for any vector field E∈Γ(TM¯).A pseudo-Riemannian submersionf:M¯→M determines tensor fields 𝒯 and 𝒜 of type (1,2) on M¯ defined by formulas [1, 2, 23]
(25)𝒯(E,F)=𝒯EF=h(∇¯vEvF)+v(∇¯vEhF),(26)𝒜(E,F)=𝒜EF=v(∇¯hEhF)+h(∇¯hEvF),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiforanyE,F∈Γ(TM¯).LetX¯,  Y¯ be horizontal vector fields and let U,  V be vertical vector fields on M¯. Then, one has
(27)𝒯UX¯=v(∇¯UX¯),𝒯UV=h(∇¯UV),(28)∇¯UX¯=𝒯UX¯+h(∇¯UX¯),(29)𝒯X¯F=0,𝒯EF=𝒯vEF,(30)∇¯UV=𝒯UV+v(∇¯UV),(31)𝒜X¯Y¯=v(∇¯X¯Y¯),𝒜X¯U=h(∇¯X¯U),(32)∇¯X¯U=𝒜X¯U+v(∇¯X¯U),(33)𝒜UF=0,𝒜EF=𝒜hEF,(34)∇¯X¯Y¯=𝒜X¯Y¯+h(∇X¯Y¯),(35)h(∇¯UX¯)=h(∇¯X¯U)=𝒜X¯U,(36)𝒜X¯Y¯=12v[X¯,Y¯],(37)𝒜X¯Y¯=-𝒜Y¯X¯,(38)𝒯UV=𝒯VU,
for all E,F∈Γ(TM¯).Moreover,𝒯UV coincides with second fundamental form of the submersion of the fibre submanifolds. The distribution ℋ is completely integrable. In view of (37) and (38), 𝒜 is alternating on the horizontal distribution and 𝒯 is symmetric on the vertical distribution.


## 3. Paracomplex Paracontact Pseudo-Riemannian Submersions

In this section, we introduce the notion of pseudo-Riemannian submersion from almost paracomplex manifolds onto almost paracontact pseudometric manifolds, illustrate examples, and study the transference of structures on total manifolds and base manifolds.Definition 2.
Let(M¯2m,J¯,g¯)  be an almost para-Hermitian manifold and let (M2n+1,ϕ,ξ,η,g) be an almost paracontact pseudometric manifold.
A pseudo-Riemannian submersionf:M¯→M is called paracomplex paracontact pseudo-Riemannian submersion if there exists a 1-form η¯ on M¯ such that
(39)f*∘J¯=ϕ∘f*+η¯⊗ξ.
Since, for eachp∈M¯,f*p is a linear isometry between horizontal spaces ℋp and tangent spaces Tf(p)M, there exists an induced almost paracontact structure (ϕ¯h,η¯h,ξ¯h,g¯) on (2n+1)-dimensional horizontal distribution ℋ such that ϕ¯|𝒟¯hhbehave just like the fundamental collineation of almost paracomplex structure J¯ on ker⁡η¯h=𝒟¯h and ϕ¯h:𝒟¯h→𝒟¯h is an endomorphism such that ϕ¯h=J¯|ker⁡η¯h and the rank of ϕ¯h=2n, where  dim⁡(𝒟¯h)=2n.
It follows that, for anyX¯h∈𝒟¯h,  η¯h(X¯h)=0, which implies that  J¯|𝒟¯h2(X¯h)=(ϕ¯h)2(X¯h)=X¯h, for any X¯h∈𝒟¯h and ℋ=𝒟¯h⊕{ξ¯h} [18].Definition 3 (see [25]).
A pseudo-Riemannian submersionf:M¯→M is called semi-J¯-invariant submersion, if there is a distribution 𝒟¯1⊆ker⁡f* such that
(40)ker⁡f*=𝒟¯1⊕𝒟¯2,(41)J¯(𝒟¯1)=𝒟¯1,J¯(𝒟¯2)⊆(ker⁡f*)⊥,
where 𝒟¯2 is orthogonal complementary to 𝒟¯1 in ker⁡f*.Proposition 4.
Letf:M¯2m→M2n+1 be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, the fibres f-1(q),  q∈M, are semi-J¯-invariant submanifolds of M¯ of dimension (2m-2n-1).Proof.
LetU¯∈𝒱. Then
(42)f*(J¯U¯)=ϕ(f*(U¯))+η¯(U¯)ξ,⟹f*{J¯(U¯)-η¯(U¯)ξ¯h}=0,
where  f*ξ¯h=ξ.
Thus, we have(43)J¯(U¯)-η¯(U¯)ξ¯h=ϕ¯(U¯),forsomeϕ¯(U¯)∈𝒱.
By (19), we get g¯(ξ¯h,J¯(ξ¯h))=0=g(ξ,f*(J¯(ξ¯h)))=0.
Asg is nondegenerate on M, we have
(44)f*(J¯(ξ¯h))=0,that isJ¯(ξ¯h)∈𝒱.
Taking U¯=J¯ξ¯h in (43), we obtain
(45)ξ¯h-η¯(J¯ξ¯h)ξ¯h=ϕ¯(J¯ξ¯h).
Since fibre f-1(q) is an odd dimensional submanifold, there exists an associated 1-form η¯v which is restriction of η¯ on fibre submanifold f-1(q),  q∈M, and a characteristic vector field ξ¯v=J¯ξ¯h such that ϕ¯(ξ¯v)=0. So, we have η¯v(ξ¯v)=1.
Let us putker⁡η¯v=𝒟¯1 and 𝒟¯2={ξ¯v}.
Then,ker⁡f*=𝒟¯1⊕𝒟¯2 and J¯(𝒟¯1)=𝒟1,  J¯(𝒟¯2)=J¯{ξ¯v}={ξ¯h}⊆(ker⁡f*)⊥.
Hence, the fibresf-1(q) are semi-J¯-invariant submanifolds of M¯.Corollary 5.
Letf:M¯2m→M2n+1 be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, the fibres f-1(q) are almost paracontact pseudometric manifolds with almost paracontact pseudo-Riemannian structures (ϕ¯v,ξ¯v,η¯v,g¯v),  q∈M, where ξ¯v=J¯(ξ¯h),  η¯v=η¯|𝒱, and g¯v=g¯.Proof.
Sincef-1(q)  are semi-J¯-invariant submanifolds of M¯ of odd dimension 2r+1=2m-2n-1, (39) implies
(46)J¯(U¯)=ϕ¯vU¯+η¯v(U¯)ξ¯h,
for any  U¯∈𝒱.
On operatingJ¯ on both sides of the above equation, we get
(47)U¯=ϕ¯v(ϕ¯v(U¯))+η¯v(ϕ¯v(U¯))ξ¯h+η¯v(U¯)ξ¯v,
where  J¯(ξ¯h)=ξ¯v.
Equating horizontal and vertical components, we have(48)U¯=ϕ¯v(ϕ¯v(U¯))+η¯v(U¯)ξ¯v,η¯v∘ϕ¯v(U¯)=0,⟹(ϕ¯v)2(U¯)=U¯-η¯v(U¯)ξ¯v;η¯v∘ϕ¯v=0;mmmmmϕ¯v(ξ¯v)=0,η¯v(ξ¯v)=1.
Hence, (ϕ¯v,ξ¯v,η¯v,g¯v) is almost paracontact pseudometric structure on the fibre f-1(q),q∈M.Proposition 6.
Letf:M¯2m→M2n+1 be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯ be basic vector fields f-related to X,  Y, respectively. Let η¯ and η be 1-forms on the total manifold M¯ and the base manifold M, respectively. Then, one has the following.(i)
The characteristic vector fieldJ¯ξ¯his a vertical vector field.(ii)
f**η=η¯h, wheref**ηis pullback ofηthroughf*.(iii)
η¯h(U¯)=0, for any vertical vector fieldU¯.(iv)
η¯v(X¯)=0, for any horizontal vector fieldX¯.Remark 7.
Results (ii) and (iv) are analogue version of results (i) and (iii) of Proposition 4 of [13].Proof.
(i) By Corollary5, (ϕ¯v,ξ¯v,η¯v,g¯v) is almost paracontact pseudometric structure on f-1(q). We have
(49)0=g¯(ξ¯h,J¯ξ¯h)=g(f*(ξ¯h),f*(J¯ξ¯h))=g(ξ,f*(J¯ξ¯h)).
Now,
(50)f*(J¯ξ¯h)=ϕ∘f*ξ¯h+η¯(ξ¯h)ξ=η¯(ξ¯h)ξ,
so we have
(51)0=g(ξ,η¯(ξ¯h)ξ)=η¯(ξ¯h)g(ξ,ξ)=η¯(ξ¯h).
Thus, f*(J¯ξ¯h)=0.
Hence,J¯ξ¯h is a vertical vector field.
(ii) Sincef:M¯2m→M2n+1 is smooth submersion, η¯h=η¯|ℋ is restriction of η¯ on the horizontal distribution ℋ, and f*p:ℋp→Tf(p)M is a linear isometry, for any X¯p∈ℋp,  we get
(52)η¯ph(X¯p)=εg¯p(ξ¯ph,X¯p)=gf(p)(f*pξ¯ph,f*pX¯p)=gf(p)(ξf(p),Xf(p))=ηf(p)(Xf(p))=f**ηp(X¯p).
Hence, pullback f**η=η¯h.
Results (iii) and (iv) immediately follow from the previous results.Example 8.
Let(ℝ24,J¯,g¯) be a paracomplex pseudometric manifold and let (ℝ13,ϕ,ξ,η,g) be an almost paracontact pseudometric manifold.
Define a submersionf:{ℝ24;(x1,x2,y1,y2)t}→{ℝ13;(u,v,w)t} by
(53)f((x1,x2,y1,y2)t)⟼(x1+x2+3y1+2y2,hhhhhhhhhhhhhhhhhhhh3x1+2x2+y1+y2,hhhhhhhhhhhhhhhhhhhh5x1+3x2+5y1+3y2)t.
Then, the kernel of f* is
(54)𝒱=ker⁡f*=Span⁡{V1=∂∂x1-2∂∂x2-∂∂y1+2∂∂y2},
which is the vertical distribution admitting one lightlike vector field; that is, fibre is degenerate submanifold of ℝ24.
The horizontal distribution is(55)ℋ=(ker⁡f*)⊥=Span⁡{X¯1=∂∂x1-∂∂y1,X¯2=∂∂x2+2∂∂y1,iiiiiiiiiiiiiiX¯3=2∂∂y1+∂∂y2}.
For any real k, the horizontal characteristic vector field ξ¯h is given by
(56)ξ¯h=k∂∂x1-(2k-13)∂∂x2-(k-1)∂∂y1+(2k-53)∂∂y2,
which is f-related to the characteristic vector field ξ=∂/∂w.
Moreover, there exists one formη¯=5dx1+3dx2+5dy1+3dy2 on (ℝ24,J¯,g¯) such that the submersion satisfies (39).Example 9.
Let(ℝ36,J¯,g¯) be an almost paracomplex pseudo-Riemannian manifold and let (ℝ13,ϕ,ξ,η,g) be an almost paracontact pseudo-Riemannian manifold. Consider a submersion f:{ℝ36;(x1,x2,x3,y1,y2,y3)t}→{ℝ13;(u,v,w)t}, defined by
(57)f((x1,x2,x3,y1,y2,y3)t)⟼(x1+x22,y1+y22,y2+y32)t.
Then, there exists one form η¯=(dx2+dx3)/2 on (ℝ36,J¯,g¯) such that (39) is satisfied. The kernel of f* is
(58)𝒱=ker⁡f*=Span⁡{V1=∂∂x1-∂∂x2,V2=∂∂y1-∂∂y2+∂∂y3,hhhhihhhV3=∂∂x3},
which is vertical distribution admitting non-lightlike vector fields; that is, the fibre is nondegenerate submanifold of (ℝ36,J¯,g¯).
The horizontal distribution is(59)ℋ=Span⁡{X¯1=∂∂x1+∂∂x2,X¯2=-∂∂y1+∂∂y3,hhhihhhhhhX¯3=∂∂y1+∂∂y2}.Example 10.
Let(ℝ24,J¯,g¯) be a paracomplex pseudometric manifold and let (ℝ13,ϕ,ξ,η,g) be an almost paracontact pseudometric manifold.
Consider a submersionf:{ℝ24;(x1,x2,y1,y2)t}→{ℝ13;(u,v,w)t}, defined by
(60)f((x1,x2,y1,y2)t)⟼(x1,y1,y2)t.
Then, the kernel of f* is
(61)𝒱=ker⁡f*=Span⁡{V1=∂∂x2},
which is the vertical distribution and the restriction of g¯ to the fibres of f is nondegenerate.
The horizontal distribution is(62)ℋ=(ker⁡f*)⊥=Span⁡{X¯=∂∂x1,Y¯=∂∂y1,ξ¯h=∂∂y2}.
The characteristic vector field ξ=∂/∂w on ℝ13 has unique horizontal lift ξ¯h, which is the characteristic vector field on horizontal distribution ℋ of ℝ24.
We also have(63)g¯(X¯,X¯)=g(f*X¯,f*X¯)=-1,g¯(Y¯,Y¯)=g(f*Y¯,f*Y¯)=1,g¯(ξ¯h,ξ¯h)=g(f*ξ¯h,f*ξ¯h)=g(ξ,ξ)=1.
Thus, the smooth map f is a pseudo-Riemannian submersion.
Moreover, we obtain that there exists a 1-formη¯=dx2 on ℝ24 such that η¯(J¯ξ¯h)=1,  η¯(ξ¯h)=0 and the map f satisfies
(64)f*J¯X¯=ϕf*X¯+η¯(X¯)ξ,f*J¯Y¯=ϕf*Y¯+η¯(Y¯)ξ,f*J¯ξ¯h=ϕf*ξ¯h+η¯(ξ¯h)ξ.
Hence, the map f is a paracomplex paracontact pseudo-Riemannian submersion from ℝ24 on to ℝ13.Proposition 11.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯ be basic vector fields f-related to X,  Y, respectively. Then, J¯(X¯)-εg¯(X¯,ξ¯h)ξ¯h is f-related to ϕX.Proof.
SinceX¯ is f-related to vector field X on M, we have
(65)η¯(X¯)={η¯v+η¯h}(X¯)=0+η¯h(X¯)=εg¯(X¯,ξ¯h),⟹f*(J¯X¯)=ϕX+η¯h(X¯)ξ,⟹f*{J¯X¯-εg¯(X¯,ξ¯h)ξ¯h}=ϕX.
Hence, J¯(X¯)-εg¯(X¯,ξ¯h)ξ¯h is f-related to ϕX.Proposition 12.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let 𝒱 and ℋ be the vertical and horizontal distributions, respectively. If ξ¯h is the basic characteristic vector field of horizontal distribution f-related to the characteristic vector field ξ of base manifold, then(i)
J¯𝒱⊂𝒟¯v⊕{J¯ξ¯h}⊕{ξ¯h},(ii)
J¯ℋ⊂𝒟¯h⊕{ξ¯h}⊕{J¯ξ¯h}.Proof.
(i) LetU¯∈𝒱. Then, U¯=aU¯|𝒟¯v+bJ¯ξ¯h, for a,b∈C∞(M¯), as J¯ξ¯h=ξ¯v is characteristic vector field on odd dimensional fibre submanifold f-1(q) of M¯,q∈M. We get (66)JU¯=aJ¯U¯|𝒟¯v+bJ¯2ξ¯h=aJ¯U¯|𝒟¯v+bξ¯h∈𝒱⊕{ξ¯h},⟹J¯𝒱⊂𝒱⊕{ξ¯h}.
Again, let V¯∈𝒱⊕{ξ¯h}. Then  V¯=aV¯|𝒟¯v+bJ¯ξ¯h+cξ¯h, where η¯v(V¯|𝒟¯v)=0,  𝒟¯v=ker⁡ηv,  aV¯|𝒟¯v+bJ¯ξ¯h∈𝒱, and a,b,c∈C∞(M¯). We have
(67)J¯V¯=aJ¯V¯|𝒟¯v+bξ¯h+cJ¯ξ¯h=(aJ¯V¯|𝒟¯v+cJ¯ξ¯h)︸∈𝒱+bξ¯h︸∈{ξ¯h}∈𝒱⊕{ξ¯h}.
Now, by (39), we get
(68)f*J¯V¯=ϕ(f*V¯)+η¯(f*(V¯))ξ=c{ϕ(f*ξ¯h)+η(ξ)ξ}=cξ∈{ξ}⊈J¯𝒱.
We get  J¯V∉𝒱.
Hence,J¯𝒱⊂𝒱⊕{ξ¯h};  that is,  J¯𝒱⊂𝒟¯v⊕{J¯ξ¯h}⊕{ξ¯h}.
(ii) LetX¯=aX¯|𝒟¯h+bξ¯h∈ℋ,  where ℋ=𝒟¯h⊕{ξ¯h},  ker⁡η¯h=𝒟¯h,   and  a,b∈C∞(M¯). Then
(69)J¯X¯=aJ¯X¯|𝒟¯h+bJ¯ξ¯h∈ℋ⊕{J¯ξ¯h},
which implies that  J¯ℋ⊂ℋ⊕{J¯ξ¯h}.
Again, letY¯∈ℋ⊕{J¯ξ¯h}. Then, Y¯=aY¯|𝒟¯h+bξ¯h+cJ¯ξ¯h∉ℋ, for a,b,c∈C∞(M¯). We have
(70)J¯Y¯=aJ¯Y¯|𝒟¯h+bJ¯ξ¯h+cJ¯2ξ¯h=aJ¯Y¯|𝒟¯h+bJ¯ξ¯h+cξ¯h=Z¯+bJ¯ξ¯h∈ℋ⊕{J¯ξ¯h},forsomeZ¯=aJ¯Y¯|𝒟¯h+cξ¯h∈ℋ.
We obtain  J¯Y¯∉ℋ.
Hence,J¯ℋ⊂ℋ⊕{J¯(ξ¯h)};  that is,  J¯ℋ⊂𝒟¯h⊕{ξ¯h}⊕{J¯ξ¯h}.Example 13.
Let(ℝ36,J¯,g¯) be an almost paracomplex pseudo-Riemannian manifold and let (ℝ13,ϕ,ξ,η,g) be an almost paracontact pseudo-Riemannian manifold. Consider a submersion f:{ℝ36;(x1,x2,x3,y1,y2,y3)t}→{ℝ13;(u,v,w)t}, defined by
(71)f((x1,x2,x3,y1,y2,y3)t)⟼(x1+x22,y1+y22,y3)t.
Then, the kernel of f* is
(72)𝒱=ker⁡f*=Span⁡{V1=∂∂x1-∂∂x2,V2=∂∂y1-∂∂y2,hhhhhhhξ¯v=∂∂x3}
which is the vertical distribution and the restriction of g¯ to the fibres of f is nondegenerate.
The horizontal distribution is(73)ℋ=(ker⁡f*)⊥=Span⁡{X¯1=∂∂x1+∂∂x2,X¯2=∂∂y1+∂∂y2,hhhhhhhξ¯h=∂∂y3}.
The characteristic vector field ξ=∂/∂w on ℝ13 has unique horizontal lift ξ¯h, which is the characteristic vector field on the horizontal distribution ℋ of ℝ36.
We also have(74)g¯(X¯1,X¯1)=g(f*X¯1,f*X¯1)=-2,g¯(X¯2,X¯2)=g(f*X¯2,f*X¯2)=2,g¯(ξ¯h,ξ¯h)=g(f*ξ¯h,f*ξ¯h)=g(ξ,ξ)=1.
Thus, the smooth map f is a pseudo-Riemannian submersion.
Also, we obtain that there exists a 1-formη¯=dx3 on ℝ36 such that η¯(J¯ξ¯h)=1,η¯(ξ¯h)=0 and the map f satisfies
(75)f*J¯X¯1=ϕf*X¯1+η¯(X¯1)ξ,f*J¯X¯2=ϕf*X¯2+η¯(X¯2)ξ,f*J¯ξ¯h=ϕf*ξ¯h+η¯(ξ¯h)ξ.
Hence, the map f is a paracomplex paracontact pseudo-Riemannian submersion from ℝ36 onto ℝ13.
Moreover, we observe that, for this submersionf, we have
(76)J¯𝒱⊂𝒱⊕{ξ¯h},J¯ℋ⊂ℋ⊕{J¯ξ¯h},
which verifies Proposition 12.Proposition 14.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯ be basic vector fields f-related to X,  Y, respectively. Let F and Φ be the second fundamental forms and let  ∇¯ and ∇ be the Levi-Civita connection on the total manifold M¯ and base manifold M, respectively. Then, one has (i)
f*((∇¯X¯J¯)Y¯)=(∇Xϕ)Y+εg(Y,∇Xξ)ξ+η(Y)∇Xξ,(ii)
F=f**Φ+εη¯⊗η¯,(iii)
f*((∇¯X¯F)(Y¯,Z¯))=(∇XΦ)(Y,Z)+η(Y)g(Z,∇Xξ)+η(Z)g(Y,∇Xξ).Proof.
(i) In view of Definition2 and Proposition 11, we have
(77)f*((∇¯X¯J¯)Y¯)=f*(∇¯X¯(J¯Y¯)-J¯(∇¯X¯Y¯))=∇X(f*(J¯Y¯))-f*(J¯(∇¯X¯Y¯))=∇X(ϕY)+∇X(η(Y)ξ)-ϕ(∇XY)-η(∇XY)ξ=(∇Xϕ)Y+∇X(εg(Y,ξ)ξ)-η(∇XY)ξ=(∇Xϕ)Y+εg(∇XY,ξ)ξ+εg(Y,∇Xξ)ξ+εg(Y,ξ)∇Xξ-εg(∇XY,ξ)ξ=(∇Xϕ)Y+εg(Y,∇Xξ)ξ+η(Y)∇Xξ.
(ii) Sincef**Φ is pullback of Φ through the linear map f*, we get
(78)f**Φ(X¯,Y¯)=Φ(X,Y)∘f=g(X,ϕY)∘f=g¯(X¯,J¯Y¯)-εη¯(Y¯)η¯(X¯)=F(X¯,Y¯)-εη¯(X¯)η¯(Y¯),
which implies  F=f**Φ+εη¯⊗η¯.
(iii) By (23), we have
(79)f*((∇¯X¯F)(Y¯,Z¯))=g(f*(Y¯),f*((∇¯XJ¯)Z¯)).
Now, using (i) in the above equation, we get (iii).Theorem 15.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯ be basic vector fields f-related to X,  Y, respectively. If the total space is para-Hermitian manifold, then the almost paracontact structure of base space is normal.
Moreover, if the almost paracontact structure of base space is normal, then the Nijenhuis tensor of total space is vertical.Proof.
The Nijenhuis tensorsNJ¯ and Nϕ of almost paracomplex structure J¯ and almost paracontact structure ϕ are, respectively, defined by (8) and (11).
Using Definition2 and properties of Sections 2.1 and 2.2, we get the following identity:
(80)f*(NJ¯(X¯,Y¯))=N(1)(X,Y)+2dη(ϕX,Y)ξ-2dη(ϕY,X)ξ+2η(X)dη(ξ,Y)ξ-2η(Y)dη(ξ,X)ξ-η(Y)N(3)(X)+η(X)N(3)(Y).
Using (12), (13), (14), and (15), (80) reduces to
(81)f*(NJ¯(X¯,Y¯))=N(1)(X,Y)+N(2)(X,Y)ξ+η(X)N(4)(Y)ξ-η(Y)N(4)(X)ξ-η(Y)N(3)(X)+η(X)N(3)(Y).
Since NJ¯(X¯,Y¯)=0, it follows from (81) that tensors  N(1),  N(2),  N(3),  and  N(4) vanish together.
Hence, the almost paracontact structure of base space is normal.
Conversely, let the almost paracontact structure of the base space be normal.
Then, (81) implies that f*(NJ¯(X¯,Y¯))=0.
Hence,NJ¯(X¯,Y¯) is vertical.Corollary 16.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯ be basic vector fields f-related to X,Y, respectively. Let the total space be para-Hermitian manifold and N(1) vanishes. Then, the base space is paracontact pseudometric manifold if and only if ξ is killing.Proof.
Let the total space be para-Hermitian andN(1) vanishes. Then, from (80), we have
(82)0=2dη(ϕX,Y)ξ-2dη(ϕY,X)ξ+2η(X)dη(ξ,Y)ξ-2η(Y)dη(ξ,X)ξ-η(Y)(ℒξϕ)X+η(X)(ℒξϕ)Y.
If ξ is killing, then we have ℒξϕ=0. It immediately follows from (82) that
(83)dη(ϕX,Y)-η(ϕY,X)+η(X)dη(ξ,Y)-η(Y)dη(ξ,X)=0.
In view of (6) and (7), the above equation gives dη=Φ.
Conversely, let the base space be paracontact. Then,dη=Φ.
Using (6), (7), and (82), we get ℒξϕ=0.
Hence, the characteristic vector fieldξ is killing.Theorem 17.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯ be basic vector fields f-related to X,  Y, respectively. If the total space is para-Kähler, then the base space is paracosymplectic. The converse is true if ∇¯X¯J¯ is vertical.Proof.
We have, for anyX,Y∈Γ(TM),(∇Xϕ)Y=0, which gives g(Z,(∇Xϕ)Y)=0, for any Z∈Γ(TM).
From Proposition14, we have
(84){g(Z,(∇Xϕ)Y)+εη(Y)(∇Xη)Z+εη(Z)(∇Xη)Y}∘f=g¯(Z¯,(∇¯X¯J¯)Y¯).
Let ∇¯J¯=0; that is, the total space is para-Kähler. Then, from (84), we obtain ∇ϕ=0 and ∇η=0. Hence, the base space is paracosymplectic.
Again, let(∇Xϕ)Y=0 and ∇η=0. Then, g¯(Z¯,(∇¯X¯J¯)Y¯)=0,  which implies that (∇¯X¯J¯)Y¯ is a vertical vector field.Theorem 18.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let X¯,  Y¯, and Z¯ be basic vector fields f-related to X,  Y, and Z, respectively. If η(X)Z(η(Y))+η(Y)X(η(Z))-η(X)Y(η(Z))=0, then the total space is almost para-Kähler if and only if the base space M is an almost paracosymplectic manifold.Proof.
We have the following equation:(85)3dF(X¯,Y¯,Z¯)=3(f**dΦ)(X¯,Y¯,Z¯)+2εη¯(Z¯)dη¯(X¯,Y¯)-2εη¯(Y¯)dη¯(X¯,Z¯)+2εη¯(X¯)dη¯(Y¯,Z¯)+2εη¯(X¯)Z¯(η¯(Y¯))+2εη¯(Y¯)X¯(η¯(Z¯))-2εη¯(X¯)Y¯(η¯(Z¯)).
If  dη=0,  dΦ=0, and η(X)Z(η(Y))+η(Y)X(η(Z))-η(X)Y(η(Z))=0, then, from (85), we have dF=0. Hence, the total space is almost para-Kähler.
Conversely, letdF=0 and η(X)Z(η(Y))+η(Y)X(η(Z))-η(X)Y(η(Z))=0.
By using the above equation in (85), we have dη=0 and dΦ=0.
Hence, the base space is almost paracosymplectic.Now, we investigate the properties of fundamental tensors𝒯 and 𝒜 of a pseudo-Riemannian submersion.Lemma 19.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, for any horizontal vector fields X¯,  Y¯ and for any vertical vector fields U,  V on M¯, one has (i)
𝒜X¯(J¯Y¯)=J¯(𝒜X¯Y¯),(ii)
𝒜J¯X¯(Y¯)=J¯(𝒜X¯Y¯),(iii)
𝒯U(J¯V)=J¯(𝒯UV),(iv)
𝒯J¯UV=J¯(𝒯UV).Proof.
The proof follows using similar steps as in Lemmas 3 and 4 of [13], so we omit it.Lemma 20.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of  M¯. Then, for any vector fields E,  F on M¯, one has (i)
𝒜E(J¯F)=J¯(𝒜EF),(ii)
𝒯E(J¯F)=J¯(𝒯EF).Proof.
The proof follows from (37) and (38).Theorem 21.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, the horizontal distribution is integrable.Proof.
For any vertical vector fieldU, we have
(86)g¯(J¯(𝒜X¯Y¯),U)=g¯(𝒜X¯J¯Y¯,U)=-g¯(J¯Y¯,𝒜X¯U)=-g¯(J¯Y¯,h(∇¯UX¯))=g¯(Y¯,h(J¯(∇¯UX¯)))=g¯(Y¯,h{(-∇¯UJ¯)X¯hhhhhhhh+∇¯U(J¯X¯)})=g¯(Y¯,h{∇¯U(J¯X¯)})=g¯(Y¯,𝒜J¯X¯U)=-g¯(𝒜J¯X¯Y¯,U)=-g¯(J¯(𝒜X¯Y¯),U).
Thus  g¯(J¯(𝒜X¯Y¯),U)=0, which is true for all X¯ and Y¯.
So,𝒜X¯Y¯=0.
Hence, the horizontal distribution is integrable.Theorem 22.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, the submersion is an affine map on ℋ.Proof.
The second fundamental form off is defined by
(87)(∇¯f*)(E,F)=(∇Eff*(F))∘f-f*(∇¯EF),
where E,F∈Γ(TM¯) and ∇f is pullback connection of Levi-Civita connection ∇ on M with respect to f.
We have, for anyX¯,Y¯∈ℋ,
(88)(∇¯f*)(X¯,Y¯)=(∇X¯ff*(Y¯))∘f-f*(∇¯X¯Y¯).
By using Lemma 1, we have f*(h(∇¯X¯Y¯))=(∇XY)∘f, which implies ∇¯f*=0.
Hence, the submersionf is an affine map on ℋ.Theorem 23.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from a para-Hermitian manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, the submersion is an affine map on 𝒱 if and only if the fibres of f are totally geodesic.Proof.
We have, for anyU,V∈𝒱,
(89)(∇¯f*)(U,V)=-f*(h(∇¯UV)),
which, in view of (27), gives
(90)(∇¯f*)(U,V)=-f*(𝒯UV).
Let the fibres of f be totally geodesic. Then, 𝒯=0. Consequently, from the above equation, we have ∇¯f*=0.
Thus, the mapf is affine on 𝒱.
Conversely, let the submersionf be an affine map on 𝒱. Then, ∇¯f*=0, which implies 𝒯=0.
Hence, the fibres off are totally geodesic.Theorem 24.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from a para-Hermitian manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Then, the submersion is an affine map if and only if h(∇¯EhF)+𝒜hEvF+𝒯vEvF is f-related to ∇XY, for any E,F∈Γ(TM¯).Proof.
For anyE,F∈Γ(TM¯) with f*hE=X∘f and f*vF=Y∘f, we have
(91)(∇¯f*)(E,F)=(∇f*hE(f*hF))∘f-f*(h(∇¯EF))=(∇XY)∘f-f*(h(∇¯hEhF+∇¯hEvFhhhhhhhhhhhhhihhh+∇¯vEhF+∇¯vEhF)).
By using (27) and (31) in the above equation, we have
(92)(∇¯f*)(E,F)=(∇XY)∘f-f*(h(∇¯EhF)+𝒜hEvFhhhhhhhhhihhhhh+𝒯vEvF∇¯).
Let the submersion map be affine. Then, for any E,F∈Γ(TM¯),  (∇¯f*)(E,F)=0. Equation (92) implies (∇XY)∘f=f*(h(∇¯EhF)+𝒜hEvF+𝒯vEhF).
Conversely, leth(∇¯EF)+𝒜hEvF+𝒯vEhF be f-related to ∇XY, for any E,F∈Γ(TM¯). Then, from (92), we have (∇¯f*)(E,F)=0.
Hence, the submersion mapf is affine.


## 4. Curvature Properties

In this section, the paraholomorphic bisectional curvatures and paraholomorphic sectional curvatures of total manifold, base manifold, and fibres of paracomplex paracontact pseudo-Riemannian submersion and their curvature properties are studied.Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold (M¯,J¯,g¯) onto an almost paracontact pseudometric manifold (M,ϕ,ξ,η,g).Suppose that the vector fieldsE,  F span the 2-dimensional plane at point p of M¯ and let ℛ¯ be the Riemannian curvature tensor of M¯. The paraholomorphic bisectional curvature B¯(E,F) of M¯ for any pair of nonzero non-lightlike vector fields E,  F on M¯ is defined by the formula
(93)B¯(E,F)=ℛ¯(E,J¯E,F,J¯F)g¯(E,E)g¯(F,F).For a nonzero non-lightlike vector fieldE, the vector field J¯E is also non-lightlike and {E,J¯E} span the 2-dimensional plane. Then the paraholomorphic sectional curvature H¯(E) is defined as
(94)H¯(E)=B¯(E,E)=ℛ¯(E,J¯E,E,J¯E)g¯(E,E)g¯(E,E).The curvature properties of Riemannian submersion and semi-Riemannian submersion have been extensively studied in the work of O’Neill [1] and Gray [3].LetB¯h and B¯v be the paraholomorphic bisectional curvatures of horizontal and vertical spaces, respectively. Let H¯h and H¯v be the paraholomorphic sectional curvatures of horizontal and vertical spaces, respectively. Let B and H be the paraholomorphic bisectional and sectional curvatures of the base manifold, respectively.Proposition 25.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let U,  V be non-lightlike unit vertical vector fields and let X¯,  Y¯ be non-lightlike unit horizontal vector fields on M¯. Then, one has
(95)B¯(U,V)=B¯v(U,V)+g¯(𝒯U(J¯V),𝒯J¯UV)-g¯(𝒯J¯U(J¯V),𝒯UV),(96)B¯(X¯,U)=g¯((∇¯U𝒜)X¯J¯X¯,J¯U)-g¯(𝒜X¯J¯U,𝒜J¯X¯U)+g¯(𝒜X¯U,𝒜J¯X¯J¯U)-g¯((∇¯J¯U𝒜)X¯J¯X¯,U)+g¯(𝒯J¯UX¯,𝒯U(J¯X¯))-g¯(𝒯UX¯,𝒯J¯U(J¯X¯)),(97)B¯(X¯,Y¯)=B¯h(X¯,Y¯)-2g¯(𝒜X¯(J¯X¯),𝒜Y¯(J¯Y¯))+g¯(𝒜J¯X¯Y¯,𝒜X¯(J¯Y¯))-g¯(𝒜X¯Y¯,𝒜J¯X¯(J¯Y¯)).Proof.
Using Definitions (93) and (94) of paraholomorphic sectional curvature and fundamental equations of submersion obtained by O’Neill [1], we have (95), (96), and (97).Corollary 26.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold. If the fibres of f are totally geodesic pseudo-Riemannian submanifolds of M¯, then for any non-lightlike unit vertical vector fields U and V, one has
(98)B¯(U,V)=B¯v(U,V).Corollary 27.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of f be totally geodesic pseudo-Riemannian submanifolds of M¯. If the horizontal distribution is integrable, then, for any non-lightlike unit horizontal vector fields X¯ and Y¯, one has
(99)B¯(X¯,Y¯)=B¯h(X¯,Y¯).Proposition 28.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of f be pseudo-Riemannian submanifolds of M¯. Let U and X¯ be non-lightlike unit vertical vector field and non-lightlike unit horizontal vector field, respectively. Then, one has
(100)H¯(U)=H¯v(U)+∥𝒯U(J¯U)∥2-g¯(𝒯J¯U(J¯U),𝒯UU),(101)H¯(X¯)=H(X)∘f-3∥𝒜X¯(J¯X¯)∥2.Proof.
The proof is straightforward. If we takeU=V in (95) and X¯=Y¯ in (97), we have (98) and (99).Corollary 29.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold. If the fibres of f are totally geodesic pseudo-Riemannian submanifolds of M¯, then the total manifold and fibres of f have the same paraholomorphic sectional curvatures.Proof.
Since the fibres are totally geodesic,𝒯=0;  consequently we have
(102)H¯(U)=H¯v(U).Corollary 30.
Letf:M¯→M be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of f be totally geodesic pseudo-Riemannian submanifolds of M¯. If the horizontal distribution is integrable, then the base manifold and horizontal distribution have the same paraholomorphic sectional curvatures.Proof.
Since the horizontal distribution is integrable,𝒜=0; consequently, we have
(103)H¯(X¯)=H(X)∘f.Theorem 31.
Letf:M¯m→Mn be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. If U,  V are the non-lightlike unit vertical vector fields and X¯,  Y¯ are the non-lightlike unit horizontal vector fields, then one has
(104)B¯(U,V)=B¯v(U,V),(105)B¯(X¯,U)=-2∥𝒯UX¯∥2,(106)B¯(X¯,Y¯)=B(X,Y)∘f.Proof.
Using results of Lemma19 in (95), we have
(107)B¯(U,V)=B¯v(U,V)-g¯(J¯(𝒯UV),J¯(𝒯UV))-g¯(J¯2(𝒯UV),𝒯UV)=B¯(U,V)+g¯(𝒯UV,𝒯UV)-g¯(𝒯UV,𝒯UV)=B¯v(U,V).
Applying results of Lemma 19 in (96), we have
(108)B¯(X¯,U)=g¯((∇¯U𝒜)X¯(J¯X¯),J¯U)-g¯((∇¯J¯U𝒜)X¯(J¯X¯),U)+2∥𝒜X¯U∥2-2∥𝒯UX¯∥2.
Since by Theorem 21 the horizontal distribution is integrable, we have 𝒜=0, which implies
(109)B¯(X¯,U)=-2∥𝒯UX¯∥2.
In view of 𝒜=0, (104) follows from (97).Theorem 32.
Letf:M¯m→Mn be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M and let the fibres of f be pseudo-Riemannian submanifolds of M¯. If U,  X¯ are non-lightlike unit vertical and non-lightlike unit horizontal vector fields, respectively, then one has
(110)H¯(U)=H¯v(U)-2∥𝒯UU∥2,(111)H¯(X¯)=H(X)∘f.Proof.
Sincef is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold M¯ onto an almost paracontact pseudometric manifold M, by (16) and equations of Lemma 19, we have
(112)g¯(𝒯J¯U(J¯U),𝒯UU)=g¯(J¯2(𝒯UU),𝒯UU)=∥𝒯UU∥2,g¯(𝒯U(J¯U),𝒯U(J¯U))=-g¯(𝒯UU,𝒯UU)=-∥𝒯UU∥2
and by using the above results in (100), we have
(113)H¯(U)=H¯v(U)-∥𝒯UU∥2-∥𝒯UU∥2=H¯v(U)-2∥𝒯UU∥2.
Again, since horizontal distribution is integrable, we have 𝒜=0, and putting it in (101), we obtain (111).



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