Research Article Paracomplex Paracontact Pseudo-Riemannian … · 2016. 9. 7. · paracontact pseudo-Riemannian submersion and let the b res of be pseudo-Riemannian submanifolds of

Research ArticleParacomplex Paracontact Pseudo-Riemannian Submersions

S. S. Shukla and Uma Shankar Verma

Department of Mathematics, University of Allahabad, Allahabad 211002, India

Correspondence should be addressed to Uma Shankar Verma; [email protected]

Received 25 February 2014; Accepted 7 April 2014; Published 7 May 2014

Academic Editor: Bennett Palmer

Copyright © 2014 S. S. Shukla and U. S. Verma.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

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

1. Introduction

The theory of Riemannian submersion was introduced byO’Neill [1, 2] and Gray [3]. It is known that the applications ofsuch Riemannian submersion are extensively used in Kaluza-Klein theories [4, 5], Yang-Mill equations [6, 7], the theory ofrobotics [8], and supergravity and superstring theories [5, 9].

There is detailed literature on the Riemannian submer-sion with suitable smooth surjective map followed by differ-ent conditions applied to total space and on the fibres of sur-jective map. The Riemannian submersions between almostHermitian manifolds have been studied by Watson [10]. TheRiemannian submersions between almost contact manifoldswere studied by Chinea [11]. He also concluded that if 𝑀 isan almost Hermitian manifold with structure (𝐽, 𝑔) and𝑀 isan almost contact metric manifold with structure (𝜙, 𝜉, 𝜂, 𝑔),then there does not exist a Riemannian submersion𝑓 : 𝑀 →𝑀 which commutes with the structures on 𝑀 and 𝑀; thatis, we cannot have the condition 𝑓

∗∘ 𝐽 = 𝜙 ∘ 𝑓

∗. Chinea

also defined the Riemannian submersion between almostcomplexmanifolds and almost contactmanifolds and studiedsome properties and interrelations between them [12]. In[13], Gündüzalp and Sahin gave the concept of paracontactparacomplex semi-Riemannian submersion between almostparacontact metric manifolds and almost para-Hermitianmanifolds submersion giving an example and studied somegeometric properties of such submersions.

An almost paracontact structure on a differentiable man-ifold was introduced by Sato [14], which is an analogue ofan almost contact structure and is closely related to almostproduct structure. An almost contact manifold is always odddimensional but an almost paracontact manifold could beeven dimensional as well.

The paracomplex geometry has been studied since thefirst papers by Rashevskij [15], Libermann [16], and Patterson[17] until now, from several different points of view. Thesubject has applications to several topics such as negativelycurved manifolds, mechanics, elliptic geometry, and pseudo-Riemannian space forms. Paracomplex and paracontactgeometries are topics with many analogies and also withdifferences with complex and contact geometries.

This motivated us to study the pseudo-Riemannian sub-mersion between pseudo-Riemannian manifolds equippedwith paracomplex and paracontact structures.

In this paper, we give the notion of paracomplex paracon-tact pseudo-Riemannian submersion between almost para-complex manifolds and almost paracontact pseudometricmanifolds giving some examples and study the geometricproperties and interrelations under such submersions.

The composition of the paper is as follows. In Section 2,we collect some basic definitions, formulas, and results onalmost paracomplex manifolds, almost paracontact pseu-dometric manifolds, and pseudo-Riemannian submersion.

Hindawi Publishing CorporationGeometryVolume 2014, Article ID 616487, 12 pageshttp://dx.doi.org/10.1155/2014/616487

2 Geometry

In Section 3, we define paracomplex paracontact pseudo-Riemannian submersion giving some relevant examples andinvestigate transference of structures on the total manifoldsand 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. Let 𝑀 be a (2𝑛 + 1)-dimensional Riemannianmanifold, 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 paracontactstructure on𝑀 if

𝜙2

𝑋 = 𝑋 − 𝜂 (𝑋) 𝜉; 𝜂 (𝜉) = 1, (1)

and the tensor field 𝜙 induces an almost paracomplexstructure on the distributionD = ker(𝜂) [18, 19].

𝑀 is said to be an almost paracontact manifold, if it isequipped with an almost paracontact structure. Again,𝑀 iscalled an almost paracontact pseudometric manifold if it isendowed with a pseudo-Riemannian metric 𝑔 of signature(−, −, −, . . . , −⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(𝑛-times), +, +, +, . . . , +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(𝑛+1)-times)) such that

𝑔 (𝜙𝑋, 𝜙𝑌) = 𝑔 (𝑋, 𝑌) − 𝜀𝜂 (𝑋) 𝜂 (𝑌) , ∀𝑋, 𝑌 ∈ Γ (𝑇𝑀) ,

(2)

where 𝜀 = 1 or −1 according to the characteristic vector field𝜉 is spacelike or timelike. It follows that

𝑔 (𝜉, 𝜉) = 𝜀, (3)𝑔 (𝜉, 𝑋) = 𝜀𝜂 (𝑋) , (4)

𝑔 (𝑋, 𝜙𝑌) = 𝑔 (𝜙𝑋, 𝑌) , ∀𝑋, 𝑌 ∈ Γ (𝑇𝑀) . (5)

In particular, if 𝑖𝑛𝑑𝑒𝑥(𝑔) = 1, then the manifold(𝑀2𝑛+1

, 𝜙, 𝜉, 𝜂, 𝑔, 𝜀) is called a Lorentzian almost paracontactmanifold.

If the metric 𝑔 is positive definite, then the manifold(𝑀2𝑛+1

, 𝜙, 𝜉, 𝜂, 𝑔) is the usual almost paracontact metricmanifold [14].

The fundamental 2-formΦ on𝑀 is defined by

Φ (𝑋, 𝑌) = 𝑔 (𝑋, 𝜙𝑌) . (6)

Let 𝑀2𝑛+1 be an almost paracontact manifold with thestructure (𝜙, 𝜉, 𝜂). An almost paracomplex structure 𝐽 on𝑀2𝑛+1

×R1 is defined by

𝐽 (𝑋, 𝑓𝑑

𝑑𝑡) = (𝜙𝑋 + 𝑓𝜉, 𝜂 (𝑋)

𝑑

𝑑𝑡) , (7)

where𝑋 is tangent to𝑀2𝑛+1, 𝑡 is the coordinate onR1, and 𝑓is a smooth function on𝑀2𝑛+1.

An almost paracontact structure (𝜙, 𝜉, 𝜂) is said to benormal, if the Nijenhuis tensor 𝑁

𝐽of almost paracomplex

structure 𝐽 defined as

𝑁𝐽(𝑋, 𝑌) = [𝐽, 𝐽] (𝑋, 𝑌) = [𝐽𝑋, 𝐽𝑌] + 𝐽

2

[𝑋, 𝑌]

− 𝐽 [𝐽𝑋, 𝑌] − 𝐽 [𝑋, 𝐽𝑌] ,(8)

for any vector fields𝑋,𝑌 ∈ Γ(𝑇𝑀), vanishes.

If 𝑋 and 𝑌 are vector fields on𝑀2𝑛+1, then we have [18–20]

𝑁𝐽((𝑋, 0) , (𝑌, 0))

= (𝑁𝜙(𝑋, 𝑌) − 2𝑑𝜂 (𝑋, 𝑌) 𝜉,

{(L𝜙𝑋𝜂)𝑌 − (L

𝜙𝑌𝜂)𝑋}

𝑑

𝑑𝑡) ,

(9)

𝑁𝐽((𝑋, 0) , (0,

𝑑

𝑑𝑡)) = −((L

𝜉𝜙)𝑋, ((L

𝜉𝜂)𝑋)

𝑑

𝑑𝑡) ,

(10)

where 𝑁𝜙is Nijenhuis tensor of 𝜙,L

𝑋is Lie derivative with

respect to a vector field𝑋, and𝑁(1), 𝑁(2), 𝑁(3), and 𝑁(4) aredefined as

𝑁𝜙(𝑋, 𝑌)

= [𝜙, 𝜙] (𝑋, 𝑌)

= [𝜙𝑋, 𝜙𝑌] + 𝜙2

[𝑋, 𝑌] − 𝜙 [𝜙𝑋, 𝑌] − 𝜙 [𝑋, 𝜙𝑌] ,

(11)

𝑁(1)

(𝑋, 𝑌) = 𝑁𝜙(𝑋, 𝑌) − 2𝑑𝜂 (𝑋, 𝑌) 𝜉, (12)

𝑁(2)

(𝑋, 𝑌) = (L𝜙𝑋𝜂)𝑌 − (L

𝜙𝑌𝜂)𝑋, (13)

𝑁(3)

(𝑋) = (L𝜉𝜙)𝑋, (14)

𝑁(4)

(𝑋) = (L𝜉𝜂)𝑋. (15)

The almost paracontact structure (𝜙, 𝜉, 𝜂) is normal if andonly if the four tensors𝑁(1), 𝑁(2), 𝑁(3), and 𝑁(4) vanish.

For an almost paracontact structure (𝜙, 𝜉, 𝜂), vanishing of𝑁(1) implies the vanishing of𝑁(2), 𝑁(3), and𝑁(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, if Φ = 𝑑𝜂,(iii) 𝐾-paracontact, if𝑀 is paracontact and 𝜉 is killing,(iv) paracosymplectic, if ∇Φ = 0, which implies ∇𝜂 = 0,

where ∇ is the Levi-Civita connection on𝑀,(v) almost paracosymplectic, if 𝑑𝜂 = 0 and 𝑑Φ = 0,(vi) weakly paracosymplectic, if𝑀 is almost paracosym-

plectic and [𝑅(𝑋, 𝑌), 𝜙] = 𝑅(𝑋, 𝑌)𝜙 − 𝜙𝑅(𝑋, 𝑌) = 0,where 𝑅 is Riemannian curvature tensor,

(vii) para-Sasakian, if Φ = 𝑑𝜂 and𝑀 is normal,(viii) quasi-para-Sasakian, if 𝑑𝜙 = 0 and𝑀 is normal.

2.2. Almost Paracomplex Manifolds. A (1, 1)-type tensor field𝐽 on 2𝑚-dimensional smooth manifold 𝑀 is said to be analmost paracomplex structure if 𝐽2 = 𝐼 and (𝑀2𝑚, 𝐽) is calledalmost paracomplex manifold.

Geometry 3

An almost paracomplex manifold (𝑀, 𝐽) is such thatthe two eigenbundles 𝑇+𝑀 and 𝑇−𝑀 corresponding torespective eigenvalues +1 and −1 of 𝐽 have the same rank[21, 22].

An almost para-Hermitianmanifold (𝑀, 𝐽, 𝑔) is a smoothmanifold endowed with an almost paracomplex structure 𝐽and a pseudo-Riemannian metric 𝑔 such that

𝑔 (𝐽𝑋, 𝐽𝑌) = −𝑔 (𝑋, 𝑌) , ∀𝑋, 𝑌 ∈ Γ (𝑇𝑀) . (16)

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

𝐹 (𝑋, 𝑌) = 𝑔 (𝑋, 𝐽𝑌) . (17)

We have the following properties [21, 22]:

𝑔 (𝐽𝑋, 𝑌) = −𝑔 (𝑋, 𝐽𝑌) , (18)

𝐹 (𝑋, 𝑌) = −𝐹 (𝑌,𝑋) , (19)

𝐹 (𝐽𝑋, 𝐽𝑌) = −𝐹 (𝑋, 𝑌) , (20)

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) ,

(21)

(∇𝑋𝐹) (𝑌, 𝑍) = 𝑔 (𝑌, (∇

𝑋𝐽) 𝑍) = −𝑔 (𝑍, (∇

𝑋𝐽) 𝑌) , (22)

3𝑑𝐹 (𝑋, 𝑌, 𝑍) = (∇𝑋𝐹) (𝑌, 𝑍) + (∇

𝑌𝐹) (𝑍,𝑋)

+ (∇𝑍𝐹) (𝑋, 𝑌) ,

(23)

the co-differential, (𝛿𝐹) (𝑋) =2𝑚

∑

𝑖=1

𝜀𝑖(∇𝑒𝑖𝐹) (𝑒𝑖, 𝑋) . (24)

An almost para-Hermitian manifold is called

(i) para-Hermitian, if 𝑁𝐽= 0; equivalently, (∇

𝐽𝑋𝐽)𝐽𝑌 +

(∇𝑋𝐽)𝑌 = 0,

(ii) para-Kähler, if, for any 𝑋 ∈ Γ(𝑇𝑀), ∇𝑋𝐽 = 0; that is,

∇𝐽 = 0,(iii) almost para-Kähler, if 𝑑𝐹 = 0,(iv) nearly para-Kähler, if (∇

𝑋𝐽)𝑋 = 0,

(v) almost semi-para-Kähler, if 𝛿𝐹 = 0,(vi) semi-para-Kähler, if 𝛿𝐹 = 0 and𝑁

𝐽= 0.

2.3. Pseudo-Riemannian Submersion. Let (𝑀𝑚, 𝑔) and(𝑀𝑛

, 𝑔) be two connected pseudo-Riemannian manifolds ofindices 𝑠 (0 ≤ 𝑠 ≤ 𝑚) and 𝑠 (0 ≤ 𝑠 ≤ 𝑛), respectively, with𝑠 ≥ 𝑠.

A pseudo-Riemannian submersion is a smooth map 𝑓 :𝑀𝑚

→ 𝑀𝑛, which is onto and satisfies the following

conditions [2, 3, 23, 24].

(i) The derivative map 𝑓∗𝑝

: 𝑇𝑝𝑀 → 𝑇

𝑓(𝑝)𝑀 is

surjective at each point 𝑝 ∈ 𝑀.

(ii) The fibres 𝑓−1(𝑞) of 𝑓 over 𝑞 ∈ 𝑀 are eitherpseudo-Riemannian submanifolds of 𝑀 ofdimension (𝑚 − 𝑛) and index ] or the degeneratesubmanifolds of 𝑀 of dimension (𝑚 − 𝑛) andindex ] with degenerate metric 𝑔

|𝑓−1(𝑞)

of type(0, 0, 0, . . . , 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝜇-times, −, −, −, . . . , −⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

]-times, +, +, +, . . . , +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(𝑚−𝑛−𝜇−])-times)), where

𝜇 = dim(V𝑝∩H𝑝) and ] = 𝑠 − 𝑠 = index of 𝑔

|𝑓−1(𝑞)

.

(iii) 𝑓∗preserves the length of horizontal vectors.

We denote the vertical and horizontal projections of avector field 𝐸 on 𝑀 by 𝐸V (or by V𝐸) and 𝐸ℎ (or by ℎ𝐸),respectively. A horizontal vector field 𝑋 on 𝑀 is said to bebasic if 𝑋 is 𝑓-related to a vector field 𝑋 on𝑀. Thus, everyvector field𝑋 on𝑀 has a unique horizontal lift𝑋 on𝑀.

Lemma 1 (see [1, 23]). If 𝑓 : 𝑀 → 𝑀 is a pseudo-Riemannian submersion and𝑋, 𝑌 are basic vector fields on𝑀that are 𝑓-related to the vector fields 𝑋, 𝑌 on𝑀, 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(∇

𝑋𝑌) is a basic vector field 𝑓-related to ∇

𝑋𝑌, where

∇ and ∇ are the Levi-Civita connections on𝑀 and𝑀,respectively,

(iv) [𝐸, 𝑈] ∈ V, for any vector field 𝑈 ∈ V and for anyvector field 𝐸 ∈ Γ(𝑇𝑀).

A pseudo-Riemannian submersion 𝑓 : 𝑀 → 𝑀determines tensor fieldsT andA of type (1, 2) on𝑀 definedby formulas [1, 2, 23]

T (𝐸, 𝐹) = T𝐸𝐹 = ℎ (∇V𝐸V𝐹) + V (∇V𝐸ℎ𝐹) , (25)

A (𝐸, 𝐹) = A𝐸𝐹 = V (∇

ℎ𝐸ℎ𝐹) + ℎ (∇

ℎ𝐸V𝐹) ,

for any 𝐸, 𝐹 ∈ Γ (𝑇𝑀) .(26)

Let 𝑋, 𝑌 be horizontal vector fields and let 𝑈, 𝑉 bevertical vector fields on𝑀. Then, one has

T𝑈𝑋 = V (∇

𝑈𝑋) , T

𝑈𝑉 = ℎ (∇

𝑈𝑉) , (27)

∇𝑈𝑋 = T

𝑈𝑋 + ℎ (∇

𝑈𝑋) , (28)

T𝑋𝐹 = 0, T

𝐸𝐹 = TV𝐸𝐹, (29)

∇𝑈𝑉 = T

𝑈𝑉 + V (∇

𝑈𝑉) , (30)

A𝑋𝑌 = V (∇

𝑋𝑌) , A

𝑋𝑈 = ℎ (∇

𝑋𝑈) , (31)

∇𝑋𝑈 = A

𝑋𝑈 + V (∇

𝑋𝑈) , (32)

A𝑈𝐹 = 0, A

𝐸𝐹 = A

ℎ𝐸𝐹, (33)

∇𝑋𝑌 = A

𝑋𝑌 + ℎ (∇

𝑋𝑌) , (34)

4 Geometry

ℎ (∇𝑈𝑋) = ℎ (∇

𝑋𝑈) = A

𝑋𝑈, (35)

A𝑋𝑌 =

1

2V [𝑋, 𝑌] , (36)

A𝑋𝑌 = −A

𝑌𝑋, (37)

T𝑈𝑉 = T

𝑉𝑈, (38)

for all 𝐸, 𝐹 ∈ Γ(𝑇𝑀).Moreover,T

𝑈𝑉 coincideswith second fundamental form

of the submersion of the fibre submanifolds.The distributionH is completely integrable. In view of (37) and (38), A isalternating on the horizontal distribution andT is symmetricon the vertical distribution.

3. Paracomplex Paracontact Pseudo-Riemannian Submersions

In this section, we introduce the notion of pseudo-Riemannian submersion from almost paracomplex mani-folds onto almost paracontact pseudometric manifolds, illus-trate examples, and study the transference of structures ontotal manifolds and base manifolds.

Definition 2. Let (𝑀2𝑚, 𝐽, 𝑔) be an almost para-Hermitianmanifold and let (𝑀2𝑛+1, 𝜙, 𝜉, 𝜂, 𝑔) be an almost paracontactpseudometric manifold.

A pseudo-Riemannian submersion 𝑓 : 𝑀 → 𝑀 is calledparacomplex paracontact pseudo-Riemannian submersion ifthere exists a 1-form 𝜂 on𝑀 such that

𝑓∗∘ 𝐽 = 𝜙 ∘ 𝑓

∗+ 𝜂 ⊗ 𝜉. (39)

Since, for each 𝑝 ∈ 𝑀,𝑓∗𝑝

is a linear isometry betweenhorizontal spacesH

𝑝and tangent spaces𝑇

𝑓(𝑝)𝑀, there exists

an induced almost paracontact structure (𝜙ℎ

, 𝜂ℎ

, 𝜉ℎ

, 𝑔) on(2𝑛 + 1)-dimensional horizontal distribution H such that𝜙ℎ

|

Dℎbehave just like the fundamental collineation of almost

paracomplex structure 𝐽 on ker 𝜂ℎ = Dℎ

and 𝜙ℎ

: Dℎ

→ Dℎ

is an endomorphism such that 𝜙ℎ

= 𝐽|ker 𝜂ℎ

and the rank of

𝜙ℎ

= 2𝑛, where dim(Dℎ

) = 2𝑛.It follows that, for any 𝑋ℎ ∈ D

ℎ

, 𝜂ℎ(𝑋ℎ) = 0, whichimplies that 𝐽2

|

Dℎ(𝑋ℎ

) = (𝜙ℎ

)2

(𝑋ℎ

) = 𝑋ℎ, for any 𝑋ℎ ∈ D

ℎ

andH = Dℎ

⊕ {𝜉ℎ

} [18].

Definition 3 (see [25]). A pseudo-Riemannian submersion𝑓 : 𝑀 → 𝑀 is called semi-𝐽-invariant submersion, if thereis a distributionD

1⊆ ker𝑓

∗such that

ker𝑓∗= D1⊕D2, (40)

𝐽 (D1) = D

1, 𝐽 (D

2) ⊆ (ker𝑓

∗)⊥

, (41)

whereD2is orthogonal complementary toD

1in ker𝑓

∗.

Proposition 4. Let𝑓 : 𝑀2𝑚 → 𝑀2𝑛+1 be a paracomplex par-acontact pseudo-Riemannian submersion and let the fibres of𝑓 be pseudo-Riemannian submanifolds of𝑀. Then, the fibres𝑓−1

(𝑞), 𝑞 ∈ 𝑀, are semi-𝐽-invariant submanifolds of 𝑀 ofdimension (2𝑚 − 2𝑛 − 1).

Proof. Let 𝑈 ∈V. Then

𝑓∗(𝐽𝑈) = 𝜙 (𝑓

∗(𝑈)) + 𝜂 (𝑈) 𝜉,

⇒ 𝑓∗{𝐽 (𝑈) − 𝜂 (𝑈) 𝜉

ℎ

} = 0,

(42)

where 𝑓∗𝜉ℎ

= 𝜉.Thus, we have

𝐽 (𝑈) − 𝜂 (𝑈) 𝜉ℎ

= 𝜙 (𝑈) , for some 𝜙 (𝑈) ∈V. (43)

By (19), we get 𝑔(𝜉ℎ

, 𝐽(𝜉ℎ

)) = 0 = 𝑔(𝜉, 𝑓∗(𝐽(𝜉ℎ

))) = 0.As 𝑔 is nondegenerate on𝑀, we have

𝑓∗(𝐽 (𝜉ℎ

)) = 0, that is 𝐽 (𝜉ℎ

) ∈V. (44)

Taking 𝑈 = 𝐽𝜉ℎ

in (43), we obtain

𝜉ℎ

− 𝜂 (𝐽𝜉ℎ

) 𝜉ℎ

= 𝜙(𝐽𝜉ℎ

) . (45)

Since fibre 𝑓−1(𝑞) is an odd dimensional submanifold, thereexists an associated 1-form 𝜂V which is restriction of 𝜂 on fibresubmanifold 𝑓−1(𝑞), 𝑞 ∈ 𝑀, and a characteristic vector field𝜉V= 𝐽𝜉ℎ

such that 𝜙(𝜉V) = 0. So, we have 𝜂V(𝜉

V) = 1.

Let us put ker 𝜂V = D1andD

2= {𝜉

V}.

Then, ker𝑓∗= D1⊕ D2and 𝐽(D

1) = D

1, 𝐽(D

2) =

𝐽{𝜉V} = {𝜉

ℎ

} ⊆ (ker𝑓∗)⊥.

Hence, the fibres 𝑓−1(𝑞) are semi-𝐽-invariant submani-folds of𝑀.

Corollary 5. Let 𝑓 : 𝑀2𝑚 → 𝑀2𝑛+1 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 𝑓 be pseudo-Riemannian submanifolds of 𝑀. Then, thefibres 𝑓−1(𝑞) are almost paracontact pseudometric mani-folds with almost paracontact pseudo-Riemannian structures(𝜙

V, 𝜉

V, 𝜂

V, 𝑔

V), 𝑞 ∈ 𝑀, where 𝜉

V= 𝐽(𝜉

ℎ

), 𝜂V = 𝜂|V, and

𝑔V= 𝑔.

Proof. Since 𝑓−1(𝑞) are semi-𝐽-invariant submanifolds of𝑀of odd dimension 2𝑟 + 1 = 2𝑚 − 2𝑛 − 1, (39) implies

𝐽 (𝑈) = 𝜙V𝑈 + 𝜂

V(𝑈) 𝜉ℎ

, (46)

for any 𝑈 ∈V.

Geometry 5

On operating 𝐽 on both sides of the above equation, weget

𝑈 = 𝜙V(𝜙

V(𝑈)) + 𝜂

V(𝜙

V(𝑈)) 𝜉

ℎ

+ 𝜂V(𝑈) 𝜉

V, (47)

where 𝐽(𝜉ℎ

) = 𝜉V.

Equating horizontal and vertical components, we have

𝑈 = 𝜙V(𝜙

V(𝑈)) + 𝜂

V(𝑈) 𝜉

V, 𝜂

V∘ 𝜙

V(𝑈) = 0,

⇒ (𝜙V)2

(𝑈) = 𝑈 − 𝜂V(𝑈) 𝜉

V; 𝜂

V∘ 𝜙

V= 0;

𝜙V(𝜉

V) = 0, 𝜂

V(𝜉

V) = 1.

(48)

Hence, (𝜙V, 𝜉

V, 𝜂

V, 𝑔

V) is almost paracontact pseudometric

structure on the fibre 𝑓−1(𝑞), 𝑞 ∈ 𝑀.

Proposition 6. Let 𝑓 : 𝑀2𝑚 → 𝑀2𝑛+1 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 𝑓 be pseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌 bebasic vector fields 𝑓-related to 𝑋, 𝑌, respectively. Let 𝜂 and 𝜂be 1-forms on the total manifold𝑀 and the base manifold𝑀,respectively. Then, one has the following.

(i) The characteristic vector field 𝐽𝜉ℎ

is a vertical vectorfield.

(ii) 𝑓∗∗𝜂 = 𝜂ℎ, where 𝑓∗

∗𝜂 is pullback of 𝜂 through 𝑓

∗.

(iii) 𝜂ℎ(𝑈) = 0, for any vertical vector field 𝑈.(iv) 𝜂V(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 Corollary 5, (𝜙V, 𝜉

V, 𝜂

V, 𝑔

V) is almost paracontact

pseudometric structure on 𝑓−1(𝑞). We have

0 = 𝑔 (𝜉ℎ

, 𝐽𝜉ℎ

) = 𝑔(𝑓∗(𝜉ℎ

) , 𝑓∗(𝐽𝜉ℎ

))

= 𝑔(𝜉, 𝑓∗(𝐽𝜉ℎ

)) .

(49)

Now,

𝑓∗(𝐽𝜉ℎ

) = 𝜙 ∘ 𝑓∗𝜉ℎ

+ 𝜂 (𝜉ℎ

) 𝜉 = 𝜂 (𝜉ℎ

) 𝜉, (50)

so we have

0 = 𝑔(𝜉, 𝜂 (𝜉ℎ

) 𝜉) = 𝜂 (𝜉ℎ

)𝑔 (𝜉, 𝜉) = 𝜂 (𝜉ℎ

) . (51)

Thus, 𝑓∗(𝐽𝜉ℎ

) = 0.Hence, 𝐽𝜉

ℎ

is a vertical vector field.(ii) Since 𝑓 : 𝑀2𝑚 → 𝑀2𝑛+1 is smooth submersion,

𝜂ℎ

= 𝜂|H

is restriction of 𝜂 on the horizontal distribution

H, and 𝑓∗𝑝

: H𝑝→ 𝑇𝑓(𝑝)

𝑀 is a linear isometry, for any𝑋𝑝∈H𝑝, we get

𝜂ℎ

𝑝(𝑋𝑝) = 𝜀𝑔

𝑝(𝜉ℎ

𝑝, 𝑋𝑝) = 𝑔

𝑓(𝑝)(𝑓∗𝑝𝜉ℎ

𝑝, 𝑓∗𝑝𝑋𝑝)

= 𝑔𝑓(𝑝)

(𝜉𝑓(𝑝)

, 𝑋𝑓(𝑝)

) = 𝜂𝑓(𝑝)

(𝑋𝑓(𝑝)

) = 𝑓∗

∗𝜂𝑝(𝑋𝑝) .

(52)

Hence, pullback 𝑓∗∗𝜂 = 𝜂ℎ.

Results (iii) and (iv) immediately follow from the previ-ous results.

Example 8. Let (R42, 𝐽, 𝑔) be a paracomplex pseudometric

manifold and let (R31, 𝜙, 𝜉, 𝜂, 𝑔) be an almost paracontact

pseudometric manifold.Define a submersion 𝑓 : {R4

2; (𝑥1, 𝑥2, 𝑦1, 𝑦2)𝑡

} →

{R31; (𝑢, V, 𝑤)𝑡} by

𝑓 ((𝑥1, 𝑥2, 𝑦1, 𝑦2)𝑡

) → (𝑥1+ 𝑥2+ 3𝑦1+ 2𝑦2,

3𝑥1+ 2𝑥2+ 𝑦1+ 𝑦2,

5𝑥1+ 3𝑥2+ 5𝑦1+ 3𝑦2)𝑡

.

(53)

Then, the kernel of 𝑓∗is

V = ker𝑓∗= Span{𝑉

1=

𝜕

𝜕𝑥1

− 2𝜕

𝜕𝑥2

−𝜕

𝜕𝑦1

+ 2𝜕

𝜕𝑦2

} ,

(54)

which is the vertical distribution admitting one lightlikevector field; that is, fibre is degenerate submanifold of R4

2.

The horizontal distribution is

H = (ker𝑓∗)⊥

= Span{𝑋1=

𝜕

𝜕𝑥1

−𝜕

𝜕𝑦1

, 𝑋2=

𝜕

𝜕𝑥2

+ 2𝜕

𝜕𝑦1

,

𝑋3= 2

𝜕

𝜕𝑦1

+𝜕

𝜕𝑦2

} .

(55)

For any real 𝑘, the horizontal characteristic vector field 𝜉ℎ

isgiven by

𝜉ℎ

= 𝑘𝜕

𝜕𝑥1

− (2𝑘 −1

3)

𝜕

𝜕𝑥2

− (𝑘 − 1)𝜕

𝜕𝑦1

+ (2𝑘 −5

3)

𝜕

𝜕𝑦2

,

(56)

which is 𝑓-related to the characteristic vector field 𝜉 = 𝜕/𝜕𝑤.Moreover, there exists one form 𝜂 = 5𝑑𝑥

1+3𝑑𝑥

2+5𝑑𝑦1+

3𝑑𝑦2on (R4

2, 𝐽, 𝑔) such that the submersion satisfies (39).

Example 9. Let (R63, 𝐽, 𝑔) be an almost paracomplex pseudo-

Riemannian manifold and let (R31, 𝜙, 𝜉, 𝜂, 𝑔) be an almost

6 Geometry

paracontact pseudo-Riemannian manifold. Consider a sub-mersion 𝑓 : {R6

3; (𝑥1, 𝑥2, 𝑥3, 𝑦1, 𝑦2, 𝑦3)𝑡

} → {R31; (𝑢, V, 𝑤)𝑡},

defined by

𝑓 ((𝑥1, 𝑥2, 𝑥3, 𝑦1, 𝑦2, 𝑦3)𝑡

)

→ (𝑥1+ 𝑥2

√2

,𝑦1+ 𝑦2

√2

,𝑦2+ 𝑦3

√2

)

𝑡

.

(57)

Then, there exists one form 𝜂 = (𝑑𝑥2+ 𝑑𝑥3)/√2 on (R6

3, 𝐽, 𝑔)

such that (39) is satisfied. The kernel of 𝑓∗is

V = ker𝑓∗

= Span{𝑉1=

𝜕

𝜕𝑥1

−𝜕

𝜕𝑥2

, 𝑉2=

𝜕

𝜕𝑦1

−𝜕

𝜕𝑦2

+𝜕

𝜕𝑦3

,

𝑉3=

𝜕

𝜕𝑥3

} ,

(58)

which is vertical distribution admitting non-lightlike vectorfields; that is, the fibre is nondegenerate submanifold of(R63, 𝐽, 𝑔).The horizontal distribution is

H = Span{𝑋1=

𝜕

𝜕𝑥1

+𝜕

𝜕𝑥2

, 𝑋2= −

𝜕

𝜕𝑦1

+𝜕

𝜕𝑦3

,

𝑋3=

𝜕

𝜕𝑦1

+𝜕

𝜕𝑦2

} .

(59)

Example 10. Let (R42, 𝐽, 𝑔) be a paracomplex pseudometric

manifold and let (R31, 𝜙, 𝜉, 𝜂, 𝑔) be an almost paracontact

pseudometric manifold.Consider a submersion 𝑓 : {R4

2; (𝑥1, 𝑥2, 𝑦1, 𝑦2)𝑡

} → {R31;

(𝑢, V, 𝑤)𝑡}, defined by

𝑓 ((𝑥1, 𝑥2, 𝑦1, 𝑦2)𝑡

) → (𝑥1, 𝑦1, 𝑦2)𝑡

. (60)


V = ker𝑓∗= Span{𝑉

1=

𝜕

𝜕𝑥2

} , (61)

which is the vertical distribution and the restriction of 𝑔 tothe fibres of 𝑓 is nondegenerate.


H = (ker𝑓∗)⊥

= Span{𝑋 = 𝜕𝜕𝑥1

, 𝑌 =𝜕

𝜕𝑦1

, 𝜉ℎ

=𝜕

𝜕𝑦2

} .

(62)

The characteristic vector field 𝜉 = 𝜕/𝜕𝑤 on R31has unique

horizontal lift 𝜉ℎ

, which is the characteristic vector field onhorizontal distributionH of R4

2.

We also have

𝑔 (𝑋,𝑋) = 𝑔 (𝑓∗𝑋,𝑓∗𝑋) = −1,

𝑔 (𝑌, 𝑌) = 𝑔 (𝑓∗𝑌, 𝑓∗𝑌) = 1,

𝑔 (𝜉ℎ

, 𝜉ℎ

) = 𝑔(𝑓∗𝜉ℎ

, 𝑓∗𝜉ℎ

) = 𝑔 (𝜉, 𝜉) = 1.

(63)

Thus, the smooth map 𝑓 is a pseudo-Riemannian submer-sion.

Moreover, we obtain that there exists a 1-form 𝜂 = 𝑑𝑥2on

R42such that 𝜂(𝐽𝜉

ℎ

) = 1, 𝜂(𝜉ℎ

) = 0 and the map 𝑓 satisfies

𝑓∗𝐽𝑋 = 𝜙𝑓

∗𝑋 + 𝜂 (𝑋) 𝜉,

𝑓∗𝐽𝑌 = 𝜙𝑓

∗𝑌 + 𝜂 (𝑌) 𝜉,

𝑓∗𝐽𝜉ℎ

= 𝜙𝑓∗𝜉ℎ

+ 𝜂 (𝜉ℎ

) 𝜉.

(64)

Hence, the map 𝑓 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1.

Proposition 11. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 𝑓 be pseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌 bebasic vector fields 𝑓-related to𝑋, 𝑌, respectively. Then, 𝐽(𝑋) −𝜀𝑔(𝑋, 𝜉

ℎ

)𝜉ℎ

is 𝑓-related to 𝜙𝑋.

Proof. Since𝑋 is 𝑓-related to vector field𝑋 on𝑀, we have

𝜂 (𝑋) = {𝜂V+ 𝜂ℎ

} (𝑋) = 0 + 𝜂ℎ

(𝑋) = 𝜀𝑔 (𝑋, 𝜉ℎ

) ,

⇒ 𝑓∗(𝐽𝑋) = 𝜙𝑋 + 𝜂

ℎ

(𝑋) 𝜉,

⇒ 𝑓∗{𝐽𝑋 − 𝜀𝑔 (𝑋, 𝜉

ℎ

) 𝜉ℎ

} = 𝜙𝑋.

(65)

Hence, 𝐽(𝑋) − 𝜀𝑔(𝑋, 𝜉ℎ

)𝜉ℎ

is 𝑓-related to 𝜙𝑋.

Proposition 12. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 𝑓be pseudo-Riemannian submanifolds of 𝑀. Let V and H bethe vertical and horizontal distributions, respectively. If 𝜉

ℎ

isthe basic characteristic vector field of horizontal distribution𝑓-related to the characteristic vector field 𝜉 of base manifold,then

(i) 𝐽V ⊂ DV⊕ {𝐽𝜉ℎ

} ⊕ {𝜉ℎ

},

(ii) 𝐽H ⊂ Dℎ

⊕ {𝜉ℎ

} ⊕ {𝐽𝜉ℎ

}.

Proof. (i) Let 𝑈 ∈ V. Then, 𝑈 = 𝑎𝑈|DV + 𝑏𝐽𝜉

ℎ

, for 𝑎, 𝑏 ∈

𝐶∞

(𝑀), as 𝐽𝜉ℎ

= 𝜉Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 𝑓−1(𝑞) of𝑀, 𝑞 ∈ 𝑀. We get

𝐽𝑈 = 𝑎𝐽𝑈|DV + 𝑏𝐽

2

𝜉ℎ

= 𝑎𝐽𝑈|DV + 𝑏𝜉

ℎ

∈V ⊕ {𝜉ℎ

} ,

⇒ 𝐽V ⊂V ⊕ {𝜉ℎ

} .

(66)

Again, let 𝑉 ∈V ⊕ {𝜉ℎ

}. Then 𝑉 = 𝑎𝑉|DV + 𝑏𝐽𝜉

ℎ

+ 𝑐𝜉ℎ

, where

𝜂V(𝑉|DV ) = 0, D

V= ker 𝜂V, 𝑎𝑉

|DV + 𝑏𝐽𝜉

ℎ

∈ V, and 𝑎, 𝑏, 𝑐 ∈𝐶∞

(𝑀). We have

𝐽𝑉 = 𝑎𝐽𝑉|DV + 𝑏𝜉

ℎ

+ 𝑐𝐽𝜉ℎ

= (𝑎𝐽𝑉|DV + 𝑐𝐽𝜉

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

∈V

+ 𝑏𝜉ℎ

⏟⏟⏟⏟⏟⏟⏟

∈{𝜉

ℎ

}

∈V ⊕ {𝜉ℎ

} .(67)

Now, by (39), we get

𝑓∗𝐽𝑉 = 𝜙 (𝑓

∗𝑉) + 𝜂 (𝑓

∗(𝑉)) 𝜉

= 𝑐 {𝜙 (𝑓∗𝜉ℎ

) + 𝜂 (𝜉) 𝜉}

= 𝑐𝜉 ∈ {𝜉} ̸⊆ 𝐽V.

(68)

We get 𝐽𝑉 ∉V.Hence, 𝐽V ⊂V ⊕ {𝜉

ℎ

}; that is, 𝐽V ⊂ DV⊕ {𝐽𝜉ℎ

} ⊕ {𝜉ℎ

}.(ii) Let 𝑋 = 𝑎𝑋

|

Dℎ+ 𝑏𝜉ℎ

∈ H, where H = Dℎ

⊕

{𝜉ℎ

}, ker 𝜂ℎ = Dℎ

, and 𝑎, 𝑏 ∈ 𝐶∞(𝑀). Then

𝐽𝑋 = 𝑎𝐽𝑋|

Dℎ+ 𝑏𝐽𝜉ℎ

∈H ⊕ {𝐽𝜉ℎ

} , (69)

which implies that 𝐽H ⊂H ⊕ {𝐽𝜉ℎ

}.Again, let𝑌 ∈H⊕{𝐽𝜉

ℎ

}.Then,𝑌 = 𝑎𝑌|

Dℎ+𝑏𝜉ℎ

+𝑐𝐽𝜉ℎ

∉H,for 𝑎, 𝑏, 𝑐 ∈ 𝐶∞(𝑀). We have

𝐽 𝑌 = 𝑎𝐽𝑌|


+ 𝑐𝐽2

𝜉ℎ

= 𝑎𝐽𝑌|


+ 𝑐𝜉ℎ

= 𝑍 + 𝑏𝐽𝜉ℎ

∈H ⊕ {𝐽𝜉ℎ

} ,

for some 𝑍 = 𝑎𝐽𝑌|

Dℎ+ 𝑐𝜉ℎ

∈H.

(70)

We obtain 𝐽 𝑌 ∉H.Hence, 𝐽H ⊂ H ⊕ {𝐽(𝜉

ℎ

)}; that is, 𝐽H ⊂ Dℎ

⊕ {𝜉ℎ

} ⊕

{𝐽𝜉ℎ

}.

Example 13. Let (R63, 𝐽, 𝑔) be an almost paracomplex pseudo-

Riemannian manifold and let (R31, 𝜙, 𝜉, 𝜂, 𝑔) be an almost

paracontact pseudo-Riemannian manifold. Consider a sub-mersion 𝑓 : {R6

3; (𝑥1, 𝑥2, 𝑥3, 𝑦1, 𝑦2, 𝑦3)𝑡

} → {R31; (𝑢, V, 𝑤)𝑡},

defined by

𝑓 ((𝑥1, 𝑥2, 𝑥3, 𝑦1, 𝑦2, 𝑦3)𝑡

) → (𝑥1+ 𝑥2

√2

,𝑦1+ 𝑦2

√2

, 𝑦3)

𝑡

.

(71)


V = ker𝑓∗

= Span{𝑉1=

𝜕

𝜕𝑥1

−𝜕

𝜕𝑥2

, 𝑉2=

𝜕

𝜕𝑦1

−𝜕

𝜕𝑦2

,

𝜉V=

𝜕

𝜕𝑥3

}

(72)

which is the vertical distribution and the restriction of 𝑔 tothe fibres of 𝑓 is nondegenerate.


H = (ker𝑓∗)⊥

= Span{𝑋1=

𝜕

𝜕𝑥1

+𝜕

𝜕𝑥2

, 𝑋2=

𝜕

𝜕𝑦1

+𝜕

𝜕𝑦2

,

𝜉ℎ

=𝜕

𝜕𝑦3

} .

(73)

The characteristic vector field 𝜉 = 𝜕/𝜕𝑤 on R31has unique

horizontal lift 𝜉ℎ

, which is the characteristic vector field onthe horizontal distributionH of R6

3.

We also have

𝑔 (𝑋1, 𝑋1) = 𝑔 (𝑓

∗𝑋1, 𝑓∗𝑋1) = −2,

𝑔 (𝑋2, 𝑋2) = 𝑔 (𝑓

∗𝑋2, 𝑓∗𝑋2) = 2,

𝑔 (𝜉ℎ

, 𝜉ℎ

) = 𝑔(𝑓∗𝜉ℎ

, 𝑓∗𝜉ℎ

) = 𝑔 (𝜉, 𝜉) = 1.

(74)

Thus, the smooth map 𝑓 is a pseudo-Riemannian submer-sion.

Also, we obtain that there exists a 1-form 𝜂 = 𝑑𝑥3on R63

such that 𝜂(𝐽𝜉ℎ

) = 1, 𝜂(𝜉ℎ

) = 0 and the map 𝑓 satisfies

𝑓∗𝐽𝑋1= 𝜙𝑓∗𝑋1+ 𝜂 (𝑋

1) 𝜉,

𝑓∗𝐽𝑋2= 𝜙𝑓∗𝑋2+ 𝜂 (𝑋

2) 𝜉,

𝑓∗𝐽𝜉ℎ

= 𝜙𝑓∗𝜉ℎ

+ 𝜂 (𝜉ℎ

) 𝜉.

(75)

Hence, the map 𝑓 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1.

Moreover, we observe that, for this submersion𝑓, we have

𝐽V ⊂V ⊕ {𝜉ℎ

} , 𝐽H ⊂H ⊕ {𝐽𝜉ℎ

} , (76)

which verifies Proposition 12.

8 Geometry

Proposition 14. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 𝑓 be pseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌 bebasic vector fields 𝑓-related to 𝑋, 𝑌, respectively. Let 𝐹 and Φbe the second fundamental forms and let ∇ and ∇ be the Levi-Civita connection on the total manifold𝑀 and base manifold𝑀, respectively. Then, one has

(i) 𝑓∗((∇𝑋𝐽)𝑌) = (∇

𝑋𝜙)𝑌 + 𝜀𝑔(𝑌, ∇

𝑋𝜉)𝜉 + 𝜂(𝑌)∇

𝑋𝜉,

(ii) 𝐹 = 𝑓∗∗Φ + 𝜀𝜂 ⊗ 𝜂,

(iii) 𝑓∗((∇𝑋𝐹)(𝑌, 𝑍)) = (∇

𝑋Φ)(𝑌, 𝑍) + 𝜂(𝑌)𝑔(𝑍, ∇

𝑋𝜉) +

𝜂(𝑍)𝑔(𝑌, ∇𝑋𝜉).

Proof. (i) In view of Definition 2 and Proposition 11, we have

𝑓∗((∇𝑋𝐽) 𝑌) = 𝑓

∗(∇𝑋(𝐽 𝑌) − 𝐽 (∇

𝑋𝑌))

= ∇𝑋(𝑓∗(𝐽 𝑌)) − 𝑓

∗(𝐽 (∇𝑋𝑌))

= ∇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.

(77)

(ii) Since𝑓∗∗Φ is pullback ofΦ through the linearmap𝑓

∗,

we get

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) ,

(78)

which implies 𝐹 = 𝑓∗∗Φ + 𝜀𝜂 ⊗ 𝜂.

(iii) By (23), we have

𝑓∗((∇𝑋𝐹) (𝑌, 𝑍)) = 𝑔 (𝑓

∗(𝑌) , 𝑓

∗((∇𝑋𝐽)𝑍)) . (79)

Now, using (i) in the above equation, we get (iii).

Theorem 15. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 𝑓 be pseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌 bebasic vector fields 𝑓-related to 𝑋, 𝑌, respectively. If the totalspace is para-Hermitian manifold, then the almost paracontactstructure of base space is normal.

Moreover, if the almost paracontact structure of base spaceis normal, then the Nijenhuis tensor of total space is vertical.

Proof. The Nijenhuis tensors 𝑁𝐽and 𝑁

𝜙of almost para-

complex structure 𝐽 and almost paracontact structure 𝜙 are,respectively, defined by (8) and (11).

Using Definition 2 and properties of Sections 2.1 and 2.2,we get the following identity:

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) .

(80)

Using (12), (13), (14), and (15), (80) reduces to

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) .

(81)

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 paracontactpseudo-Riemannian submersion and let the fibres of 𝑓 bepseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌 be basicvector fields 𝑓-related to 𝑋, 𝑌, respectively. Let the total spacebe para-Hermitian manifold and𝑁(1) vanishes. Then, the basespace is paracontact pseudometric manifold if and only if 𝜉 iskilling.

Proof. Let the total space be para-Hermitian and 𝑁(1) van-ishes. Then, from (80), we have

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) (L

𝜉𝜙)𝑌.

(82)

If 𝜉 is killing, then we have L𝜉𝜙 = 0. It immediately follows

from (82) that

𝑑𝜂 (𝜙𝑋, 𝑌) − 𝜂 (𝜙𝑌,𝑋) + 𝜂 (𝑋) 𝑑𝜂 (𝜉, 𝑌)

− 𝜂 (𝑌) 𝑑𝜂 (𝜉, 𝑋) = 0.

(83)

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 getL

𝜉𝜙 = 0.

Hence, the characteristic vector field 𝜉 is killing.

Geometry 9

Theorem 17. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 𝑓 bepseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌 be basicvector fields 𝑓-related to 𝑋, 𝑌, respectively. If the total spaceis para-Kähler, then the base space is paracosymplectic. Theconverse is true if ∇

𝑋𝐽 is vertical.

Proof. We have, for any 𝑋,𝑌 ∈ Γ(𝑇𝑀), (∇𝑋𝜙)𝑌 = 0, which

gives 𝑔(𝑍, (∇𝑋𝜙)𝑌) = 0, for any 𝑍 ∈ Γ(𝑇𝑀).

From Proposition 14, we have

{𝑔 (𝑍, (∇𝑋𝜙)𝑌) + 𝜀𝜂 (𝑌) (∇

𝑋𝜂)𝑍 + 𝜀𝜂 (𝑍) (∇

𝑋𝜂) 𝑌} ∘ 𝑓

= 𝑔 (𝑍, (∇𝑋𝐽) 𝑌) .

(84)

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 isparacosymplectic.

Again, let (∇𝑋𝜙)𝑌 = 0 and ∇𝜂 = 0. Then, 𝑔(𝑍, (∇

𝑋𝐽)𝑌) =

0, which implies that (∇𝑋𝐽)𝑌 is a vertical vector field.

Theorem 18. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 𝑓 bepseudo-Riemannian submanifolds of 𝑀. Let 𝑋, 𝑌, and 𝑍 bebasic vector fields 𝑓-related to 𝑋, 𝑌, and 𝑍, respectively. If𝜂(𝑋)𝑍(𝜂(𝑌))+𝜂(𝑌)𝑋(𝜂(𝑍))−𝜂(𝑋)𝑌(𝜂(𝑍)) = 0, then the totalspace is almost para-Kähler if and only if the base space𝑀 isan almost paracosymplectic manifold.

Proof. We have the following equation:

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)) .

(85)

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, let 𝑑𝐹 = 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 and𝑑Φ = 0.

Hence, the base space is almost paracosymplectic.

Now,we investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion.

Lemma 19. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kähler manifold𝑀 onto an almost paracontact pseudometric manifold𝑀 and

let the fibres of 𝑓 be pseudo-Riemannian submanifolds of 𝑀.Then, for any horizontal vector fields𝑋, 𝑌 and for any verticalvector fields 𝑈, 𝑉 on𝑀, one has

(i) A𝑋(𝐽 𝑌) = 𝐽(A

𝑋𝑌),

(ii) A𝐽𝑋(𝑌) = 𝐽(A

𝑋𝑌),

(iii) T𝑈(𝐽 𝑉) = 𝐽(T

𝑈𝑉),

(iv) T𝐽𝑈𝑉 = 𝐽(T

𝑈𝑉).

Proof. The proof follows using similar steps as in Lemmas 3and 4 of [13], so we omit it.

Lemma 20. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kähler manifold𝑀 onto an almost paracontact pseudometric manifold𝑀 andlet the fibres of 𝑓 be pseudo-Riemannian submanifolds of 𝑀.Then, for any vector fields 𝐸, 𝐹 on𝑀, one has

(i) A𝐸(𝐽𝐹) = 𝐽(A

𝐸𝐹),

(ii) T𝐸(𝐽𝐹) = 𝐽(T

𝐸𝐹).

Proof. The proof follows from (37) and (38).

Theorem 21. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kähler manifold𝑀 onto an almost paracontact pseudometric manifold𝑀 andlet the fibres of 𝑓 be pseudo-Riemannian submanifolds of 𝑀.Then, the horizontal distribution is integrable.

Proof. For any vertical vector field 𝑈, we have

𝑔 (𝐽 (A𝑋𝑌) ,𝑈) = 𝑔 (A

𝑋𝐽 𝑌,𝑈)

= −𝑔 (𝐽𝑌,A𝑋𝑈)

= −𝑔 (𝐽𝑌, ℎ (∇𝑈𝑋))

= 𝑔 (𝑌, ℎ (𝐽 (∇𝑈𝑋)))

= 𝑔 (𝑌, ℎ {(−∇𝑈𝐽)𝑋

+∇𝑈(𝐽𝑋)})

= 𝑔 (𝑌, ℎ {∇𝑈(𝐽𝑋)}) = 𝑔 (𝑌,A

𝐽𝑋𝑈)

= −𝑔 (A𝐽𝑋𝑌,𝑈) = −𝑔 (𝐽 (A

𝑋𝑌) , 𝑈) .

(86)

Thus 𝑔(𝐽(A𝑋𝑌), 𝑈) = 0, which is true for all𝑋 and 𝑌.

So,A𝑋𝑌 = 0.

Hence, the horizontal distribution is integrable.

Theorem 22. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kähler manifold𝑀 onto an almost paracontact pseudometric manifold𝑀 andlet the fibres of 𝑓 be pseudo-Riemannian submanifolds of 𝑀.Then, the submersion is an affine map onH.

10 Geometry

Proof. The second fundamental form of 𝑓 is defined by

(∇𝑓∗) (𝐸, 𝐹) = (∇

𝑓

𝐸𝑓∗(𝐹)) ∘ 𝑓 − 𝑓

∗(∇𝐸𝐹) , (87)

where 𝐸, 𝐹 ∈ Γ(𝑇𝑀) and ∇𝑓 is pullback connection of Levi-Civita connection ∇ on𝑀 with respect to 𝑓.

We have, for any𝑋,𝑌 ∈H,

(∇𝑓∗) (𝑋, 𝑌) = (∇

𝑓

𝑋

𝑓∗(𝑌)) ∘ 𝑓 − 𝑓

∗(∇𝑋𝑌) . (88)

By using Lemma 1, we have 𝑓∗(ℎ(∇𝑋𝑌)) = (∇

𝑋𝑌) ∘ 𝑓, which

implies ∇𝑓∗= 0.

Hence, the submersion 𝑓 is an affine map onH.

Theorem 23. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold𝑀 onto an almost paracontact pseudometric manifold𝑀and let the fibres of 𝑓 be pseudo-Riemannian submanifolds of𝑀. Then, the submersion is an affine map on V if and only ifthe fibres of 𝑓 are totally geodesic.

Proof. We have, for any 𝑈,𝑉 ∈V,

(∇𝑓∗) (𝑈, 𝑉) = −𝑓

∗(ℎ (∇𝑈𝑉)) , (89)

which, in view of (27), gives

(∇𝑓∗) (𝑈, 𝑉) = −𝑓

∗(T𝑈𝑉) . (90)

Let the fibres of 𝑓 be totally geodesic. Then, T = 0.Consequently, from the above equation, we have ∇𝑓

∗= 0.

Thus, the map 𝑓 is affine onV.Conversely, let the submersion 𝑓 be an affine map onV.

Then, ∇𝑓∗= 0, which impliesT = 0.

Hence, the fibres of 𝑓 are totally geodesic.

Theorem 24. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold𝑀 onto an almost paracontact pseudometric manifold𝑀and let the fibres of 𝑓 be pseudo-Riemannian submanifoldsof 𝑀. Then, the submersion is an affine map if and only ifℎ(∇𝐸ℎ𝐹)+A

ℎ𝐸V𝐹+TV𝐸V𝐹 is 𝑓-related to ∇𝑋𝑌, for any 𝐸, 𝐹 ∈

Γ(𝑇𝑀).

Proof. For any 𝐸, 𝐹 ∈ Γ(𝑇𝑀) with 𝑓∗ℎ𝐸 = 𝑋 ∘ 𝑓 and 𝑓

∗V𝐹 =

𝑌 ∘ 𝑓, we have

(∇𝑓∗) (𝐸, 𝐹) = (∇

𝑓∗ℎ𝐸(𝑓∗ℎ𝐹)) ∘ 𝑓 − 𝑓

∗(ℎ (∇𝐸𝐹))

= (∇𝑋𝑌) ∘ 𝑓 − 𝑓

∗(ℎ (∇ℎ𝐸ℎ𝐹 + ∇

ℎ𝐸V𝐹

+∇V𝐸ℎ𝐹 + ∇V𝐸ℎ𝐹)) .

(91)

By using (27) and (31) in the above equation, we have

(∇𝑓∗) (𝐸, 𝐹) = (∇

𝑋𝑌) ∘ 𝑓 − 𝑓

∗(ℎ (∇𝐸ℎ𝐹) +A

ℎ𝐸V𝐹

+TV𝐸V𝐹) .(92)

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) +A

ℎ𝐸V𝐹 +TV𝐸ℎ𝐹).

Conversely, let ℎ(∇𝐸𝐹) + A

ℎ𝐸V𝐹 + TV𝐸ℎ𝐹 be 𝑓-related

to ∇𝑋𝑌, for any 𝐸, 𝐹 ∈ Γ(𝑇𝑀). Then, from (92), we have

(∇𝑓∗)(𝐸, 𝐹) = 0.Hence, the submersion map 𝑓 is affine.

4. Curvature Properties

In this section, the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold, base manifold, and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied.

Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (𝑀, 𝐽, 𝑔) onto an almost paracontact pseudometricmanifold (𝑀, 𝜙, 𝜉, 𝜂, 𝑔).

Suppose that the vector fields 𝐸, 𝐹 span the 2-dimensional plane at point 𝑝 of 𝑀 and let R be theRiemannian curvature tensor of 𝑀. The paraholomorphicbisectional curvature 𝐵(𝐸, 𝐹) of 𝑀 for any pair of nonzeronon-lightlike vector fields 𝐸, 𝐹 on 𝑀 is defined by theformula

𝐵 (𝐸, 𝐹) =

R (𝐸, 𝐽𝐸, 𝐹, 𝐽𝐹)

𝑔 (𝐸, 𝐸) 𝑔 (𝐹, 𝐹). (93)

For a nonzero non-lightlike vector field 𝐸, the vector field𝐽𝐸 is also non-lightlike and {𝐸, 𝐽𝐸} span the 2-dimensionalplane. Then the paraholomorphic sectional curvature 𝐻(𝐸)is defined as

𝐻(𝐸) = 𝐵 (𝐸, 𝐸) =

R (𝐸, 𝐽𝐸, 𝐸, 𝐽𝐸)

𝑔 (𝐸, 𝐸) 𝑔 (𝐸, 𝐸). (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of O’Neill [1] and Gray [3].

Let 𝐵ℎ and 𝐵V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces, respectively. Let𝐻ℎ and 𝐻V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces, respectively. Let 𝐵 and 𝐻 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold, respectively.

Proposition 25. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 𝑓 be pseudo-Riemanniansubmanifolds of 𝑀. Let 𝑈, 𝑉 be non-lightlike unit verticalvector fields and let𝑋, 𝑌 be non-lightlike unit horizontal vectorfields on𝑀. Then, one has

𝐵 (𝑈,𝑉) = 𝐵V(𝑈, 𝑉) + 𝑔 (T

𝑈(𝐽𝑉) ,T

𝐽𝑈𝑉)

− 𝑔 (T𝐽𝑈(𝐽𝑉) ,T

𝑈𝑉) ,

(95)

Geometry 11

𝐵 (𝑋,𝑈) = 𝑔 ((∇𝑈A)𝑋

𝐽𝑋, 𝐽𝑈) − 𝑔 (A𝑋𝐽𝑈,A

𝐽𝑋𝑈)

+ 𝑔 (A𝑋𝑈,A𝐽𝑋𝐽𝑈) − 𝑔 ((∇

𝐽𝑈A)𝑋

𝐽𝑋,𝑈)

+ 𝑔 (T𝐽𝑈𝑋,T𝑈(𝐽𝑋))

− 𝑔 (T𝑈𝑋,T𝐽𝑈(𝐽𝑋)) ,

(96)

𝐵 (𝑋, 𝑌) = 𝐵ℎ

(𝑋, 𝑌) − 2𝑔 (A𝑋(𝐽𝑋) ,A

𝑌(𝐽 𝑌))

+ 𝑔 (A𝐽𝑋𝑌,A𝑋(𝐽 𝑌))

− 𝑔 (A𝑋𝑌,A𝐽𝑋(𝐽 𝑌)) .

(97)

Proof. Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by O’Neill [1], we have (95), (96), and (97).

Corollary 26. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold. If the fibres of 𝑓 are totally geodesic pseudo-Riemannian submanifolds of𝑀, then for any non-lightlike unitvertical vector fields 𝑈 and 𝑉, one has

𝐵 (𝑈,𝑉) = 𝐵V(𝑈, 𝑉) . (98)

Corollary 27. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 𝑓 be totally geodesic pseudo-Riemannian submanifolds of𝑀. If the horizontal distributionis integrable, then, for any non-lightlike unit horizontal vectorfields𝑋 and 𝑌, one has

𝐵 (𝑋, 𝑌) = 𝐵ℎ

(𝑋, 𝑌) . (99)

Proposition 28. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 𝑓 be pseudo-Riemanniansubmanifolds of𝑀. Let 𝑈 and 𝑋 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector field,respectively. Then, one has

𝐻(𝑈) = 𝐻V(𝑈) +

T𝑈(𝐽𝑈)

2

− 𝑔 (T𝐽𝑈(𝐽𝑈) ,T

𝑈𝑈) ,

(100)

𝐻(𝑋) = 𝐻 (𝑋) ∘ 𝑓 − 3A𝑋(𝐽𝑋)

2

. (101)

Proof. The proof is straightforward. If we take 𝑈 = 𝑉 in (95)and𝑋 = 𝑌 in (97), we have (98) and (99).

Corollary 29. Let 𝑓 : 𝑀 → 𝑀 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold. If the fibres of 𝑓 are totally geodesic pseudo-Riemannian submanifolds of 𝑀, then the total manifold andfibres of𝑓have the same paraholomorphic sectional curvatures.

Proof. Since the fibres are totally geodesic, T = 0; conse-quently we have

𝐻(𝑈) = 𝐻V(𝑈) . (102)

Corollary 30. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 𝑓 be totally geodesic pseudo-Riemannian submanifolds of𝑀. If the horizontal distributionis integrable, then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures.

Proof. Since the horizontal distribution is integrable,A = 0;consequently, we have

𝐻(𝑋) = 𝐻 (𝑋) ∘ 𝑓. (103)

Theorem 31. Let 𝑓 : 𝑀𝑚 → 𝑀𝑛 be a paracomplexparacontact pseudo-Riemannian submersion from a para-Kähler manifold𝑀 onto an almost paracontact pseudometricmanifold 𝑀 and let the fibres of 𝑓 be pseudo-Riemanniansubmanifolds of 𝑀. If 𝑈, 𝑉 are the non-lightlike unit verticalvector fields and 𝑋, 𝑌 are the non-lightlike unit horizontalvector fields, then one has

𝐵 (𝑈,𝑉) = 𝐵V(𝑈, 𝑉) , (104)

𝐵 (𝑋,𝑈) = −2T𝑈𝑋

2

, (105)

𝐵 (𝑋, 𝑌) = 𝐵 (𝑋, 𝑌) ∘ 𝑓. (106)

Proof. Using results of Lemma 19 in (95), we have

𝐵 (𝑈,𝑉) = 𝐵V(𝑈, 𝑉) − 𝑔 (𝐽 (T

𝑈𝑉) , 𝐽 (T

𝑈𝑉))

− 𝑔 (𝐽2

(T𝑈𝑉) ,T

𝑈𝑉)

= 𝐵 (𝑈,𝑉) + 𝑔 (T𝑈𝑉,T𝑈𝑉) − 𝑔 (T

𝑈𝑉,T𝑈𝑉)

= 𝐵V(𝑈, 𝑉) .

(107)

Applying results of Lemma 19 in (96), we have

𝐵 (𝑋,𝑈) = 𝑔 ((∇𝑈A)𝑋

(𝐽𝑋) , 𝐽𝑈) − 𝑔 ((∇𝐽𝑈A)𝑋

(𝐽𝑋) , 𝑈)

+ 2A𝑋𝑈

2

− 2T𝑈𝑋

2

.

(108)

Since byTheorem 21 the horizontal distribution is integrable,we haveA = 0, which implies

𝐵 (𝑋,𝑈) = −2T𝑈𝑋

2

. (109)

In view ofA = 0, (104) follows from (97).

12 Geometry

Theorem 32. Let 𝑓 : 𝑀𝑚 → 𝑀𝑛 be a paracomplexparacontact pseudo-Riemannian submersion from a para-Kähler manifold𝑀 onto an almost paracontact pseudometricmanifold 𝑀 and let the fibres of 𝑓 be pseudo-Riemanniansubmanifolds of𝑀. If 𝑈, 𝑋 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields, respectively, then onehas

𝐻(𝑈) = 𝐻V(𝑈) − 2

T𝑈𝑈

2

, (110)

𝐻(𝑋) = 𝐻 (𝑋) ∘ 𝑓. (111)

Proof. Since 𝑓 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold 𝑀onto an almost paracontact pseudometric manifold 𝑀, by(16) and equations of Lemma 19, we have

𝑔 (T𝐽𝑈(𝐽𝑈) ,T

𝑈𝑈) = 𝑔 (𝐽

2

(T𝑈𝑈) ,T

𝑈𝑈) =

T𝑈𝑈

2

,

𝑔 (T𝑈(𝐽𝑈) ,T

𝑈(𝐽𝑈)) = −𝑔 (T

𝑈𝑈,T𝑈𝑈) = −

T𝑈𝑈

2

(112)

and by using the above results in (100), we have

𝐻(𝑈) = 𝐻V(𝑈) −

T𝑈𝑈

2

−T𝑈𝑈

2

= 𝐻V(𝑈) − 2

T𝑈𝑈

2

.

(113)

Again, since horizontal distribution is integrable, we haveA = 0, and putting it in (101), we obtain (111).

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission, New Delhi, India, for financial support.

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