Paracomplex Paracontact Pseudo-Riemannian Submersions

Hindawi Publishing Corporation Geometry Volume 2014, Article ID 616487, 12 pages http://dx.doi.org/10.1155/2014/616487

Research Article Paracomplex 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 Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.

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 KaluzaKlein 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], G¨und¨uzalp 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 pseudoRiemannian 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 Section 2, we collect some basic definitions, formulas, and results on almost paracomplex manifolds, almost paracontact pseudometric manifolds, and pseudo-Riemannian submersion.

2

Geometry

In Section 3, we define paracomplex paracontact pseudoRiemannian 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.

If 𝑋 and 𝑌 are vector fields on 𝑀2𝑛+1 , then we have [18– 20] 𝑁𝐽 ((𝑋, 0) , (𝑌, 0)) = (𝑁𝜙 (𝑋, 𝑌) − 2𝑑𝜂 (𝑋, 𝑌) 𝜉,

2. Preliminaries 2.1. Almost Paracontact Manifolds. Let 𝑀 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 1form on 𝑀. Then, (𝜙, 𝜉, 𝜂) is called an almost paracontact structure on 𝑀 if 𝜙2 𝑋 = 𝑋 − 𝜂 (𝑋) 𝜉;

𝜂 (𝜉) = 1,

(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 +, +, +, . . . , +) such that (−, −, −, . . . , −, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑛-times)

(𝑛+1)-times)

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 field 𝜉 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 paracontact manifold. If the metric 𝑔 is positive definite, then the manifold (𝑀2𝑛+1 , 𝜙, 𝜉, 𝜂, 𝑔) is the usual almost paracontact metric manifold [14]. The fundamental 2-form Φ on 𝑀 is defined by Φ (𝑋, 𝑌) = 𝑔 (𝑋, 𝜙𝑌) .

(6)

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 𝐽 (𝑋, 𝑓

𝑑 𝑑 ) = (𝜙𝑋 + 𝑓𝜉, 𝜂 (𝑋) ) , 𝑑𝑡 𝑑𝑡 2𝑛+1

(7) 1

, 𝑡 is the coordinate on R , and 𝑓 where 𝑋 is tangent to 𝑀 2𝑛+1 is a smooth function on 𝑀 . An almost paracontact structure (𝜙, 𝜉, 𝜂) is said to be normal, if the Nijenhuis tensor 𝑁𝐽 of almost paracomplex structure 𝐽 defined as 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.

(8)

{(L𝜙𝑋 𝜂) 𝑌 − (L𝜙𝑌 𝜂) 𝑋} 𝑁𝐽 ((𝑋, 0) , (0,

(9) 𝑑 ), 𝑑𝑡

𝑑 𝑑 )) = − ((L𝜉 𝜙) 𝑋, ((L𝜉 𝜂) 𝑋) ) , 𝑑𝑡 𝑑𝑡 (10)

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 𝑁𝜙 (𝑋, 𝑌) = [𝜙, 𝜙] (𝑋, 𝑌)

(11)

= [𝜙𝑋, 𝜙𝑌] + 𝜙2 [𝑋, 𝑌] − 𝜙 [𝜙𝑋, 𝑌] − 𝜙 [𝑋, 𝜙𝑌] , 𝑁(1) (𝑋, 𝑌) = 𝑁𝜙 (𝑋, 𝑌) − 2𝑑𝜂 (𝑋, 𝑌) 𝜉,

(12)

𝑁(2) (𝑋, 𝑌) = (L𝜙𝑋 𝜂) 𝑌 − (L𝜙𝑌 𝜂) 𝑋,

(13)

𝑁(3) (𝑋) = (L𝜉 𝜙) 𝑋,

(14)

𝑁(4) (𝑋) = (L𝜉 𝜂) 𝑋.

(15)

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, if 𝑁𝜙 − 2𝑑𝜂 ⊗ 𝜉 = 0, (ii) paracontact, if Φ = 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, if 𝑑𝜂 = 0 and 𝑑Φ = 0, (vi) weakly paracosymplectic, if 𝑀 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, if Φ = 𝑑𝜂 and 𝑀 is normal, (viii) quasi-para-Sasakian, if 𝑑𝜙 = 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.

Geometry

3

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 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)

(ii) The fibres 𝑓−1 (𝑞) of 𝑓 over 𝑞 ∈ 𝑀 are either pseudo-Riemannian submanifolds of 𝑀 of dimension (𝑚 − 𝑛) and index ] or the degenerate submanifolds of 𝑀 of dimension (𝑚 − 𝑛) and index ] with degenerate metric 𝑔| −1 of type 𝑓 (𝑞) (0, 0, 0, . . . , 0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ −, −, −, . . . , −, +, +, +, . . . , + ), where ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜇-times

]-times

(𝑚−𝑛−𝜇−])-times)

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 a vector 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 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]). If 𝑓 : 𝑀 → 𝑀 is a pseudoRiemannian 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,

3𝑑𝐹 (𝑋, 𝑌, 𝑍) = 𝑋 (𝐹 (𝑌, 𝑍)) − 𝑌 (𝐹 (𝑋, 𝑍)) + 𝑍 (𝐹 (𝑋, 𝑌))

(21)

− 𝐹 ([𝑋, 𝑌] , 𝑍) + 𝐹 ([𝑋, 𝑍] , 𝑌) − 𝐹 ([𝑌, 𝑍] , 𝑋) , (∇𝑋 𝐹) (𝑌, 𝑍) = 𝑔 (𝑌, (∇𝑋 𝐽) 𝑍) = −𝑔 (𝑍, (∇𝑋𝐽) 𝑌) , 3𝑑𝐹 (𝑋, 𝑌, 𝑍) = (∇𝑋 𝐹) (𝑌, 𝑍) + (∇𝑌 𝐹) (𝑍, 𝑋) + (∇𝑍𝐹) (𝑋, 𝑌) ,

(22) (23)

2𝑚

the co-differential, (𝛿𝐹) (𝑋) = ∑𝜀𝑖 (∇𝑒𝑖 𝐹) (𝑒𝑖 , 𝑋) .

(24)

𝑖=1

(i) para-Hermitian, if 𝑁𝐽 = 0; equivalently, (∇𝐽𝑋𝐽)𝐽𝑌 + (∇𝑋 𝐽)𝑌 = 0, (ii) para-K¨ahler, if, for any 𝑋 ∈ Γ(𝑇𝑀), ∇𝑋 𝐽 = 0; that is, ∇𝐽 = 0, (iii) almost para-K¨ahler, if 𝑑𝐹 = 0, (iv) nearly para-K¨ahler, if (∇𝑋𝐽)𝑋 = 0,

(iv) [𝐸, 𝑈] ∈ V, for any vector field 𝑈 ∈ V and for any vector field 𝐸 ∈ Γ(𝑇𝑀). A pseudo-Riemannian submersion 𝑓 : 𝑀 → 𝑀 determines tensor fields T and A of type (1, 2) on 𝑀 defined by formulas [1, 2, 23]

A (𝐸, 𝐹) = A𝐸 𝐹 = V (∇ℎ𝐸 ℎ𝐹) + ℎ (∇ℎ𝐸 V𝐹) , for any 𝐸, 𝐹 ∈ Γ (𝑇𝑀) .

𝑚

2.3. Pseudo-Riemannian Submersion. Let (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 map 𝑓 : 𝑚 𝑀 → 𝑀𝑛 , which is onto and satisfies the following conditions [2, 3, 23, 24]. →

T𝑈𝑉 = ℎ (∇𝑈𝑉) ,

∇𝑈𝑋 = T𝑈𝑋 + ℎ (∇𝑈𝑋) ,

(vi) semi-para-K¨ahler, if 𝛿𝐹 = 0 and 𝑁𝐽 = 0.

𝑇𝑓(𝑝) 𝑀 is

(25) (26)

Let 𝑋, 𝑌 be horizontal vector fields and let 𝑈, 𝑉 be vertical vector fields on 𝑀. Then, one has T𝑈𝑋 = V (∇𝑈𝑋) ,

(v) almost semi-para-K¨ahler, if 𝛿𝐹 = 0,

surjective at each point 𝑝 ∈ 𝑀.

(iii) ℎ(∇𝑋 𝑌) is a basic vector field 𝑓-related to ∇𝑋 𝑌, where ∇ and ∇ are the Levi-Civita connections on 𝑀 and 𝑀, respectively,

T (𝐸, 𝐹) = T𝐸 𝐹 = ℎ (∇V𝐸 V𝐹) + V (∇V𝐸 ℎ𝐹) ,

An almost para-Hermitian manifold is called

(i) The derivative map 𝑓∗𝑝 : 𝑇𝑝 𝑀

(ii) ℎ[𝑋, 𝑌] is a vector field and ℎ[𝑋, 𝑌] = [𝑋, 𝑌] ∘ 𝑓,

T𝑋 𝐹 = 0,

T𝐸 𝐹 = TV𝐸 𝐹,

∇𝑈𝑉 = T𝑈𝑉 + V (∇𝑈𝑉) , A𝑋 𝑌 = V (∇𝑋 𝑌) ,

A𝑋 𝑈 = ℎ (∇𝑋 𝑈) ,

∇𝑋 𝑈 = A𝑋 𝑈 + V (∇𝑋 𝑈) , A𝑈𝐹 = 0,

A𝐸 𝐹 = Aℎ𝐸 𝐹,

∇𝑋 𝑌 = A𝑋 𝑌 + ℎ (∇𝑋𝑌) ,

(27) (28) (29) (30) (31) (32) (33) (34)

4

Geometry 2𝑚

ℎ (∇𝑈𝑋) = ℎ (∇𝑋 𝑈) = A𝑋 𝑈,

(35)

1 A𝑋 𝑌 = V [𝑋, 𝑌] , 2

(36)

A𝑋 𝑌 = −A𝑌 𝑋,

(37)

Proposition 4. Let 𝑓 : 𝑀 → 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).

T𝑈𝑉 = T𝑉𝑈,

(38)

Proof. Let 𝑈 ∈ V. Then

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 PseudoRiemannian Submersions

𝑓∗ (𝐽𝑈) = 𝜙 (𝑓∗ (𝑈)) + 𝜂 (𝑈) 𝜉,

ℎ

where 𝑓∗ 𝜉 = 𝜉. Thus, we have ℎ

𝐽 (𝑈) − 𝜂 (𝑈) 𝜉 = 𝜙 (𝑈) ,

In this section, we introduce the notion of pseudoRiemannian submersion from almost paracomplex manifolds onto almost paracontact pseudometric manifolds, illustrate examples, and study the transference of structures on total manifolds and base manifolds.

ℎ

𝑓∗ ∘ 𝐽 = 𝜙 ∘ 𝑓∗ + 𝜂 ⊗ 𝜉.

(39)

Since, for each 𝑝 ∈ 𝑀, 𝑓∗𝑝 is a linear isometry between horizontal spaces H𝑝 and tangent spaces 𝑇𝑓(𝑝) 𝑀, there exists ℎ

ℎ

an induced almost paracontact structure (𝜙 , 𝜂ℎ , 𝜉 , 𝑔) on (2𝑛 + 1)-dimensional horizontal distribution H such that ℎ 𝜙| ℎ behave just like the fundamental collineation of almost D

ℎ

ℎ

ℎ

paracomplex structure 𝐽 on ker 𝜂ℎ = D and 𝜙 : D → D ℎ

is an endomorphism such that 𝜙 = 𝐽|

ker 𝜂ℎ

ℎ

ℎ

𝜙 = 2𝑛, where dim(D ) = 2𝑛. It follows that, for any 𝑋 implies that

2 𝐽|

ℎ

D

ℎ 2

ℎ

ℎ

ℎ

ℎ

ℎ

and the rank of

ℎ

ℎ

(𝑋 ) = (𝜙 ) (𝑋 ) = 𝑋 , for any 𝑋 ∈ D

ℎ

ℎ

and H = D ⊕ {𝜉 } [18].

ℎ

ℎ

𝑓∗ (𝐽 (𝜉 )) = 0,

ℎ

that is 𝐽 (𝜉 ) ∈ V.

(44)

ℎ

Taking 𝑈 = 𝐽𝜉 in (43), we obtain ℎ

ℎ

ℎ

ℎ

𝜉 − 𝜂 (𝐽𝜉 ) 𝜉 = 𝜙 (𝐽𝜉 ) .

(45)

Since fibre 𝑓−1 (𝑞) is an odd dimensional submanifold, there exists an associated 1-form 𝜂V which is restriction of 𝜂 on fibre submanifold 𝑓−1 (𝑞), 𝑞 ∈ 𝑀, and a characteristic vector field V

ℎ

V

V

𝜉 = 𝐽𝜉 such that 𝜙(𝜉 ) = 0. So, we have 𝜂V (𝜉 ) = 1. V Let us put ker 𝜂V = D1 and D2 = {𝜉 }. Then, ker 𝑓∗ = D1 ⊕ D2 and 𝐽(D1 ) = D1 , 𝐽(D2 ) = V

ℎ

𝐽{𝜉 } = {𝜉 } ⊆ (ker 𝑓∗ )⊥ .

Hence, the fibres 𝑓−1 (𝑞) are semi-𝐽-invariant submanifolds of 𝑀.

Corollary 5. Let 𝑓 : 𝑀 → 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 V

Definition 3 (see [25]). A pseudo-Riemannian submersion 𝑓 : 𝑀 → 𝑀 is called semi-𝐽-invariant submersion, if there is a distribution D1 ⊆ ker 𝑓∗ such that ker 𝑓∗ = D1 ⊕ D2 , 𝐽 (D1 ) = D1 ,

ℎ

(43)

2𝑚

ℎ

∈ D , 𝜂ℎ (𝑋 ) = 0, which ℎ

for some 𝜙 (𝑈) ∈ V.

By (19), we get 𝑔(𝜉 , 𝐽(𝜉 )) = 0 = 𝑔(𝜉, 𝑓∗ (𝐽(𝜉 ))) = 0. As 𝑔 is nondegenerate on 𝑀, we have

2𝑚

Definition 2. Let (𝑀 , 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 submersion 𝑓 : 𝑀 → 𝑀 is called paracomplex paracontact pseudo-Riemannian submersion if there exists a 1-form 𝜂 on 𝑀 such that

(42)

ℎ

󳨐⇒ 𝑓∗ {𝐽 (𝑈) − 𝜂 (𝑈) 𝜉 } = 0,

(40) ⊥

𝐽 (D2 ) ⊆ (ker 𝑓∗ ) ,

where D2 is orthogonal complementary to D1 in ker 𝑓∗ .

V

(𝜙 , 𝜉 , 𝜂V , 𝑔V ), 𝑞 ∈ 𝑀, where 𝜉 𝑔V = 𝑔.

V

ℎ

= 𝐽(𝜉 ), 𝜂V = 𝜂|V , and

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

ℎ

𝐽 (𝑈) = 𝜙 𝑈 + 𝜂V (𝑈) 𝜉 ,

(41) for any 𝑈 ∈ V.

(46)

Geometry

5

On operating 𝐽 on both sides of the above equation, we get

H, and 𝑓∗𝑝 : H𝑝 → 𝑇𝑓(𝑝) 𝑀 is a linear isometry, for any 𝑋𝑝 ∈ H𝑝 , we get

V

V

ℎ

V

V

𝑈 = 𝜙 (𝜙 (𝑈)) + 𝜂V (𝜙 (𝑈)) 𝜉 + 𝜂V (𝑈) 𝜉 , ℎ

(47)

ℎ

ℎ

𝜂ℎ𝑝 (𝑋𝑝 ) = 𝜀𝑔𝑝 (𝜉𝑝 , 𝑋𝑝 ) = 𝑔𝑓(𝑝) (𝑓∗𝑝 𝜉𝑝 , 𝑓∗𝑝 𝑋𝑝 )

V

where 𝐽(𝜉 ) = 𝜉 . Equating horizontal and vertical components, we have V

V

V

= 𝑔𝑓(𝑝) (𝜉𝑓(𝑝) , 𝑋𝑓(𝑝) ) = 𝜂𝑓(𝑝) (𝑋𝑓(𝑝) ) = 𝑓∗∗ 𝜂𝑝 (𝑋𝑝 ) . (52)

V

𝑈 = 𝜙 (𝜙 (𝑈)) + 𝜂V (𝑈) 𝜉 ,

𝜂V ∘ 𝜙 (𝑈) = 0, (48)

Hence, pullback 𝑓∗∗ 𝜂 = 𝜂ℎ . Results (iii) and (iv) immediately follow from the previous results.

Hence, (𝜙 , 𝜉 , 𝜂V , 𝑔V ) is almost paracontact pseudometric structure on the fibre 𝑓−1 (𝑞), 𝑞 ∈ 𝑀.

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 : {R42 ; (𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 )𝑡 } → 3 {R1 ; (𝑢, V, 𝑤)𝑡 } by

V 2

V

V

󳨐⇒ (𝜙 ) (𝑈) = 𝑈 − 𝜂 (𝑈) 𝜉 ; V

V

V

V

𝜂 ∘ 𝜙 = 0; V

𝜂V (𝜉 ) = 1.

𝜙 (𝜉 ) = 0, V

V

2𝑚

Proposition 6. Let 𝑓 : 𝑀 → 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.

𝑡

𝑓 ((𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ) ) 󳨃󳨀→ (𝑥1 + 𝑥2 + 3𝑦1 + 2𝑦2 , 3𝑥1 + 2𝑥2 + 𝑦1 + 𝑦2 , 𝑡

5𝑥1 + 3𝑥2 + 5𝑦1 + 3𝑦2 ) .

ℎ

(i) The characteristic vector field 𝐽𝜉 is a vertical vector field. (ii) 𝑓∗∗ 𝜂 = 𝜂ℎ , where 𝑓∗∗ 𝜂 is pullback of 𝜂 through 𝑓∗ .

Then, the kernel of 𝑓∗ is V = ker 𝑓∗ = Span {𝑉1 =

(iii) 𝜂ℎ (𝑈) = 0, for any vertical vector field 𝑈. (iv) 𝜂V (𝑋) = 0, for any horizontal vector field 𝑋. Remark 7. Results (ii) and (iv) are analogue version of results (i) and (iii) of Proposition 4 of [13]. V

V

ℎ

ℎ

⊥

H = (ker 𝑓∗ )

= Span {𝑋1 =

ℎ

0 = 𝑔 (𝜉 , 𝐽𝜉 ) = 𝑔 (𝑓∗ (𝜉 ) , 𝑓∗ (𝐽𝜉 ))

𝑋3 = 2

= 𝑔 (𝜉, 𝑓∗ (𝐽𝜉 )) . Now, ℎ

ℎ

ℎ

ℎ

(50)

𝜕 𝜕 + }. 𝜕𝑦1 𝜕𝑦2

For any real 𝑘, the horizontal characteristic vector field 𝜉 is given by ℎ

𝜉 =𝑘

so we have ℎ

ℎ

ℎ

0 = 𝑔 (𝜉, 𝜂 (𝜉 ) 𝜉) = 𝜂 (𝜉 ) 𝑔 (𝜉, 𝜉) = 𝜂 (𝜉 ) .

(51)

ℎ

Thus, 𝑓∗ (𝐽𝜉 ) = 0. ℎ

𝜂

(55)

ℎ

𝑓∗ (𝐽𝜉 ) = 𝜙 ∘ 𝑓∗ 𝜉 + 𝜂 (𝜉 ) 𝜉 = 𝜂 (𝜉 ) 𝜉,

ℎ

𝜕 𝜕 𝜕 𝜕 − ,𝑋 = +2 , 𝜕𝑥1 𝜕𝑦1 2 𝜕𝑥2 𝜕𝑦1

(49)

ℎ

𝜕 𝜕 𝜕 𝜕 −2 − +2 }, 𝜕𝑥1 𝜕𝑥2 𝜕𝑦1 𝜕𝑦2 (54)

which is the vertical distribution admitting one lightlike vector field; that is, fibre is degenerate submanifold of R42 . The horizontal distribution is

Proof. (i) By Corollary 5, (𝜙 , 𝜉 , 𝜂V , 𝑔V ) is almost paracontact pseudometric structure on 𝑓−1 (𝑞). We have ℎ

(53)

Hence, 𝐽𝜉 is a vertical vector field. 2𝑚 (ii) Since 𝑓 : 𝑀 → 𝑀2𝑛+1 is smooth submersion, = 𝜂|H is restriction of 𝜂 on the horizontal distribution

𝜕 𝜕 1 𝜕 5 𝜕 − (2𝑘 − ) − (𝑘 − 1) + (2𝑘 − ) , 𝜕𝑥1 3 𝜕𝑥2 𝜕𝑦1 3 𝜕𝑦2 (56)

which is 𝑓-related to the characteristic vector field 𝜉 = 𝜕/𝜕𝑤. Moreover, there exists one form 𝜂 = 5𝑑𝑥1 + 3𝑑𝑥2 + 5𝑑𝑦1 + 3𝑑𝑦2 on (R42 , 𝐽, 𝑔) such that the submersion satisfies (39). Example 9. Let (R63 , 𝐽, 𝑔) be an almost paracomplex pseudoRiemannian manifold and let (R31 , 𝜙, 𝜉, 𝜂, 𝑔) be an almost

6

Geometry

paracontact pseudo-Riemannian manifold. Consider a submersion 𝑓 : {R63 ; (𝑥1 , 𝑥2 , 𝑥3 , 𝑦1 , 𝑦2 , 𝑦3 )𝑡 } → {R31 ; (𝑢, V, 𝑤)𝑡 }, defined by

ℎ

ℎ

R42 such that 𝜂(𝐽𝜉 ) = 1, 𝜂(𝜉 ) = 0 and the map 𝑓 satisfies

𝑡

𝑓 ((𝑥1 , 𝑥2 , 𝑥3 , 𝑦1 , 𝑦2 , 𝑦3 ) ) 󳨃󳨀→ (

Thus, the smooth map 𝑓 is a pseudo-Riemannian submersion. Moreover, we obtain that there exists a 1-form 𝜂 = 𝑑𝑥2 on

(57)

𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑦2 + 𝑦3 𝑡 , , ). √2 √2 √2

Then, there exists one form 𝜂 = (𝑑𝑥2 + 𝑑𝑥3 )/√2 on such that (39) is satisfied. The kernel of 𝑓∗ is

𝑓∗ 𝐽𝑋 = 𝜙𝑓∗ 𝑋 + 𝜂 (𝑋) 𝜉, (R63 , 𝐽, 𝑔)

𝑓∗ 𝐽𝑌 = 𝜙𝑓∗ 𝑌 + 𝜂 (𝑌) 𝜉,

V = ker 𝑓∗

ℎ

= Span {𝑉1 =

𝜕 }, 𝜕𝑥3

Hence, the map 𝑓 is a paracomplex paracontact pseudo-Riemannian submersion from R42 on to R31 .

which is vertical distribution admitting non-lightlike vector fields; that is, the fibre is nondegenerate submanifold of (R63 , 𝐽, 𝑔). The horizontal distribution is H = Span {𝑋1 =

ℎ

𝑓∗ 𝐽𝜉 = 𝜙𝑓∗ 𝜉 + 𝜂 (𝜉 ) 𝜉.

𝜕 𝜕 𝜕 𝜕 𝜕 − ,𝑉 = − + , 𝜕𝑥1 𝜕𝑥2 2 𝜕𝑦1 𝜕𝑦2 𝜕𝑦3 (58)

𝑉3 =

ℎ

(64)

𝜕 𝜕 𝜕 𝜕 + ,𝑋 = − + , 𝜕𝑥1 𝜕𝑥2 2 𝜕𝑦1 𝜕𝑦3

𝜕 𝜕 𝑋3 = + }. 𝜕𝑦1 𝜕𝑦2

ℎ

𝑡

𝑡

ℎ

𝜀𝑔(𝑋, 𝜉 )𝜉 is 𝑓-related to 𝜙𝑋. (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 : {R42 ; (𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 )𝑡 } → {R31 ; (𝑢, V, 𝑤)𝑡 }, defined by 𝑓 ((𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ) ) 󳨃󳨀→ (𝑥1 , 𝑦1 , 𝑦2 ) .

Proposition 11. Let 𝑓 : 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) −

Proof. Since 𝑋 is 𝑓-related to vector field 𝑋 on 𝑀, we have

ℎ

𝜂 (𝑋) = {𝜂V + 𝜂ℎ } (𝑋) = 0 + 𝜂ℎ (𝑋) = 𝜀𝑔 (𝑋, 𝜉 ) , 󳨐⇒ 𝑓∗ (𝐽 𝑋) = 𝜙𝑋 + 𝜂ℎ (𝑋) 𝜉, ℎ

(65)

ℎ

(60)

󳨐⇒ 𝑓∗ {𝐽 𝑋 − 𝜀𝑔 (𝑋, 𝜉 ) 𝜉 } = 𝜙𝑋.

(61)

Hence, 𝐽(𝑋) − 𝜀𝑔(𝑋, 𝜉 )𝜉 is 𝑓-related to 𝜙𝑋.

Then, the kernel of 𝑓∗ is V = ker 𝑓∗ = Span {𝑉1 =

𝜕 }, 𝜕𝑥2

ℎ

ℎ

which is the vertical distribution and the restriction of 𝑔 to the fibres of 𝑓 is nondegenerate. The horizontal distribution is

Proposition 12. Let 𝑓 : 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

𝜕 𝜕 ℎ 𝜕 ,𝑌 = ,𝜉 = }. 𝜕𝑥1 𝜕𝑦1 𝜕𝑦2 (62)

the vertical and horizontal distributions, respectively. If 𝜉 is the basic characteristic vector field of horizontal distribution 𝑓-related to the characteristic vector field 𝜉 of base manifold, then

⊥

H = (ker 𝑓∗ ) = Span {𝑋 =

The characteristic vector field 𝜉 = 𝜕/𝜕𝑤 on R31 has unique

ℎ

ℎ

horizontal lift 𝜉 , which is the characteristic vector field on horizontal distribution H of R42 . We also have 𝑔 (𝑋, 𝑋) = 𝑔 (𝑓∗ 𝑋, 𝑓∗ 𝑋) = −1, 𝑔 (𝑌, 𝑌) = 𝑔 (𝑓∗ 𝑌, 𝑓∗ 𝑌) = 1, ℎ

ℎ

ℎ

ℎ

𝑔 (𝜉 , 𝜉 ) = 𝑔 (𝑓∗ 𝜉 , 𝑓∗ 𝜉 ) = 𝑔 (𝜉, 𝜉) = 1.

ℎ

V

ℎ

(i) 𝐽V ⊂ D ⊕ {𝐽𝜉 } ⊕ {𝜉 }, ℎ

ℎ

ℎ

(ii) 𝐽H ⊂ D ⊕ {𝜉 } ⊕ {𝐽𝜉 }. (63)

Proof. (i) Let 𝑈 ∈ V. Then, 𝑈 = 𝑎𝑈| 𝐶∞ (𝑀), as 𝐽𝜉

ℎ

V

D

ℎ

V

+ 𝑏𝐽𝜉 , for 𝑎, 𝑏 ∈

= 𝜉 is characteristic vector field on odd

Geometry

7

dimensional fibre submanifold 𝑓−1 (𝑞) of 𝑀, 𝑞 ∈ 𝑀. We get 2 ℎ

𝐽𝑈 = 𝑎𝐽 𝑈|

V

+ 𝑏𝐽 𝜉

= 𝑎𝐽 𝑈|

V

+ 𝑏𝜉 ∈ V ⊕ {𝜉 } ,

D

ℎ

D

ℎ

(66)

paracontact pseudo-Riemannian manifold. Consider a submersion 𝑓 : {R63 ; (𝑥1 , 𝑥2 , 𝑥3 , 𝑦1 , 𝑦2 , 𝑦3 )𝑡 } → {R31 ; (𝑢, V, 𝑤)𝑡 }, defined by 𝑡

𝑓 ((𝑥1 , 𝑥2 , 𝑥3 , 𝑦1 , 𝑦2 , 𝑦3 ) ) 󳨃󳨀→ (

ℎ

󳨐⇒ 𝐽V ⊂ V ⊕ {𝜉 } .

Then, the kernel of 𝑓∗ is

ℎ

Again, let 𝑉 ∈ V ⊕ {𝜉 }. Then 𝑉 = 𝑎𝑉|

D

V

𝜂V (𝑉| V ) = 0, D = ker 𝜂V , 𝑎𝑉| D

∞

D

𝐶 (𝑀). We have 𝐽 𝑉 = 𝑎𝐽 𝑉|

ℎ

D

V

+ 𝑏𝜉 + 𝑐𝐽𝜉

ℎ

ℎ

+ 𝑏𝐽𝜉 + 𝑐𝜉 , where

V

= Span {𝑉1 =

ℎ

V

𝜉 =

ℎ

ℎ

(67)

ℎ

= (𝑎𝐽 𝑉| V + 𝑐𝐽𝜉 ) + ⏟⏟𝑏𝜉 ⏟⏟⏟⏟⏟ ∈ V ⊕ {𝜉 } . D ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ℎ ∈{𝜉 }

∈V

V = ker 𝑓∗

ℎ

+ 𝑏𝐽𝜉 ∈ V, and 𝑎, 𝑏, 𝑐 ∈

V

Now, by (39), we get

(72)

𝜕 } 𝜕𝑥3

which is the vertical distribution and the restriction of 𝑔 to the fibres of 𝑓 is nondegenerate. The horizontal distribution is ⊥

= Span {𝑋1 =

ℎ

= 𝑐 {𝜙 (𝑓∗ 𝜉 ) + 𝜂 (𝜉) 𝜉}

(68) ℎ

𝜉 =

= 𝑐𝜉 ∈ {𝜉} ⊈ 𝐽V. We get 𝐽𝑉 ∉ V.

ℎ

ℎ

V

ℎ

Hence, 𝐽V ⊂ V ⊕ {𝜉 }; that is, 𝐽V ⊂ D ⊕ {𝐽𝜉 } ⊕ {𝜉 }. (ii) Let 𝑋 = 𝑎𝑋| ℎ

𝜕 𝜕 𝜕 𝜕 − ,𝑉 = − , 𝜕𝑥1 𝜕𝑥2 2 𝜕𝑦1 𝜕𝑦2

H = (ker 𝑓∗ )

𝑓∗ 𝐽 𝑉 = 𝜙 (𝑓∗ 𝑉) + 𝜂 (𝑓∗ (𝑉)) 𝜉

ℎ

𝑡 𝑥1 + 𝑥2 𝑦1 + 𝑦2 , , 𝑦3 ) . √2 √2 (71)

D

+ 𝑏𝜉

ℎ

ℎ

ℎ

∈ H, where H = D ⊕

{𝜉 }, ker 𝜂ℎ = D , and 𝑎, 𝑏 ∈ 𝐶∞ (𝑀). Then 𝐽 𝑋 = 𝑎𝐽 𝑋|

ℎ

D

(69)

ℎ

ℎ

ℎ

ℎ

D

ℎ

2 ℎ

ℎ

+ 𝑏𝐽𝜉 + 𝑐𝐽 𝜉

= 𝑎𝐽 𝑌|

ℎ

+ 𝑏𝐽𝜉 + 𝑐𝜉

D

D

ℎ

ℎ

(74)

ℎ

Thus, the smooth map 𝑓 is a pseudo-Riemannian submersion. Also, we obtain that there exists a 1-form 𝜂 = 𝑑𝑥3 on R63 ℎ

𝑓∗ 𝐽 𝑋1 = 𝜙𝑓∗ 𝑋1 + 𝜂 (𝑋1 ) 𝜉, ℎ

(70)

ℎ

ℎ

ℎ

ℎ

ℎ

ℎ

D

𝑔 (𝑋2 , 𝑋2 ) = 𝑔 (𝑓∗ 𝑋2 , 𝑓∗ 𝑋2 ) = 2,

such that 𝜂(𝐽𝜉 ) = 1, 𝜂(𝜉 ) = 0 and the map 𝑓 satisfies

= 𝑍 + 𝑏𝐽𝜉 ∈ H ⊕ {𝐽𝜉 } , for some 𝑍 = 𝑎𝐽 𝑌|

ℎ

horizontal lift 𝜉 , which is the characteristic vector field on the horizontal distribution H of R63 . We also have

𝑔 (𝜉 , 𝜉 ) = 𝑔 (𝑓∗ 𝜉 , 𝑓∗ 𝜉 ) = 𝑔 (𝜉, 𝜉) = 1.

Again, let 𝑌 ∈ H⊕{𝐽𝜉 }. Then, 𝑌 = 𝑎𝑌| ℎ +𝑏𝜉 +𝑐𝐽𝜉 ∉ H,

𝐽 𝑌 = 𝑎𝐽 𝑌|

𝜕 }. 𝜕𝑦3

The characteristic vector field 𝜉 = 𝜕/𝜕𝑤 on R31 has unique

ℎ

which implies that 𝐽H ⊂ H ⊕ {𝐽𝜉 }. for 𝑎, 𝑏, 𝑐 ∈ 𝐶∞ (𝑀). We have

(73)

𝑔 (𝑋1 , 𝑋1 ) = 𝑔 (𝑓∗ 𝑋1 , 𝑓∗ 𝑋1 ) = −2, ℎ

+ 𝑏𝐽𝜉 ∈ H ⊕ {𝐽𝜉 } ,

ℎ

𝜕 𝜕 𝜕 𝜕 + ,𝑋 = + , 𝜕𝑥1 𝜕𝑥2 2 𝜕𝑦1 𝜕𝑦2

+ 𝑐𝜉 ∈ H.

We obtain 𝐽 𝑌 ∉ H. ℎ ℎ ℎ Hence, 𝐽H ⊂ H ⊕ {𝐽(𝜉 )}; that is, 𝐽H ⊂ D ⊕ {𝜉 } ⊕ ℎ

{𝐽𝜉 }. Example 13. Let (R63 , 𝐽, 𝑔) be an almost paracomplex pseudoRiemannian manifold and let (R31 , 𝜙, 𝜉, 𝜂, 𝑔) be an almost

𝑓∗ 𝐽 𝑋2 = 𝜙𝑓∗ 𝑋2 + 𝜂 (𝑋2 ) 𝜉, ℎ

ℎ

(75)

ℎ

𝑓∗ 𝐽𝜉 = 𝜙𝑓∗ 𝜉 + 𝜂 (𝜉 ) 𝜉. Hence, the map 𝑓 is a paracomplex paracontact pseudoRiemannian submersion from R63 onto R31 . Moreover, we observe that, for this submersion 𝑓, we have ℎ

𝐽V ⊂ V ⊕ {𝜉 } , which verifies Proposition 12.

ℎ

𝐽H ⊂ H ⊕ {𝐽𝜉 } ,

(76)

8

Geometry

Proposition 14. Let 𝑓 : 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 LeviCivita connection on the total manifold 𝑀 and base manifold 𝑀, respectively. Then, one has (i) 𝑓∗ ((∇𝑋 𝐽)𝑌) = (∇𝑋 𝜙)𝑌 + 𝜀𝑔(𝑌, ∇𝑋 𝜉)𝜉 + 𝜂(𝑌)∇𝑋𝜉,

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) .

(ii) 𝐹 = 𝑓∗∗ Φ + 𝜀𝜂 ⊗ 𝜂,

(80)

(iii) 𝑓∗ ((∇𝑋 𝐹)(𝑌, 𝑍)) = (∇𝑋 Φ)(𝑌, 𝑍) + 𝜂(𝑌)𝑔(𝑍, ∇𝑋 𝜉) + 𝜂(𝑍)𝑔(𝑌, ∇𝑋 𝜉). Proof. (i) In view of Definition 2 and Proposition 11, we have

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

𝑓∗ ((∇𝑋 𝐽) 𝑌) = 𝑓∗ (∇𝑋 (𝐽 𝑌) − 𝐽 (∇𝑋 𝑌)) = ∇𝑋 (𝑓∗ (𝐽 𝑌)) − 𝑓∗ (𝐽 (∇𝑋 𝑌))

− 𝜂 (𝑌) 𝑁(4) (𝑋) 𝜉 − 𝜂 (𝑌) 𝑁(3) (𝑋)

= ∇𝑋 (𝜙𝑌) + ∇𝑋 (𝜂 (𝑌) 𝜉) − 𝜙 (∇𝑋 𝑌)

+ 𝜂 (𝑋) 𝑁(3) (𝑌) .

− 𝜂 (∇𝑋𝑌) 𝜉 = (∇𝑋 𝜙) 𝑌 + ∇𝑋 (𝜀𝑔 (𝑌, 𝜉) 𝜉) − 𝜂 (∇𝑋𝑌) 𝜉 = (∇𝑋 𝜙) 𝑌 + 𝜀𝑔 (∇𝑋 𝑌, 𝜉) 𝜉 + 𝜀𝑔 (𝑌, ∇𝑋 𝜉) 𝜉 + 𝜀𝑔 (𝑌, 𝜉) ∇𝑋 𝜉 − 𝜀𝑔 (∇𝑋 𝑌, 𝜉) 𝜉 = (∇𝑋 𝜙) 𝑌 + 𝜀𝑔 (𝑌, ∇𝑋 𝜉) 𝜉 + 𝜂 (𝑌) ∇𝑋 𝜉. (77) (ii) Since 𝑓∗∗ Φ is pullback of Φ through the linear map 𝑓∗ , we get 𝑓∗∗ Φ (𝑋, 𝑌) = Φ (𝑋, 𝑌) ∘ 𝑓 = 𝑔 (𝑋, 𝜙𝑌) ∘ 𝑓 = 𝑔 (𝑋, 𝐽 𝑌) − 𝜀𝜂 (𝑌) 𝜂 (𝑋)

(78)

= 𝐹 (𝑋, 𝑌) − 𝜀𝜂 (𝑋) 𝜂 (𝑌) ,

= 0, it follows from (81) that Since 𝑁𝐽 (𝑋, 𝑌) (1) (2) (3) tensors 𝑁 , 𝑁 , 𝑁 , 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. Let 𝑓 : 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 and 𝑁(1) vanishes. Then, from (80), we have

which implies 𝐹 = 𝑓∗∗ Φ + 𝜀𝜂 ⊗ 𝜂. (iii) By (23), we have 𝑓∗ ((∇𝑋 𝐹) (𝑌, 𝑍)) = 𝑔 (𝑓∗ (𝑌) , 𝑓∗ ((∇𝑋 𝐽) 𝑍)) .

(81)

(79)

Now, using (i) in the above equation, we get (iii). Theorem 15. Let 𝑓 : 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 tensors 𝑁𝐽 and 𝑁𝜙 of almost paracomplex structure 𝐽 and almost paracontact structure 𝜙 are, respectively, defined by (8) and (11).

0 = 2𝑑𝜂 (𝜙𝑋, 𝑌) 𝜉 − 2𝑑𝜂 (𝜙𝑌, 𝑋) 𝜉 + 2𝜂 (𝑋) 𝑑𝜂 (𝜉, 𝑌) 𝜉 − 2𝜂 (𝑌) 𝑑𝜂 (𝜉, 𝑋) 𝜉 − 𝜂 (𝑌) (L𝜉 𝜙) 𝑋 + 𝜂 (𝑋) (L𝜉 𝜙) 𝑌. (82) If 𝜉 is killing, then we have L𝜉 𝜙 = 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 get L𝜉 𝜙 = 0. Hence, the characteristic vector field 𝜉 is killing.

Geometry

9

Theorem 17. Let 𝑓 : 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-K¨ahler, then the base space is paracosymplectic. The converse is true if ∇𝑋 𝐽 is vertical. Proof. We have, for any 𝑋, 𝑌 ∈ Γ(𝑇𝑀), (∇𝑋 𝜙)𝑌 = 0, which gives 𝑔(𝑍, (∇𝑋 𝜙)𝑌) = 0, for any 𝑍 ∈ Γ(𝑇𝑀). From Proposition 14, we have {𝑔 (𝑍, (∇𝑋𝜙) 𝑌) + 𝜀𝜂 (𝑌) (∇𝑋 𝜂) 𝑍 + 𝜀𝜂 (𝑍) (∇𝑋 𝜂) 𝑌} ∘ 𝑓 = 𝑔 (𝑍, (∇𝑋 𝐽) 𝑌) . (84) Let ∇ 𝐽 = 0; that is, the total space is para-K¨ahler. 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.

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) T𝐽𝑈𝑉 = 𝐽(T𝑈𝑉). Proof. The proof follows using similar steps as in Lemmas 3 and 4 of [13], so we omit it. Lemma 20. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from a para-K¨ahler 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 (𝐽𝐹) = 𝐽(A𝐸 𝐹), (ii) T𝐸 (𝐽𝐹) = 𝐽(T𝐸 𝐹).

Theorem 18. Let 𝑓 : 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-K¨ahler if and only if the base space 𝑀 is an almost paracosymplectic manifold.

Theorem 21. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from a para-K¨ahler 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. We have the following equation:

Proof. For any vertical vector field 𝑈, we have

3𝑑𝐹 (𝑋, 𝑌, 𝑍)

𝑔 (𝐽 (A𝑋 𝑌) , 𝑈) = 𝑔 (A𝑋 𝐽 𝑌, 𝑈)

= 3 (𝑓∗∗ 𝑑Φ) (𝑋, 𝑌, 𝑍) + 2𝜀𝜂 (𝑍) 𝑑𝜂 (𝑋, 𝑌)

= −𝑔 (𝐽 𝑌, A𝑋 𝑈)

− 2𝜀𝜂 (𝑌) 𝑑𝜂 (𝑋, 𝑍) + 2𝜀𝜂 (𝑋) 𝑑𝜂 (𝑌, 𝑍) + 2𝜀𝜂 (𝑋) 𝑍 (𝜂 (𝑌))

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

(85)

+ 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-K¨ahler. 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 and 𝑑Φ = 0. Hence, the base space is almost paracosymplectic. Now, we investigate the properties of fundamental tensors T and A of a pseudo-Riemannian submersion. Lemma 19. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from a para-K¨ahler manifold 𝑀 onto an almost paracontact pseudometric manifold 𝑀 and

= −𝑔 (𝐽 𝑌, ℎ (∇𝑈𝑋)) = 𝑔 (𝑌, ℎ (𝐽 (∇𝑈𝑋))) = 𝑔 (𝑌, ℎ {(−∇𝑈𝐽) 𝑋 +∇𝑈 (𝐽 𝑋)}) = 𝑔 (𝑌, ℎ {∇𝑈 (𝐽𝑋)}) = 𝑔 (𝑌, A𝐽 𝑋 𝑈) = −𝑔 (A𝐽 𝑋 𝑌, 𝑈) = −𝑔 (𝐽 (A𝑋 𝑌) , 𝑈) . (86) 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. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from a para-K¨ahler 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.

10

Geometry

Proof. The second fundamental form of 𝑓 is defined by 𝑓

(∇𝑓∗ ) (𝐸, 𝐹) = (∇𝐸 𝑓∗ (𝐹)) ∘ 𝑓 − 𝑓∗ (∇𝐸 𝐹) ,

(87)

where 𝐸, 𝐹 ∈ Γ(𝑇𝑀) and ∇𝑓 is pullback connection of LeviCivita connection ∇ on 𝑀 with respect to 𝑓. We have, for any 𝑋, 𝑌 ∈ H, 𝑓

(∇𝑓∗ ) (𝑋, 𝑌) = (∇𝑋 𝑓∗ (𝑌)) ∘ 𝑓 − 𝑓∗ (∇𝑋 𝑌) .

(88)

By using Lemma 1, we have 𝑓∗ (ℎ(∇𝑋 𝑌)) = (∇𝑋 𝑌) ∘ 𝑓, which implies ∇𝑓∗ = 0. Hence, the submersion 𝑓 is an affine map on H. Theorem 23. Let 𝑓 : 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 any 𝑈, 𝑉 ∈ V, (∇𝑓∗ ) (𝑈, 𝑉) = −𝑓∗ (ℎ (∇𝑈𝑉)) ,

(89)

which, in view of (27), gives (∇𝑓∗ ) (𝑈, 𝑉) = −𝑓∗ (T𝑈𝑉) .

Theorem 24. Let 𝑓 : 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(∇𝐸 ℎ𝐹) + Aℎ𝐸 V𝐹 + TV𝐸 V𝐹 is 𝑓-related to ∇𝑋 𝑌, 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 (∇𝐸 𝐹)) = (∇𝑋 𝑌) ∘ 𝑓 − 𝑓∗ (ℎ (∇ℎ𝐸 ℎ𝐹 + ∇ℎ𝐸 V𝐹 +∇V𝐸 ℎ𝐹 + ∇V𝐸 ℎ𝐹)) . (91) By using (27) and (31) in the above equation, we have

+TV𝐸 V𝐹) .

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. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudoRiemannian 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 fields 𝐸, 𝐹 span the 2dimensional 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

(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 on V. Conversely, let the submersion 𝑓 be an affine map on V. Then, ∇𝑓∗ = 0, which implies T = 0. Hence, the fibres of 𝑓 are totally geodesic.

(∇𝑓∗ ) (𝐸, 𝐹) = (∇𝑋 𝑌) ∘ 𝑓 − 𝑓∗ (ℎ (∇𝐸 ℎ𝐹) + Aℎ𝐸 V𝐹

Let the submersion map be affine. Then, for any 𝐸, 𝐹 ∈ Γ(𝑇𝑀), (∇𝑓∗ )(𝐸, 𝐹) = 0. Equation (92) implies (∇𝑋 𝑌) ∘ 𝑓 = 𝑓∗ (ℎ(∇𝐸 ℎ𝐹) + Aℎ𝐸 V𝐹 + TV𝐸 ℎ𝐹). Conversely, let ℎ(∇𝐸 𝐹) + Aℎ𝐸 V𝐹 + TV𝐸 ℎ𝐹 be 𝑓-related to ∇𝑋 𝑌, for any 𝐸, 𝐹 ∈ Γ(𝑇𝑀). Then, from (92), we have (∇𝑓∗ )(𝐸, 𝐹) = 0. Hence, the submersion map 𝑓 is affine.

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 field 𝐽𝐸 is also non-lightlike and {𝐸, 𝐽𝐸} span the 2-dimensional plane. 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 and semi-Riemannian submersion have been extensively studied in the work of O’Neill [1] and Gray [3]. ℎ

V

Let 𝐵 and 𝐵 be the paraholomorphic bisectional curvatures of horizontal and vertical spaces, respectively. Let ℎ

V

𝐻 and 𝐻 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. Let 𝑓 : 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 V

(92)

𝐵 (𝑈, 𝑉) = 𝐵 (𝑈, 𝑉) + 𝑔 (T𝑈 (𝐽 𝑉) , T𝐽𝑈𝑉) − 𝑔 (T𝐽𝑈 (𝐽 𝑉) , T𝑈𝑉) ,

(95)

Geometry

11

𝐵 (𝑋, 𝑈) = 𝑔 ((∇𝑈A)𝑋 𝐽 𝑋, 𝐽 𝑈) − 𝑔 (A𝑋 𝐽𝑈, A𝐽 𝑋 𝑈) + 𝑔 (A𝑋 𝑈, A𝐽 𝑋 𝐽𝑈) − 𝑔 ((∇𝐽𝑈A)𝑋 𝐽 𝑋, 𝑈)

Proof. Since the fibres are totally geodesic, T = 0; consequently we have V

𝐻 (𝑈) = 𝐻 (𝑈) .

+ 𝑔 (T𝐽𝑈𝑋, T𝑈 (𝐽 𝑋)) − 𝑔 (T𝑈𝑋, T𝐽𝑈 (𝐽 𝑋)) , (96) ℎ

𝐵 (𝑋, 𝑌) = 𝐵 (𝑋, 𝑌) − 2𝑔 (A𝑋 (𝐽 𝑋) , A𝑌 (𝐽 𝑌)) + 𝑔 (A𝐽 𝑋 𝑌, A𝑋 (𝐽 𝑌))

(97)

− 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. Let 𝑓 : 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 pseudoRiemannian submanifolds of 𝑀, then for any non-lightlike unit vertical vector fields 𝑈 and 𝑉, one has V

𝐵 (𝑈, 𝑉) = 𝐵 (𝑈, 𝑉) .

(98)

Corollary 27. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from an almost paraHermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of 𝑓 be totally geodesic pseudoRiemannian submanifolds of 𝑀. If the horizontal distribution is integrable, then, for any non-lightlike unit horizontal vector fields 𝑋 and 𝑌, one has ℎ

𝐵 (𝑋, 𝑌) = 𝐵 (𝑋, 𝑌) .

(99)

Proposition 28. Let 𝑓 : 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 V 󵄩2 󵄩 𝐻 (𝑈) = 𝐻 (𝑈) + 󵄩󵄩󵄩󵄩T𝑈(𝐽𝑈)󵄩󵄩󵄩󵄩 − 𝑔 (T𝐽𝑈 (𝐽𝑈) , T𝑈𝑈) , (100) 󵄩 󵄩2 𝐻 (𝑋) = 𝐻 (𝑋) ∘ 𝑓 − 3󵄩󵄩󵄩󵄩A𝑋 (𝐽 𝑋)󵄩󵄩󵄩󵄩 .

(101)

Proof. The proof is straightforward. If we take 𝑈 = 𝑉 in (95) and 𝑋 = 𝑌 in (97), we have (98) and (99). Corollary 29. Let 𝑓 : 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 pseudoRiemannian submanifolds of 𝑀, then the total manifold and fibres of 𝑓 have the same paraholomorphic sectional curvatures.

(102)

Corollary 30. Let 𝑓 : 𝑀 → 𝑀 be a paracomplex paracontact pseudo-Riemannian submersion from an almost paraHermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of 𝑓 be totally geodesic pseudoRiemannian 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 𝐻 (𝑋) = 𝐻 (𝑋) ∘ 𝑓.

(103)

𝑚

Theorem 31. Let 𝑓 : 𝑀 → 𝑀𝑛 be a paracomplex paracontact pseudo-Riemannian submersion from a paraK¨ahler 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 V

𝐵 (𝑈, 𝑉) = 𝐵 (𝑈, 𝑉) ,

(104)

󵄩 󵄩2 𝐵 (𝑋, 𝑈) = −2󵄩󵄩󵄩󵄩T𝑈𝑋󵄩󵄩󵄩󵄩 ,

(105)

𝐵 (𝑋, 𝑌) = 𝐵 (𝑋, 𝑌) ∘ 𝑓.

(106)

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

𝐵 (𝑈, 𝑉) = 𝐵 (𝑈, 𝑉) − 𝑔 (𝐽 (T𝑈𝑉) , 𝐽 (T𝑈𝑉)) 2

− 𝑔 (𝐽 (T𝑈𝑉) , T𝑈𝑉) = 𝐵 (𝑈, 𝑉) + 𝑔 (T𝑈𝑉, T𝑈𝑉) − 𝑔 (T𝑈𝑉, T𝑈𝑉) V

= 𝐵 (𝑈, 𝑉) . (107) Applying results of Lemma 19 in (96), we have 𝐵 (𝑋, 𝑈) = 𝑔 ((∇𝑈A)𝑋 (𝐽 𝑋) , 𝐽𝑈) − 𝑔 ((∇𝐽𝑈A)𝑋 (𝐽𝑋) , 𝑈) 󵄩 󵄩2 󵄩 󵄩2 + 2󵄩󵄩󵄩A𝑋 𝑈󵄩󵄩󵄩 − 2󵄩󵄩󵄩󵄩T𝑈𝑋󵄩󵄩󵄩󵄩 .

(108)

Since by Theorem 21 the horizontal distribution is integrable, we have A = 0, which implies 󵄩 󵄩2 𝐵 (𝑋, 𝑈) = −2󵄩󵄩󵄩󵄩T𝑈𝑋󵄩󵄩󵄩󵄩 . In view of A = 0, (104) follows from (97).

(109)

12

Geometry 𝑚

Theorem 32. Let 𝑓 : 𝑀 → 𝑀𝑛 be a paracomplex paracontact pseudo-Riemannian submersion from a paraK¨ahler 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 V 󵄩2 󵄩 (110) 𝐻 (𝑈) = 𝐻 (𝑈) − 2󵄩󵄩󵄩T𝑈𝑈󵄩󵄩󵄩 , 𝐻 (𝑋) = 𝐻 (𝑋) ∘ 𝑓.

(111)

Proof. Since 𝑓 is the paracomplex paracontact pseudoRiemannian submersion from a para-K¨ahler manifold 𝑀 onto an almost paracontact pseudometric manifold 𝑀, by (16) and equations of Lemma 19, we have 2 󵄩 󵄩2 𝑔 (T𝐽𝑈 (𝐽𝑈) , T𝑈𝑈) = 𝑔 (𝐽 (T𝑈𝑈) , T𝑈𝑈) = 󵄩󵄩󵄩T𝑈𝑈󵄩󵄩󵄩 ,

󵄩 󵄩2 𝑔 (T𝑈 (𝐽𝑈) , T𝑈 (𝐽𝑈)) = −𝑔 (T𝑈𝑈, T𝑈𝑈) = −󵄩󵄩󵄩T𝑈𝑈󵄩󵄩󵄩 (112) and by using the above results in (100), we have V 󵄩2 󵄩 󵄩2 󵄩 𝐻 (𝑈) = 𝐻 (𝑈) − 󵄩󵄩󵄩T𝑈𝑈󵄩󵄩󵄩 − 󵄩󵄩󵄩T𝑈𝑈󵄩󵄩󵄩 V 󵄩2 󵄩 = 𝐻 (𝑈) − 2󵄩󵄩󵄩T𝑈𝑈󵄩󵄩󵄩 .

(113)

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

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment Uma Shankar Verma is thankful to University Grant Commission, New Delhi, India, for financial support.

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[7] B. Watson, “G, 𝐺 -Riemannian submersions and nonlinear gauge field equations of general relativity,” in Global Analysis— Analysis on Manifolds, T. Rassias and M. Morse, Eds., vol. 57 of Teubner-Texte zur Mathematik, pp. 324–349, Teubner, Leipzig, Germany, 1983. [8] C. Altafini, “Redundant robotic chains on Riemannian submersions,” IEEE Transactions on Robotics and Automation, vol. 20, no. 2, pp. 335–340, 2004. [9] M. T. Mustafa, “Applications of harmonic morphisms to gravity,” Journal of Mathematical Physics, vol. 41, no. 10, pp. 6918– 6929, 2000. [10] B. Watson, “Almost Hermitian submersions,” Journal of Differential Geometry, vol. 11, no. 1, pp. 147–165, 1976. [11] D. Chinea, “Almost contact metric submersions,” Rendiconti del Circolo Matematico di Palermo II, vol. 34, no. 1, pp. 319–330, 1984. [12] D. Chinea, “Transference of structures on almost complex contact metric submersions,” Houston Journal of Mathematics, vol. 14, no. 1, pp. 9–22, 1988. [13] Y. G¨und¨uzalp and B. Sahin, “Para-contact para-complex pseudo-Riemannian submersions,” Bulletin of the Malaysian Mathematical Sciences Society. [14] I. Sato, “On a structure similar to the almost contact structure,” Tensor, vol. 30, no. 3, pp. 219–224, 1976. [15] P. K. Rashevskij, “The scalar field in a stratified space,” Trudy Seminara po Vektornomu i Tenzornomu Analizu s ikh Prilozheniyami k Geometrii, Mekhanike i Fizike, vol. 6, pp. 225– 248, 1948. [16] P. Libermann, “Sur les structures presque paracomplexes,” Comptes Rendus de l’Acad´emie des Sciences I , vol. 234, pp. 2517– 2519, 1952. [17] E. M. Patterson, “Riemann extensions which have K¨ahler metrics,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 64, pp. 113–126, 1954. [18] S. Sasaki, “On differentiable manifolds with certain structures which are closely related to almost contact structure I,” The Tohoku Mathematical Journal, vol. 12, pp. 459–476, 1960. [19] S. Zamkovoy, “Canonical connections on paracontact manifolds,” Annals of Global Analysis and Geometry, vol. 36, no. 1, pp. 37–60, 2009. [20] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 23 of Progress in Mathematics, Birkh¨auser, Boston, Mass, USA, 2002. [21] V. Cruceanu, P. Fortuny, and P. M. Gadea, “A survey on paracomplex geometry,” The Rocky Mountain Journal of Mathematics, vol. 26, no. 1, pp. 83–115, 1996. [22] P. M. Gadea and J. M. Masque, “Classification of almost paraHermitian manifolds,” Rendiconti di Matematica e delle sue Applicazioni VII, vol. 7, no. 11, pp. 377–396, 1991. [23] M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian Submersions and Related Topics, World Scientific, River Edge, NJ, USA, 2004. [24] E. G. Rio and D. N. Kupeli, Semi-Riemannian Maps and Their Applications, Kluwer Academic Publisher, Dordrecht, The Netherlands, 1999. [25] B. Sahin, “Semi-invariant submersions from almost Hermitian manifolds,” Canadian Mathematical Bulletin, vol. 54, no. 3, 2011.

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