Geometry

Volume 2014 (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

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

Academic Editor: Bennett Palmer

Copyright © 2014 S. S. Shukla and Uma Shankar 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.

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.

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 is an almost Hermitian manifold with structure () and is an almost contact metric manifold with structure (), then there does not exist a Riemannian submersion which commutes with the structures on and ; that is, we cannot have the condition . 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 Section 2, 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

Let be a -dimensional Riemannian manifold, a (1,1)-type tensor field, a vector field, called characteristic vector field, and a 1-form on . Then, is called an almost paracontact structure on if

and the tensor field induces an almost paracomplex structure on the distribution [18, 19].

is said to be an almost paracontact manifold, if it is equipped with an almost paracontact structure. Again, is called an almost paracontact pseudometric manifold if it is endowed with a pseudo-Riemannian metric of signature such that

where or according to the characteristic vector field is spacelike or timelike. It follows that

In particular, if , then the manifold is called a Lorentzian almost paracontact manifold.

If the metric is positive definite, then the manifold is the usual almost paracontact metric manifold [14].

The fundamental 2-form on is defined by

Let be an almost paracontact manifold with the structure . An almost paracomplex structure on is defined by

where is tangent to , is the coordinate on , and is a smooth function on .

An almost paracontact structure is said to be normal, if the Nijenhuis tensor of almost paracomplex structure defined as

for any vector fields , vanishes.

If and are vector fields on , then we have [18–20]

where is Nijenhuis tensor of is Lie derivative with respect to a vector field , and ,  ,  ,  and   are defined as

The almost paracontact structure is normal if and only if the four tensors ,  ,, and   vanish.

For an almost paracontact structure , vanishing of implies the vanishing of ,  , and . Moreover, vanishes if and only if is a killing vector field.

An almost paracontact pseudometric manifold is called

(i)normal, if ,(ii)paracontact, if ,(iii)-paracontact, if is paracontact and is killing,(iv)paracosymplectic, if , which implies , where is the Levi-Civita connection on ,(v)almost paracosymplectic, if and ,(vi)weakly paracosymplectic, if is almost paracosymplectic and , where is Riemannian curvature tensor,(vii)para-Sasakian, if and is normal,(viii)quasi-para-Sasakian, if and is normal.

2.2. Almost Paracomplex Manifolds

A -type tensor field on -dimensional smooth manifold is said to be an almost paracomplex structure if and is called almost paracomplex manifold.

An almost paracomplex manifold is such that the two eigenbundles and corresponding to respective eigenvalues and of have the same rank [21, 22].

An almost para-Hermitian manifold is a smooth manifold endowed with an almost paracomplex structure and a pseudo-Riemannian metric such that

Here, the metric is neutral; that is, has signature .

The fundamental 2-form of the almost para-Hermitian manifold is defined by

We have the following properties [21, 22]:

An almost para-Hermitian manifold is called

(i)para-Hermitian, if ; equivalently, ,(ii)para-Kähler, if, for any ,  ; that is, ,(iii)almost para-Kähler, if ,(iv)nearly para-Kähler, if ,(v)almost semi-para-Kähler, if ,(vi)semi-para-Kähler, if and .

2.3. Pseudo-Riemannian Submersion

Let and be two connected pseudo-Riemannian manifolds of indices and , respectively, with .

A pseudo-Riemannian submersion is a smooth map , which is onto and satisfies the following conditions [2, 3, 23, 24].

(i)The derivative map is surjective at each point .(ii)The fibres of over are either pseudo-Riemannian submanifolds of of dimension and index or the degenerate submanifolds of of dimension and index with degenerate metric of type , where and .(iii) preserves the length of horizontal vectors.

We denote the vertical and horizontal projections of a vector field on by (or by ) and (or by ), respectively. A horizontal vector field on is said to be basic if is -related to a vector field on . Thus, every vector field on has a unique horizontal lift on .

Lemma 1 (see [1, 23]). If is a pseudo-Riemannian submersion and ,   are basic vector fields on that are -related to the vector fields ,   on , respectively, then one has the following properties:

(i),(ii) is a vector field and ,(iii) is a basic vector field -related to , where and are the Levi-Civita connections on and , respectively,(iv), for any vector field and for any vector field .

A pseudo-Riemannian submersion determines tensor fields and of type on defined by formulas [1, 2, 23]

Let ,   be horizontal vector fields and let ,   be vertical vector fields on . Then, one has

for all .

Moreover, coincides with second fundamental form of the submersion of the fibre submanifolds. The distribution is completely integrable. In view of (37) and (38), is alternating on the horizontal distribution and 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   be an almost para-Hermitian manifold and let be an almost paracontact pseudometric manifold.

A pseudo-Riemannian submersion is called paracomplex paracontact pseudo-Riemannian submersion if there exists a -form on such that

Since, for each is a linear isometry between horizontal spaces and tangent spaces , there exists an induced almost paracontact structure on -dimensional horizontal distribution such that behave just like the fundamental collineation of almost paracomplex structure on and is an endomorphism such that and the rank of , where  .

It follows that, for any ,  , which implies that  , for any and [18].

Definition 3 (see [25]). A pseudo-Riemannian submersion is called semi--invariant submersion, if there is a distribution such that

where is orthogonal complementary to in .

Proposition 4. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Then, the fibres ,  , are semi--invariant submanifolds of of dimension .

Proof. Let . Then

where  .

Thus, we have

By (19), we get .

As is nondegenerate on , we have

Taking in (43), we obtain

Since fibre is an odd dimensional submanifold, there exists an associated 1-form which is restriction of on fibre submanifold ,  , and a characteristic vector field such that . So, we have .

Let us put and .

Then, and ,  .

Hence, the fibres are semi--invariant submanifolds of .

Corollary 5. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Then, the fibres are almost paracontact pseudometric manifolds with almost paracontact pseudo-Riemannian structures ,  , where ,  , and .

Proof. Since   are semi--invariant submanifolds of of odd dimension , (39) implies

for any  .

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

where  .

Equating horizontal and vertical components, we have

Hence, is almost paracontact pseudometric structure on the fibre .

Proposition 6. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be basic vector fields -related to ,  , respectively. Let and be -forms on the total manifold and the base manifold , respectively. Then, one has the following.

(i)The characteristic vector field is a vertical vector field.(ii), where is pullback of through .(iii), for any vertical vector field .(iv), for any horizontal vector field .

Remark 7. Results (ii) and (iv) are analogue version of results (i) and (iii) of Proposition 4 of [13].

Proof. (i) By Corollary 5, is almost paracontact pseudometric structure on . We have

Now,

so we have

Thus, .

Hence, is a vertical vector field.

(ii) Since is smooth submersion, is restriction of on the horizontal distribution , and is a linear isometry, for any ,  we get

Hence, pullback .

Results (iii) and (iv) immediately follow from the previous results.

Example 8. Let be a paracomplex pseudometric manifold and let be an almost paracontact pseudometric manifold.

Define a submersion by

Then, the kernel of is

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

The horizontal distribution is

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

which is -related to the characteristic vector field .

Moreover, there exists one form on such that the submersion satisfies (39).

Example 9. Let be an almost paracomplex pseudo-Riemannian manifold and let be an almost paracontact pseudo-Riemannian manifold. Consider a submersion , defined by

Then, there exists one form on such that (39) is satisfied. The kernel of is

which is vertical distribution admitting non-lightlike vector fields; that is, the fibre is nondegenerate submanifold of .

The horizontal distribution is

Example 10. Let be a paracomplex pseudometric manifold and let be an almost paracontact pseudometric manifold.

Consider a submersion , defined by

Then, the kernel of is

which is the vertical distribution and the restriction of to the fibres of is nondegenerate.

The horizontal distribution is

The characteristic vector field on has unique horizontal lift , which is the characteristic vector field on horizontal distribution of .

We also have

Thus, the smooth map is a pseudo-Riemannian submersion.

Moreover, we obtain that there exists a 1-form on such that ,   and the map satisfies

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

Proposition 11. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be basic vector fields -related to ,  , respectively. Then, is -related to .

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

Hence, is -related to .

Proposition 12. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let and be the vertical and horizontal distributions, respectively. If is the basic characteristic vector field of horizontal distribution -related to the characteristic vector field of base manifold, then

(i),(ii).

Proof. (i) Let . Then, , for , as is characteristic vector field on odd dimensional fibre submanifold of . We get

Again, let . Then  , where ,  ,  , and . We have

Now, by (39), we get

We get  .

Hence,  ;  that is,  .

(ii) Let ,  where ,  ,   and  . Then

which implies that  .

Again, let . Then, , for . We have

We obtain  .

Hence,  ;  that is,  .

Example 13. Let be an almost paracomplex pseudo-Riemannian manifold and let be an almost paracontact pseudo-Riemannian manifold. Consider a submersion , defined by

Then, the kernel of is

which is the vertical distribution and the restriction of to the fibres of is nondegenerate.

The horizontal distribution is

The characteristic vector field on has unique horizontal lift , which is the characteristic vector field on the horizontal distribution of .

We also have

Thus, the smooth map is a pseudo-Riemannian submersion.

Also, we obtain that there exists a 1-form on such that and the map satisfies

Hence, the map is a paracomplex paracontact pseudo-Riemannian submersion from onto .

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

which verifies Proposition 12.

Proposition 14. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be basic vector fields -related to ,  , respectively. Let and be the second fundamental forms and let   and be the Levi-Civita connection on the total manifold and base manifold , respectively. Then, one has

(i),(ii),(iii).

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

(ii) Since is pullback of through the linear map , we get

which implies  .

(iii) By (23), we have

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

Theorem 15. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be basic vector fields -related to ,  , 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 and of almost paracomplex structure and almost paracontact structure are, respectively, defined by (8) and (11).

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

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

Since , it follows from (81) that tensors  ,  ,  ,  and   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 .

Hence, is vertical.

Corollary 16. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be basic vector fields -related to ,, respectively. Let the total space be para-Hermitian manifold and vanishes. Then, the base space is paracontact pseudometric manifold if and only if is killing.

Proof. Let the total space be para-Hermitian and vanishes. Then, from (80), we have

If is killing, then we have . It immediately follows from (82) that

In view of (6) and (7), the above equation gives .

Conversely, let the base space be paracontact. Then, .

Using (6), (7), and (82), we get .

Hence, the characteristic vector field is killing.

Theorem 17. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be basic vector fields -related to ,  , respectively. If the total space is para-Kähler, then the base space is paracosymplectic. The converse is true if is vertical.

Proof. We have, for any , which gives , for any .

From Proposition 14, we have

Let ; that is, the total space is para-Kähler. Then, from (84), we obtain and . Hence, the base space is paracosymplectic.

Again, let and . Then, ,  which implies that is a vertical vector field.

Theorem 18. Let be a paracomplex paracontact pseudo-Riemannian submersion and let the fibres of be pseudo-Riemannian submanifolds of . Let ,  , and be basic vector fields -related to ,  , and , respectively. If , then the total space is almost para-Kähler if and only if the base space is an almost paracosymplectic manifold.

Proof. We have the following equation:

If  ,  , and , then, from (85), we have . Hence, the total space is almost para-Kähler.

Conversely, let and .

By using the above equation in (85), we have and .

Hence, the base space is almost paracosymplectic.

Now, we investigate the properties of fundamental tensors and of a pseudo-Riemannian submersion.

Lemma 19. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Then, for any horizontal vector fields ,   and for any vertical vector fields ,   on , one has

(i), (ii), (iii),(iv).

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

Lemma 20. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of  . Then, for any vector fields ,   on , one has

(i),(ii).

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

Theorem 21. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Then, the horizontal distribution is integrable.

Proof. For any vertical vector field , we have

Thus  , which is true for all and .

So, .

Hence, the horizontal distribution is integrable.

Theorem 22. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Then, the submersion is an affine map on .

Proof. The second fundamental form of is defined by

where and is pullback connection of Levi-Civita connection on with respect to .

We have, for any ,

By using Lemma 1, we have , which implies .

Hence, the submersion is an affine map on .

Theorem 23. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Then, the submersion is an affine map on if and only if the fibres of are totally geodesic.

Proof. We have, for any ,

which, in view of (27), gives

Let the fibres of be totally geodesic. Then, . Consequently, from the above equation, we have .

Thus, the map is affine on .

Conversely, let the submersion be an affine map on . Then, , which implies .

Hence, the fibres of are totally geodesic.

Theorem 24. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Then, the submersion is an affine map if and only if is -related to , for any .

Proof. For any with and , we have

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

Let the submersion map be affine. Then, for any ,  . Equation (92) implies .

Conversely, let be -related to , for any . Then, from (92), we have .

Hence, the submersion map 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.

Let be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold .

Suppose that the vector fields ,   span the 2-dimensional plane at point of and let be the Riemannian curvature tensor of . The paraholomorphic bisectional curvature of for any pair of nonzero non-lightlike vector fields ,   on is defined by the formula

For a nonzero non-lightlike vector field , the vector field is also non-lightlike and span the 2-dimensional plane. Then the paraholomorphic sectional curvature is defined as

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

Let and be the paraholomorphic bisectional curvatures of horizontal and vertical spaces, respectively. Let and be the paraholomorphic sectional curvatures of horizontal and vertical spaces, respectively. Let and be the paraholomorphic bisectional and sectional curvatures of the base manifold, respectively.

Proposition 25. Let be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Let ,   be non-lightlike unit vertical vector fields and let ,   be non-lightlike unit horizontal vector fields on . Then, one has

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 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold. If the fibres of are totally geodesic pseudo-Riemannian submanifolds of , then for any non-lightlike unit vertical vector fields and , one has

Corollary 27. Let be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of be totally geodesic pseudo-Riemannian submanifolds of . If the horizontal distribution is integrable, then, for any non-lightlike unit horizontal vector fields and , one has

Proposition 28. Let be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . Let and be non-lightlike unit vertical vector field and non-lightlike unit horizontal vector field, respectively. Then, one has

Proof. The proof is straightforward. If we take in (95) and in (97), we have (98) and (99).

Corollary 29. Let be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold. If the fibres of are totally geodesic pseudo-Riemannian submanifolds of , then the total manifold and fibres of have the same paraholomorphic sectional curvatures.

Proof. Since the fibres are totally geodesic,  ;  consequently we have

Corollary 30. Let be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometric manifold and let the fibres of be totally geodesic pseudo-Riemannian submanifolds of . 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, ; consequently, we have

Theorem 31. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . If ,   are the non-lightlike unit vertical vector fields and ,   are the non-lightlike unit horizontal vector fields, then one has

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

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

Since by Theorem 21 the horizontal distribution is integrable, we have , which implies

In view of , (104) follows from (97).

Theorem 32. Let be a paracomplex paracontact pseudo-Riemannian submersion from a para-Kähler manifold onto an almost paracontact pseudometric manifold and let the fibres of be pseudo-Riemannian submanifolds of . If ,   are non-lightlike unit vertical and non-lightlike unit horizontal vector fields, respectively, then one has

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

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

Again, since horizontal distribution is integrable, we have , 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.

References

B. O'Neill, “The fundamental equations of a submersion,” The Michigan Mathematical Journal, vol. 13, pp. 459–469, 1966.

B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1983.

A. Gray, “Pseudo-Riemannian almost product manifolds and submersions,” vol. 16, pp. 715–737, 1967.

J. P. Bourguignon and H. B. Lawson, A Mathematicians Visit to Kaluza-Klein Theory, Rendiconti del Seminario Matematico, 1989.

S. Ianus and M. Visinescu, “Space-time compactification and Riemannian submersions,” in The Mathematical Heritage, G. Rassias and C. F. Gauss, Eds., pp. 358–371, World Scientific, River Edge, NJ, USA, 1991.

J. P. Bourguignon and H. B. Lawson, Jr., “Stability and isolation phenomena for Yang-Mills fields,” Communications in Mathematical Physics, vol. 79, no. 2, pp. 189–230, 1981.

B. Watson, “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.

C. Altafini, “Redundant robotic chains on Riemannian submersions,” IEEE Transactions on Robotics and Automation, vol. 20, no. 2, pp. 335–340, 2004.

M. T. Mustafa, “Applications of harmonic morphisms to gravity,” Journal of Mathematical Physics, vol. 41, no. 10, pp. 6918–6929, 2000.

B. Watson, “Almost Hermitian submersions,” Journal of Differential Geometry, vol. 11, no. 1, pp. 147–165, 1976.

D. Chinea, “Almost contact metric submersions,” Rendiconti del Circolo Matematico di Palermo II, vol. 34, no. 1, pp. 319–330, 1984.

D. Chinea, “Transference of structures on almost complex contact metric submersions,” Houston Journal of Mathematics, vol. 14, no. 1, pp. 9–22, 1988.

Y. Gündüzalp and B. Sahin, “Para-contact para-complex pseudo-Riemannian submersions,” Bulletin of the Malaysian Mathematical Sciences Society

I. Sato, “On a structure similar to the almost contact structure,” Tensor, vol. 30, no. 3, pp. 219–224, 1976.

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.

P. Libermann, “Sur les structures presque paracomplexes,” Comptes Rendus de l'Académie des Sciences I, vol. 234, pp. 2517–2519, 1952.

E. M. Patterson, “Riemann extensions which have Kähler metrics,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 64, pp. 113–126, 1954.

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.

S. Zamkovoy, “Canonical connections on paracontact manifolds,” Annals of Global Analysis and Geometry, vol. 36, no. 1, pp. 37–60, 2009.

D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 23 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 2002.

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.

P. M. Gadea and J. M. Masque, “Classification of almost para-Hermitian manifolds,” Rendiconti di Matematica e delle sue Applicazioni VII, vol. 7, no. 11, pp. 377–396, 1991.

M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian Submersions and Related Topics, World Scientific, River Edge, NJ, USA, 2004.

E. G. Rio and D. N. Kupeli, Semi-Riemannian Maps and Their Applications, Kluwer Academic Publisher, Dordrecht, The Netherlands, 1999.

B. Sahin, “Semi-invariant submersions from almost Hermitian manifolds,” Canadian Mathematical Bulletin, vol. 54, no. 3, 2011.