Abstract
Purpose
The central idea of this research article is to examine the characteristics of Clairaut submersions from Lorentzian trans-Sasakian manifolds of type (α, β) and also, to enhance this geometrical analysis with some specific cases, namely Clairaut submersion from Lorentzian α-Sasakian manifold, Lorentzian β-Kenmotsu manifold and Lorentzian cosymplectic manifold. Furthermore, the authors discuss some results about Clairaut Lagrangian submersions whose total space is a Lorentzian trans-Sasakian manifolds of type (α, β). Finally, the authors furnished some examples based on this study.
Design/methodology/approach
This research discourse based on classifications of submersion, mainly Clairaut submersions, whose total manifolds is Lorentzian trans-Sasakian manifolds and its all classes like Lorentzian Sasakian, Lorenztian Kenmotsu and Lorentzian cosymplectic manifolds. In addition, the authors have explored some axioms of Clairaut Lorentzian submersions and illustrates our findings with some non-trivial examples.
Findings
The major finding of this study is to exhibit a necessary and sufficient condition for a submersions to be a Clairaut submersions and also find a condition for Clairaut Lagrangian submersions from Lorentzian trans-Sasakian manifolds.
Originality/value
The results and examples of the present manuscript are original. In addition, more general results with fair value and supportive examples are provided.
Keywords
Citation
Siddiqi, M.D., Chaubey, S.K. and Siddiqui, A.N. (2024), "Clairaut anti-invariant submersions from Lorentzian trans-Sasakian manifolds", Arab Journal of Mathematical Sciences, Vol. 30 No. 2, pp. 134-149. https://doi.org/10.1108/AJMS-05-2021-0106
Publisher
:Emerald Publishing Limited
Copyright © 2021, Mohd Danish Siddiqi, Sudhakar Kumar Chaubey and Aliya Naaz Siddiqui
License
Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
The conception of Riemannian immersion is studied extensively together with starting the study of Riemannian geometry. In fact, Riemannian manifolds are studied first as surfaces imbedded in
Contrastingly, “dual” concept of Riemannian immersions is one of the famous research fields in differential geometry and is the theory of Riemannian submersions, which was first investigated by O’Neill [4] and Gray [5]. Watson [6] popularized the knowledge of Riemannian submersions considering almost Hermitian manifolds in terms of almost Hermitian submersions. Afterward, almost Hermitian submersions are discussed with in various subcategories of almost Hermitian manifolds. Also, Riemannian submersions are enhanced considering many subcategories of almost contact metric manifolds in terms of contact Riemannian submersions. Several materials about Riemannian, almost Hermitian or contact Riemannian submersions are available in reference [2].
Most of the research linked to the theory of anti-invariant Riemannian, Lagrangian submersions and Clairaut anti-invariant submersions is available in Şahin’s book [3]. Afterward, several kinds of Riemannian submersions appeared, for example: semi-invariant, slant, pointwise-slant, semi-slant, hemi-slant and generic submersions. Most of the studies related to these can also be found in Şahin’s book [3].
In 1972, Bishop [7] proposed the concept and conditions of a Clairaut submersion in terms of a natural generalization of a surface of revolution. Under these circumstances, for every geodesic σ at the surface
The concept of anti-invariant Riemannian and Clairaut anti-invariant submersion has been fitting a very progressive geometric analysis field since Şahin [8] essentially described such submersions of almost Hermitian manifolds on Riemannian manifolds. Indeed, anti-invariant Riemannian and Clairaut anti-invariant submersion have been examined in various types of geometrical manifolds, namely Kähler [8–10], almost product [11], Sasakian [12, 13], Kenmotsu [13], cosymplectic [30], paracosymplectic [14, 15] and trans-Sasakian manifolds [16–18]. Note that this concept of anti-invariant Riemannian submersion is generalized to conformal anti-invariant submersions [19–21].
In [22], Allison proposed Clairaut submersions in case the total manifold is Lorentzian. In addition, it is discovered that Clairaut submersions are used for static spacetime applications. Basically, a static spacetime can be considered as a Lorentzian manifold.
On the other hand, in 2013, De et al. [23] presented the concept of Lorentzian trans-Sasakian manifolds. Trans-Sasakian structure together with Lorentzian metric can be applied naturally at the odd dimensional manifold. Motivated by above research studies mentioned in this paper, we have examined the Clairaut anti-invariant submersions from Lorentzian trans-Sasakian manifolds.
The work is ordered as follows. Section 2 presents basic notion and definition for Lorentzian trans-Sasakian manifolds. Section 3 includes particular background of Riemannian submersions. Section 4 presents definition of anti-invariant and Lagrangian submersions. In section 5, we study anti-invariant submersions and Clairaut anti-invariant submersion from trans-Sasakian manifolds onto Riemannian manifolds admitting horizontal Reeb vector field. In section 6, we deal with some axioms of Clairaut Lagrangian submersion and provide some examples and some of their characteristic properties.
2. Lorentzian trans-Sasakian manifolds
A (2n + 1)-dimensional differentiable manifold M is named the Lorentzian Trans-Sasakian manifold [23] in case it allows (1, 1) tensor field ϕ, the global vector field ζ named Reeb vector field or contra-variant vector field, that is, in case η is a dual 1-form of ζ, and the Lorentzian metric g that satisfies [24].
If α = 0 and β ≠ 0(or β = 1), therefore the manifold turns into the Lorentzian β-Kenmotsu manifold (or Lorentzian Kenmotsu manifold) [23].
If α ≠ 0(or α = 1) and β = 0, therefore this manifold turns into the Lorentzian α-Sasakian manifold (or Lorentzian Sasakian manifold) [23].
In case α = 0 and β, therefore, the manifold turns into the Lorentzian cosymplectic manifold [23].
3. Riemannian submersions
An essential background of Riemannian submersions is given at this part.
Suppose (M, g) and (N, gN) are Riemannian manifolds, such that dim(M) > dim(N). The subjective mapping ψ: (M, g) → (N, gN) is named the Riemannian submersion [4] if:
(S1) The rank(ψ) = dim(N).
Therefore, for all q ∈ N,
The vector field at M is named vertical (resp. horizontal) in case it is still as a tangent (orthogonal) relating to the fibers. The vector field X at M is named basic in case X is horizontal and ψ-connecting to the vector field X* at N, which means ψ*(Xp) = X∗ψ(p) for any p ∈ M, where ψ* is derivative or differential map of ψ.
(S2) ψ* preserves the lengths of horizontal vectors.
This condition is equivalent to say that the derivative map ψ* of ψ, restricted to
On the other hand, from (3.1) and (3.2), we obtain
At the end, the concept of second fundamental form of the map within Riemannian manifolds is recalled. Suppose (M, g) and (N, gN) are Riemannian manifolds and f: (M, g) → (N, gN) is the smooth map. Therefore, second fundamental form of f is written as
4. Anti-invariant Riemannian submersions
We first recall idea of an anti-invariant Riemannian submersion where its total manifold is the almost contact metric manifold.
([18, 27]) Let M be (2n + 1)-dimensional almost contact metric manifold among almost contact metric constructor (ϕ, ζ, η, g) and N is the Riemannian manifold among Riemannian metric gN. Considering there is Riemannian submersion ψ: M → N where vertical distribution kerψ* defines anti-invariant with respect to ϕ, which means,
Here, horizontal distribution
It is said that the anti-invariant ψ: M → N allows vertical Reeb vector field in case Reeb vector field ξ is tangent to kerψ* and allows horizontal Reeb vector field in case Reeb vector field ξ is normal to kerψ*. Clearly, μ includes Reeb vector field ξ if ψ: M → N allows horizontal Reeb vector field ξ.
Now, we begin to study anti-invariant submersions admitting vertical Reeb vector field from Lorentzian trans-Sasakian manifolds (M, ϕ, ζ, η, g) of type (α, β) using (nontrivial) example.
Suppose M is three-dimensional Euclidean space written as
An orthonormal ϕ-basis of this structure is written as
Here, the map
Since the rank of this matrix equals 1, the map π is the submersion. Using some calculation leads to
Immediate calculations show that ψ ensures the condition (S2). Thus, ψ is the Riemannian submersion. Moreover, we got ϕ(U) = W. Therefore, ψ is the anti-invariant submersion admitting vertical Reeb vector field.
Throughout this research, as a total manifold of an anti-invariant submersion, let us consider a Lorentzian trans-Sasakian manifold (M, ϕ, ζ, η, g) of type (α, β) such that both α ≠ 0 and β ≠ 0.
Notion of Lagrangian submersion is considered the specific case from notion of anti-invariant submersion. We next recall the definition of the Lagrangian submersion from Lorentzian trans-Sasakian manifold onto a Riemannian manifold.
([12]) Let ψ be the anti-invariant Riemannian submersion from the almost contact metric manifold (M, ϕ, ξ, η, g) on the Riemannian manifold (N, gN). In case μ = {0} or μ = span{ξ}, i.e.
Let ψ is the anti-invariant submersion from the Lorentzian trans-Sasakian manifold (M, ϕ, ζ, η, g) on the Riemannian manifold (N, gN). For any
([7]) Suppose
Using geometrical analysis of Riemannian submersions, Bishop [7] described the idea of Clairaut submersion as follows.
([7]) The Riemannian submersion ψ: M → N is known as the Clairaut submersion in case there is positive function γ at M, that is, for all geodesics σ at M, the function (γ˚σ)SinΘ is constant, where, for all l, Θ(l) is an angle within
Bishop also provided the necessary and sufficient condition for the Riemannian submersion turns into the Clairaut submersion as follows.
([7]) Let ψ: M → N be the Riemannian submersion with connected fibers. Therefore ψ is the Clairaut submersion with γ = exp(ω) if and only if all fibers are totally umbilical and have the mean curvature vector field
5. Anti-invariant submersions admitting horizontal Reeb vector field from Lorentzian trans-Sasakian manifolds
The anti-invariant submersions are studied in this part from trans-Sasakian manifolds conceding horizontal Reeb vector field. First, the modern necessary and sufficient condition for similar submersions turns into a Clairaut submersion, and then a few distinctive outcomes for this sort of submersions are shown.
We observe from Definition 4.5, the source of the knowledge of a Clairaut submersion comes from geodesic on its total space. As a result, the necessary and sufficient condition of the curve on total space explored remains geodesic.
Now, the following results are given:
Let ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invaraint Riemannian submersion from Lorentzian trans-Sasakian manifold of type (α, β) onto the Riemannian manifold allowing horizontal Reeb vector field. In case
Proof.In view of Eqn (2.4), we find
Now, from a straight forward calculation, we find
In fact η (V1) = 0. By using Eqns (3.3), (3.4), (3.5) and (3.6), we find
Now capturing the vertical and horizontal components from Eqn (5.6), we find the following equations:
From equations (5.7) and (5.8), it is simply observed that σ is geodesic if and only if (5.1) and (5.2) hold. □
Using Theorem (5.1) in addition to Remark (1), the following corollaries are obtained.
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invaraint Riemannian submersion from Lorentzian α-Sasakian manifold of type (α, 0) onto the Riemannian manifold allowing horizontal Reeb vector field. In case
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invariant Riemannian submersion from Lorentzian β-Kenmotsu manifold of type (0, β) onto the Riemannian manifold admitting horizontal Reeb vector field. In case
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is an anti-invaraint Riemannian submersion from Lorentzian cosymplectic manifold of type (0, 0) onto the Riemannian manifold allowing horizontal Reeb vector field. If
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invariant Riemannian submersion from Lorentzian trans-Sasakian manifold of type (α, β) onto the Riemannian manifold allowing horizontal Reeb vector field. Therefore ψ is Clairaut submersion with γ = exp(ω) if and only if through σ
Proof. Consider σ(l) as the geodesic having the speed
Now, from Eqn (5.16), we achieve that
Using the Lorentzian trans-Sasakian structure, we find
Once again, from Eqn (2.4), we have
Thus, from Eqn (5.19), we obtain
From Eqn (5.2), we find along σ,
On contrary, ψ is Clairaut submersion with γ = exp(ω) if and only if
Now, taking the product of Eqn (5.24) with nonzero factor vSinΘ, we find
Using equations (5.23) and (5.24), we obtain
In fact
Now, the following corollaries are given:
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invariant Riemannian submersion from Lorentzian α-Sasakian manifold of type (α, 0) onto the Riemannian manifold allowing horizontal Reeb vector field. Therefore ψ is Clairaut submersion with γ = exp(ω) if and only if through σ
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invariant Riemannian submersion from Lorentzian β-Kenmotsu manifold of type (0, β) onto the Riemannian manifold allowing horizontal Reeb vector field. Therefore ψ is Clairaut submersion with γ = exp(ω) if and only if through σ
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the anti-invariant Riemannian submersion from Lorentzian cosymplectic manifold of type (α, β) onto the Riemannian manifold admitting horizontal Reeb vector field. Therefore ψ is Clairaut submersion with γ = exp(ω) if and only if along σ
Now, from Eqn (5.29), we also obtain the following conclusion.
Suppose ψ is the Clairaut anti-invariant submersion from Lorentzian trans-Sasakian manifold (M, ϕ, ζ, η, g) of type (α, β) on the Riemannian manifold (N, gN). Therefore,
Proof. Since ζ is a horizontal Reeb vector field. Setting Z1 = ζ and using the fact
Suppose ψ is the Clairaut anti-invariant submersion from Lorentzian α-Sasakian (or Lorentzian Sasakian) manifold (M, ϕ, ζ, η, g) of type (α, 0) onto the Riemannian manifold (N, gN). Therefore,
Proof. Since for Lorentzian α-Sasakian (or Lorentzian Sasakian) β = 0, and using similar fact as we have used in proof of Corollary 5.9 together, we find the desired result. □
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is a Clairaut anti-invariant submersion from Lorentzian trans-Sasakian manifold of type (α, β) onto a Riemannian manifold admitting horizontal Reeb vector field with γ = exp(ω). Then we have
Proof. Suppose ψ is the Clairaut anti-invariant submersion allowing horizontal Reeb vector field from a Lorentzian trans-Sasakian manifold onto a Riemannian manifold with γ = exp(ω). Now, by consequences of Theorem (4.6), we find
Adopting the Lorentzian trans-Sasakian structure, we notice
Once again, adopting (3.3), we turn up
Henceforth, through Eqn (5.33), we attain
On the contrary, involving Eqn (2.4), we turn up
Adopting equations (3.3) and (5.33), we get
Involving again, Eqns (5.44), (5.45) and the skew-symmetric nature of A, we turn up
Particularly if
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the Clairaut Lagrangian submersion allowing horizontal Reeb vector field from Lorentzian trans-Sasakian manifold of type (α, β) onto a Riemannian manifold with γ = exp(ω). If
6. Clairaut Lagrangian submersions
This section deals with some results of Clairaut Lagrangian submersions conceding with horizontal Reeb vector field. Moreover, when the function ω is constant,
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is a Clairaut anti-invariant submersion allowing horizontal Reeb vector field from Lorentzian trans-Sasakian manifold of type (α, β) on the Riemannian manifold with γ = exp(ω) and dim(kerψ*) > 1, then fibers of ψ are totally geodesic if and only if
Moreover, in case the submersion ψ at Theorem (5.11) is Lagrangian submersion, therefore
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the Clairaut Lagrangian submersion allowing horizontal Reeb vector field from Lorentzian trans-Sasakian manifold of type (α, β) onto a Riemannian manifold with γ = exp(ω). Therefore, fibers of ψ can be one-dimensional or totally geodesic.
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the Clairaut Lagrangian submersion admitting horizontal Reeb vector field from Lorentzian α-Sasakian manifold of type (α, 0) onto a Riemannian manifold with γ = exp(ω). Therefore fibers of ψ can be one-dimensional or totally geodesic.
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the Clairaut Lagrangian submersion allowing horizontal Reeb vector field from Lorentzian β-Kenmotsu manifold of type (0, β) onto a Riemannian manifold with γ = exp(ω). Therefore fibers of ψ can be one-dimensional or totally geodesic.
Suppose ψ: (M, ϕ, ζ, η, g) → (N, gN) is the Clairaut Lagrangian submersion allowing horizontal Reeb vector field from Lorentzian cosymplectic manifold of type (α, β) onto a Riemannian manifold with γ = exp(ω). Therefore either fibers of ψ can be one-dimensional or totally geodesic.
7. Applications
The following result is Theorem 2 stated by Gauchman in [28].
Suppose ψ; (M, g) → (N, gN) is the Clairaut submersion with γ, where M is complete, connected and simply connected, and N is simply connected. Assume that any vertical leaf of ψ has no nontrivial Killing vector field. Suppose p is the point of M. Therefore M is isometric to the warped product N ×fB, where B is the vertical leaf through p and
In [23] De and Srakar prove that trans-Sasakian structures are complete and connected. Indeed, Riemannian manifold also preserved the characteristic of simple connectedness. Therefore, the following results are obtained.
ψ: (M, ϕ, ζ, η, g, α, β) → (N, gN) is a Clairaut Lagrangian submersion with γ, where (M, ϕ, ζ, η, g) is complete, connected, and simply connected Lorentzian trans-Sasakian manifold, and Riemannian manifold (N, gN) is simply connected. Assume that any vertical leaf of ψ has no nontrivial Killing vector field. Let p be a point of (M, ϕ, ζ, η, g). Then Lorentzian trans-Sasakian manifold of (α, β) type is isometric to a warped product N ×fB, where B is the vertical leaf through p and
For particular values of α and β easily we can turn up the similar results like Theorem (7.2) for α-Lorentzian Sasakian manifold (Lorentzian Sasakian manifold), β-Lorentzian Kenmotsu manifold (Lorentzian Kenmotsu manifold), and Lorentzian cosymplectic manifold.
Now, we describe some examples of Clairaut submersion from Lorentzian trans-Sasakian manifolds (M, ϕ, ξ, η, g) of type (α, β).
Suppose M is three-dimensional Euclidean space written as
We consider the Lorentzian trans-Sasakian structure (ϕ, ξ, η, g, α, β) at M with α = 0 and β = 1 [23] given by the following:
ϕ(E1) = − E2, ϕ(E2) = − E1, ϕ(E3) = 0. An orthonormal ϕ-basis of this constructor is written
Here, the map
where g1 is the usual metric at
Adopting the Lorentzian trans-Sasakian structure results in
Using (3.5), we turn up
For any function ω of
Henceforth, using Theorem (5.26), the submersion ψ is Clairaut submersion.
Suppose M is three-dimensional Euclidean space written as
We consider the Lorentzian trans-Sasakian structure (ϕ, ξ, η, g, α, β) at M with α = − 1 and β = 0 given by the following:
ϕ(E1) = − E1, ϕ(E2) = − E2, ϕ(E3) = 0. An orthonormal ϕ-basis is written as
Moreover, we have
Here, the map
Easily, we observe that ψ is the Riemannian submersion. Moreover, we have ϕ(U) = V. Therefore, ψ is the anti-invariant submersion admitting horizontal Reeb vector field. Particularly, ψ is Lagrangian submersion. Furthermore, after all the fibers of ψ are one-dimensional, then they are simply totally umbilical. At this point, it is proved that fibers are not totally geodesic, and it is found that the function of
Adopting the Lorentzian trans-Sasakian structure, we observe that
Using (3.5), we turn up
Henceforth, by Theorem (5.26), the submersion ψ is Clairaut submersion.
References
1.Nash JN. The imbedding problem for Riemannian manifolds. Ann Math. 1956; 63(2): 20-63.
2.Falcitelli M, Ianus S, Pastore AM Riemannian submersions and related topics, River Edge, New Jerssey, NJ: World Scientific, 2004.
3.
4.O'Neill B. The fundamental equations of a submersion. Mich Math J. 1966; 13: 458-69.
5.Gray A. Pseudo-Riemannian almost product manifolds and submersion. J Math Mech. 1967; 16: 715-37.
6.Watson B. Almost Hermitian submersions. J Differ Geom. 1976; 11(1): 147-65.
7.Bishop RL, Clairaut submersions, Differential Geometry, Tokyo: Kinokuniya. 1972: 1689-703, 21-31.
8.
9.Lee J, Park JH,
10.Taştan HM. On Lagrangian submersions. Hacettepe J Math Stat. 2014; 43(6): 993-1000.
11.Gündüzalp Y. Clairaut anti-invaraiant submersions from almost product Riemannnian manifolds. Beitr Algebra Geom. 2020; 61: 605-14.
12.Taştan HM. Lagrangian submersions from normal almost contact manifolds. Filomat. 2017; 31(12): 3885-95.
13.Taştan HM, Gerdan S. Clairaut anti-invariant submersions from Sasakian and Kenmotsu manifolds. Mediterr J Math. 2017; 14: 235.
14.Gündüzalp Y. Anti-invariant Pseudo-Riemannian submersions and Clairaut anti-invaraiant submersions from Paracosymplectic manifolds. Mediterr J Math. 2019; 16: 94.
15.Siddiqi MD, Akyol MA. Anti-invariant ξ⊥-Riemannian submersions from hyperbolic β-Kenmotsu manifolds. Cubo (Temuco). 2018; 20(1): 79-94.
16.Siddiqi MD, Akyol MA. Anti-invariant ξ⊥-Riemannian submersions from almost hyperbolic contact manifolds. Int Elec J Geom. 2019; 12(1): 32-42.
17.Siddiqi MD. Submersions of contact CR-submanifolds of generalized quasi-Sasakian manifolds. J Dynamical Syst Geometric theories. 2020; 18(1): 81-95.
18.Taştan HM, Siddiqi MD. Anti-invariant and Lagrangian submersions from trans-Sasakian manifolds. Balkan J Geom App. 2020; 25(2) : 106-23.
19.Akyol MA,
20.Akyol MA, Sari R, Aksoy, E. Semi-invariant ξ⊥-Riemannian submersion from almost contact metric manifolds. Int J Geom Meth Mode Phys. 2017; 14(5), 1750074.
21.Akyol MA. Conformal anti-invariant submersions from cosymplectic manifolds. Hacettepe J Math Stat. 2017; 46(2): 177-92.
22.Allison D. Lorentzian Clairaut submersions. Geom Dedicata. 1996; 63(3): 309-19.
23.De UC and De K. On Lorentzian trans-Sasakian manifolds. Commun Fac Sci Univ Ank Ser A1. 2013; 62(2): 1303-5991.
24.Blair DE Contact manifolds in Riemannian geometry, Lecture Notes in math, Berlin-New York, NY: Springer Verlag. 1976: 509.
25.Oubina JA. New classes of almost contact metric structures. Pub. Math. Debrecen. 1980; 32: 187-93.
26.Blair DE, Oubina JA. Conformal and related changes of metric on the product of two almost contact metric manifolds. Publications Matematiques Debrecen. 1990; 34: 199-207.
27.Yano K and Kon M Structures on manifolds, Singapore: World Scientific, 1984.
28.Gauchman H. On a decomposition of Riemannian manifolds, Houston J Math. 1981; 7(3): 365-372.
Further reading
29.Eells J, Sampson JH. Harmonic mapping of Riemannian manifolds. Amer J Math. 1964; 86: 109-60.
30.Taştan HM, Gerdan S. Clairaut anti-invariant submersions from Cosymplectic manifolds. Honam Math J. 2019; 41(4): 707-24.
31.Vilms J Totally geodesic maps, J Differ Geom. 1970; 4: 73-9.
Acknowledgements
The authors are grateful to the referee for the valuable suggestions and comments toward the improvement of the paper.