Clairaut anti-invariant submersions from Lorentzian trans-Sasakian manifolds

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 nontrivial examples. Findings –Themajor 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.


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 R 3 . In 1956, Nash [1] proved that a revolution for Riemannian manifold that all Riemannian manifolds are isometrically embedded at any small part of Euclidean space. Consequently, the differential geometry of the Riemannian immersion is commonly noted, and it can be found in different text books such as ( [2,3]).
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 Ş ahin'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 S, function γSinΘ is constant through σ, here γ is a metric between the point at surface and rotation axis, also Θ defines angle within σ and meridian through σ.
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.

Lorentzian trans-Sasakian manifolds
A (2n þ 1)-dimensional differentiable manifold M is named the Lorentzian Trans-Sasakian manifold [23] in case it allows (1, 1) tensor field w, 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].
Therefore, for all q ∈ N, ψ −1 ðqÞ ¼ ψ −1 q is the k-dimensional submanifold of M and is named the fiber, with 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 * (X p ) 5 X *ψ(p) for any p ∈ M, where ψ * is derivative or differential map of ψ. V and H define the projections at vertical distribution kerψ * and horizontal distribution kerψ ⊥ * , in the same order. Usually, a manifold (M, g) is named the total manifold and (N, g N ) is named base manifold of the submersion ψ: (M, g) → (N, g N ).

Clairaut antiinvariant submersions
This condition is equivalent to say that the derivative map ψ * of ψ, restricted to kerψ ⊥ * , is the linear isometry. The geometrical description of Riemannian submersions is represented by O'Neill's tensors T and A, determined as: (3.1) for any vector fields E 1 and F 1 at M, with D is Levi-Civita connection of g. Clearly, T E 1 in addition to A E 1 are skew-symmetric operators at tangent bundle of M reversing vertical and the horizontal distributions. To sum up, tensor fields properties T as well as A, Suppose V 1 , W 1 are vertical and X 1 , Y 1 are horizontal vector fields at M, therefore On the other hand, from (3.1) and (3.2), we obtain It appears that T is acting at fibers as second fundamental form, whereas A is acting at horizontal distribution and measuring obstruction to integrability of the distribution. Further details are given in the paper of O'Neill [4] in addition to this book [2].
At the end, the concept of second fundamental form of the map within Riemannian manifolds is recalled. Suppose (M, g) and (N, g N ) are Riemannian manifolds and f: (M, g) → (N, g N ) is the smooth map. Therefore, second fundamental form of f is written as for U, V ∈ Γ(TM), with ∇ f defining the pull-back connection, and D defines the Riemannian connections of the metrics g and g N . Symmetry is widely known property of second fundamental form, and further, f is named totally geodesic [31] in case ðDf * ÞðE; FÞ ¼ 0 for any U, V ∈ Γ(TM) (as in [19, p. 119]), and f is named the harmonic map [29] in case traceðDf * Þ ¼ 0 (as in [19, p. 73]).

Anti-invariant Riemannian submersions
We first recall idea of an anti-invariant Riemannian submersion where its total manifold is the almost contact metric manifold. Therefore, Riemannian submersion π is named the anti-invariant Riemannian submersion.
Similar submersions are called the anti-invariant submersions.
with μ refers to orthogonal complementary distribution of wkerψ * at kerψ ⊥ * , and it is invariant with respect to w.
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.
We consider the Lorentzian trans-Sasakian structure (w, ζ, η, g) at M with α ¼ 1 given by the following: An orthonormal w-basis of this structure is written as Here, the map ψ : ðM ; w; ξ; η; gÞ → ðR; g 1 Þ is introduced as: where g 1 is Lorentzian metric on R. Therefore, Jacobian matrix of ψ is given as: 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 w(U) 5 W. Therefore, ψ is the anti-invariant submersion admitting vertical Reeb vector field.
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.
therefore ψ is called the Lagrangian submersion.
Definition 4.4. ( [7]) Suppose S is the revolution surface at R 3 with rotation axis L. For all q ∈ S, where γ(q) represents the distance between q and L. Choosing the geodesic σ : U ⊂ R → S on S, Suppose Θ(l) is an angle between σ(l) and the meridian curve through σ(l), l ∈ U. By the famous Clairaut's theorem, we know that for all geodesics σ on S, the product γSinΘ is constant along σ, which means the independence of l.
Using geometrical analysis of Riemannian submersions, Bishop [7] described the idea of Clairaut submersion as follows. Bishop also provided the necessary and sufficient condition for the Riemannian submersion turns into the Clairaut submersion as follows.

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: horizontal Reeb vector field. In case σ : U ⊂ R → M is regular curve and V 1 (l) in addition to Z 1 (l) defines vertical and horizontal components of the tangent vector field σðlÞ ¼ G of σ(l), in the same order, therefore σ is geodesic if and only if through σ the following equation Proof. In view of Eqn (2.4), we find Now, from a straight forward calculation, we find In fact η(V 1 ) 5 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: and 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.
Corollary 5.2. Suppose ψ: (M, w, ζ, η, g) → (N, g N ) 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 σ : U ⊂ R → M is regular curve and V 1 (l) in addition to Z 1 (l) defines vertical and horizontal components of tangent vector field σðlÞ ¼ G of σ(l), in the same order, therefore σ is geodesic if and only if through σ the following equations Clairaut antiinvariant submersions maintain, where ffiffi s p is constant speed of σ. Corollary 5.3. Suppose ψ: (M, w, ζ, η, g) → (N, g N ) 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 σ : U ⊂ R → M is the regular curve and V 1 (l) in addition to Z 1 (l) defines vertical and horizontal components of tangent vector field σðlÞ ¼ G of σ(l), in the same order, therefore σ is geodesic if and only if through σ the following equation hold, where ffiffi s p is constant speed of σ.
holds, where V 1 (l) and Z 1 (l) are vertical and horizontal components of the tangent vector field σðlÞ of the geodesic σ(l) at M, in the same order. Hence, since η(V) 5 0, g(wV 1 , ζ) 5 0 and using the fact that wV 1 is horizontal.
Clairaut antiinvariant submersions holds, where V 1 (l) and Z 1 (l) are vertical and horizontal components of tangent vector field σðlÞ of the geodesic σ(l) at M, in the same order.
holds, where V 1 (l) and Z 1 (l) are vertical and horizontal components of the tangent vector field σðlÞ of the geodesic σ(l) at M, in the same order.
Now, from Eqn (5.29), we also obtain the following conclusion.

AJMS
Adopting the Lorentzian trans-Sasakian structure, we notice Once again, adopting (3.3), we turn up Henceforth, through Eqn (5.33), we attain Putting U 1 5 V 1 and shifting U 1 with by G 1 in Eqn (5.39), we acquire G 1 j j 2 gðDω; wG 1 Þ ¼ gðU 1 ; G 1 ÞgðDω; wV 1 Þ: Adopting Eqn (5.39) with setting V 1 5 U 1 , we have On the contrary, involving Eqn (2.4), we turn up for W 1 ∈ Γ(μ) and W 1 ≠ ζ. Using Eqn (2.5), we get Adopting equations (3.3) and (5.33), we get After all wV 1 is basic vector and using the case that HD G 1 wV 1 ¼ A wV 1 G 1 , we turn up Involving again, Eqns (5.44), (5.45) and the skew-symmetric nature of A, we turn up By reason of A wV 1 wW 1 , G 1 and V 1 are vertical and ω is horizontal, we turn up expression (5.32). Particularly if Dω ∈ wðkerψ * Þ, then from (5.41) in proof of Theorem 5.11 and the equality case of Schwarz inequality, we have have that , onto a Riemannian manifold with γ 5 exp(ω). If Dω ∈ wðkerψ * Þ, then either ω is constant on w(kerψ * ) or fiber of ψ is one-dimensional.
Moreover, in case the submersion ψ at Theorem (5.11) is Lagrangian submersion, therefore A wV 1 wZ 1 is always vanish, because μ ¼ 0 f g or μ ¼ span ζ f g. Also from Corollaries 5.9 and 5.10, we have Dω ∈ wðkerψ * Þ. Hence, the following consequences of Theorem (5.11) and Corollary 5.12 are given.

Applications
The following result is Theorem 2 stated by Gauchman in [28]. 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.
Theorem 7.2. ψ: (M, w, ζ, η, g, α, β) → (N, g N ) is a Clairaut Lagrangian submersion with γ, where (M, w, ζ, η, g) is complete, connected, and simply connected Lorentzian trans-Sasakian manifold, and Riemannian manifold (N, g N ) is simply connected. Assume that any vertical leaf of ψ has no nontrivial Killing vector field. Let p be a point of (M, w, ζ, η, g). Then Lorentzian trans-Sasakian manifold of (α, β) type is isometric to a warped product N 3 f B, where B is the vertical leaf through p and f : N → R is defined by the equation γ 5 f ψ. We consider the Lorentzian trans-Sasakian structure (w, ξ, η, g, α, β) at M with α 5 0 and β 5 1 [23] given by the following: and w is the (1, 1) tensor field determined as 0. An orthonormal w-basis of this constructor is written as Here, the map ψ : ðM ; w; ξ; η; g; α; βÞ → ðR; g 1 Þ is written as: where g 1 is the usual metric at R. Now, by a straightforward computation, we turn up Easily, we observe that ψ is the Riemannian submersion. Moreover, we have w(U) 5 V. Therefore, ψ is the anti-invariant submersion allowing horizontal Reeb vector field. Particularly, ψ is Lagrangian submersion. Furthermore, after all the fibers of ψ are onedimensional, then they are simply totally umbilical. At this point, it is proved that fibers are not considered totally geodesic, and it is found that the function of R 3 obeying T U 1 U 1 ¼ −Dω. Therefore, after some sort of calculation, we turn up Adopting the Lorentzian trans-Sasakian structure results in Using (3.5), we turn up For any function ω of ðR 3 ; w; ζ; η; gÞ, the gradient of ω with respect to the metric g is Here, at this point, it is clear to observe that ω ¼ − z 2 2 for the function of z and T U 1 U 1 ¼ −Dω ¼ −ζ. Also for any U 2 ∈ (kerψ * ), we have Henceforth, using Theorem (5.26), the submersion ψ is Clairaut submersion.
Particularly, ψ is Lagrangian submersion. Furthermore, after all the fibers of ψ are onedimensional, 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 R 3 obeying T U 1 U 1 ¼ −Dω. Therefore, after some sort of calculation, we turn up Adopting the Lorentzian trans-Sasakian structure, we observe that Using (3.5), we turn up For all functions ω at ðR 3 ; w; ξ; η; gÞ, the gradient of ω with respect to the metric g is Now, at this point, it is clear to observe that ω 5 À 2z for the function of z and Also for any U 2 ∈ (kerψ * ), we have Henceforth, by Theorem (5.26), the submersion ψ is Clairaut submersion.