Skip to main content

ORIGINAL RESEARCH article

Front. Phys., 28 February 2022
Sec. Statistical and Computational Physics
This article is part of the Research Topic Differential Geometric Methods in Modern Physics View all 7 articles

Anti-Invariant Lorentzian Submersions From Lorentzian Concircular Structure Manifolds

  • 1Department of Mathematics, Faculty of Science, Jazan University, Jizan, Saudi Arabia
  • 2Department of Mathematics, College of Science, University of Tabuk University, Tabuk, Saudi Arabia
  • 3Department of Mathematics, College of Science, Taif University, Taif, Saudi Arabia
  • 4Section of Mathematics, Department of Information Technology, University of Technology and Applied Sciences-Shinas, Muscat, Oman

This research article attempts to investigate anti-invariant Lorentzian submersions and the Lagrangian Lorentzian submersions (LLS) from the Lorentzian concircular structure [in short (LCS)n] manifolds onto semi-Riemannian manifolds with relevant non-trivial examples. It is shown that the horizontal distributions of such submersions are not integrable and their fibers are not totally geodesic. As a result, they can not be totally geodesic maps. Anti-invariant and Lagrangian submersions are also explored for their harmonicity. We illustrate that if the Reeb vector field is horizontal, the anti-invariant and LLS can not be harmonic.

1 Introduction

In 2003, Shaikh [1] studied the properties of Lorentzian manifold M endowed with a concircular vector field, and he named such manifold the Lorentzian concircular structure manifold (briefly (LCS)n-manifold), which is the extension of the Lorentzian para-Sasakian (in short, LP-Sasakian) manifold developed by Matsumoto [2] and Mihai and Rosca [3]. Many researchers have looked at the characteristics of (LCS)n-manifolds, and used them in applied mathematics and mathematical physics (as an example, see [48]). In [9], Mantica and Molinari have proved that the (LCS)n-manifold coincides with generalized Robertson-Walker (GRW) spacetime, which was introduced by Alı́as, Romero and Sánchez [10] in 1995. The geometry of semi-Riemannian submersions has became a fascinating topic for research due to its involvement in physics, particularly in the theory of relativity (GR) such as Yang-Mills theory, String theory, Kaluza-Klein theory, and Hodge theory, etc.

We can develop more structures, for example, locally trivial fiber spaces include product manifolds, covering spaces, the tangent and cotangent bundles of a manifold. Thus, we can use the framework on structure preserving submersions to study the spaces with symmetries. In particular, the theory can be directly applied to study the black holes of various dimensions, Lagrangian with symmetries, and simple quantum systems with symmetrical properties.

In 1956, Nash [11] proved the embedding theorem for a Riemannian manifold. According to him, every Riemannian manifold can be isometrically embedded into some Euclidean space. Thus, the differential geometry of Riemannian immersions is well-known and available in many textbooks such as [12, 13]. On the other hand, the Lorentzian submersions are the semi-Riemannian submersions whose total space is a Lorentzian manifold [14].

The concept of semi-Riemannian submersions was given by O’Neill [15, 16] and Gray [17]. In 1983, Magid [18] described the Lorentzian submersion from anti-de Sitter spacetime. In fact, these Lorentzian submersions are generalizations of Lorentzian warped products. Various spacetimes in general relativity (GR), such as Robertson-Walker spacetimes and (LCS)4- spacetimes, are warped products. This study is closely connected to these works.

Watson [19] considered the Riemannian submersions between almost Hermitian manifolds, and he named almost Hermitian submersions. Afterwards, the almost Hermitian submersions between various subclasses of almost Hermitian manifolds are thoroughly studied in [2022]. Moreover, paracontact semi-Riemannian submersions were extensively discussed by Yilmaz and Akyol [23, 24] and Faghfouri et al. [25]. Recently, Siddiqi et al. [26, 27] discussed some properties of anti-invariant semi-Riemannian submersions which are closely related to this work. The majority of the works on semi-Riemannian, almost contact Riemannian submersions have been found in the books [12, 13].

Şahin [28] first described anti-invariant Riemannian submersions and Lagrangian submersions from almost Hermitian manifolds onto Riemannian manifolds. Since then, the topics of anti-invariant Riemannian submersions and Lagrangian submersions have become an active field for researchers. The extension of anti-invariant Riemannian submersion as various types of submersions, such as anti-invariant ξ-Riemannian submersions and Lagrangian submersions, have been studied in different forms of structures such as Kähler [28, 29], nearly Kähler [22], almost product [30], locally product Riemannian [31], Sasakian [3234], Kenmotsu [35], cosymplectic [36] and hyperbolic structures [37, 38]. Moreover, a Lagrangian submersion is a specific version of anti-invariant Riemannian submersion such that the total manifold (almost complex structure) interchanges the role of horizontal and vertical distributions [39].

The following is an overview of the paper’s content. In sections 2, 3, and 4, we reveal basic definitions and known results of (LCS)n-manifolds, Lorentzian submersions, and anti-invariant Lorentzian and LLS, respectively. In Section 5, we study anti-invariant Lorentzian submersions from (LCS)n-manifolds onto semi-Riemannian manifolds admitting the vertical Reeb vector field (VRVF). Section 6 is concerned with the study of the properties of anti-invariant submersions with the horizontal Reeb vector field. We also provide an example of anti-invariant submersions with the horizontal Reeb vector field and study its characteristic properties. In Section 7, we consider LLS admitting VRVF and investigate the geometry of vertical and horizontal distributions. We give a non-trivial example of LLS admitting a VRVF. We also give a necessary and sufficient condition for such submersions to be harmonic.

Note: Throughout the paper we used the following acronyms:

LLS: Lagrangian Lorentzian submersion.

HRVF: Horizontal reeb vector field.

VRVF: Vertical reeb vector field.

2 Lorentzian Concircular Structure Manifolds

Lorentzian manifold L of dimension n = (2m + 1) is a smooth connected manifold with a Lorentzian metric g, that is, L admits a smooth symmetric tensor field g of type (0, 2) such that for each point pL, the tensor gp:TpL×TpLR is a non-degenerate inner product of signature (−, +, …., +), where TpL denotes the tangent vector space of L at p and R is the real number space. A non-zero vector v ∈ TpL is said to be timelike, null, and spacelike, if it fulfills gp(v, v) < 0, gp(v, v) = 0, and gp(v, v) > 0, respectively.

Definition 2.1. [1] Let (L, g) be a Lorentzian manifold, a vector field Q ∈ Γ(TL) satisfying g(E, Q) = P(E), is said to be a concircular vector field if

EPF=αgE,F+ωEPF,

for any E, F ∈ Γ(TL), where α is a non-zero scalar function, ω is a closed 1-form, and ∇ is the Levi-Civita connection corresponding to the Lorentzian metric g.

Let the Lorentzian manifold L of dimension n admit a unit timelike concircular vector field ζ, it follows that

gζ,ζ=1.

Since ζ is a unit concircular vector field, consequently, there exists a non-zero 1-form η such that g(E, ζ) = η(E), then the following equations hold

EηF=αgE,F+ηEηF,α0,
Eζ=αE+ηEζ

for all vector fields E, F and α is a non-zero real valued function. Further, we have

Eα=Eα=dαE=ρηE,

here ρ is a scalar function defined as ρ = − (ζα). If we write

φE=1αEζ,

on using Eqs 2.2, 2.3, we deduce

φE=E+ηEζ,
gφE,F=gE,φF.

As a consequence, φ is a symmetric (1, 1) tensor field, which is known as the structure tensor field of L. Thus, the Lorentzian manifold L with unit timelike concircular vector field ζ, 1-form η, and (1, 1) tensor field φ is said to be Lorentzian concircular structure manifold (LCS)n-manifold). If α = 1, then the (LCS)n-manifolds become LP-Sasakian manifolds. The following tensorial equations holds on a (LCS)n-manifold [1].

φ2E=E+ηEζ,
φζ=0,ηφ=0,ηζ=1,
gφE,φF=gE,F+ηEηF,
ηRE,FG=α2ρgF,GηEgE,GηF,
RE,Fζ=α2ρηFEηEF,
EφF=αgE,Fζ+2ηEηFζ+ηFE,
Eρ=dρE=βηE.

3 Lorentzian Submersions

We provide the required foundation for Lorentzian submersions in this section.

A surjective mapping γ: (L, g) → (S, gS) between a Lorentzian manifold (L, g) and a semi-Riemannian manifold (N, gS) is called a Lorentzian submersion [15] if γ* is onto it and it satisfies.

(C1) Rank(γ) = dim(S), where dim(L) > dim(S).

In this situation, for each q ∈ S, γ1(q)=γq1 is a t-dimensional submanifold of L termed as a fiber, where t = dim(L) − dim(S).

A vector field E on L is vertical (resp. horizontal) if it is consistently tangential (resp. orthogonal) to fibers. A vector field E on L is termed basic if E is horizontal and γ-related to a vector field E* on S.

Also, γ*(Ep) = E*γ(p) for all p ∈ L, where γ* is the differential map of γ. Here V and H indicates the projections on the vertical distribution Kerγ*, and the horizontal distribution Kerγ*, respectively. Generally, the manifold (L, g) is said total manifold and the manifold (N, gN) is said the base manifold for submersion γ.

(C2) The lengths of the horizontal vectors are conserved by γ*.

This situation is analogous to saying that the derivative map γ* of γ is a linear isometry when confined to Kerγ*. O’Neill’s tensors T and A, which are formulated as follows, describe the geometry of semi-Riemannian submersions:

TE1E1=VVE1HE2+HVE1VE2,
AE1E2=VHE1HE2+HHE1VE2

for any vector fields E1 and E2 on L, where is the Levi-Civita connection of g. TE1 and AE1 are skew-symmetric operators on the tangent bundle of L inverting the vertical and the horizontal distributions, as can be shown.

The features of the tensor fields T and A are stated. On M if V1, V2 are vertical and E1, E2 are horizontal vector fields, we possess

TV1V2=TV2V1,
AE1E2=AE2E1=12VE1,E2.

Equations 3.1, 3.2, entail that

V1V2=TV1V2+̂V1V2,
V1E1=TV1E1+HV1E1,
E1V1=AE1V1+VE1V1,
E1E2=HE1E2+AE1E2,

where

̂V1V2=VV1V2,HV1E1=AE1V1.

It is easy to see that T operates on the fibers as the second fundamental form, whereas A operates on the horizontal distribution and evaluates the restriction to its integrability. We refer to O’Neill’s work [15] and book [12] for more information on the semi-Riemannian submersions.

Next, we revisit the theory of map between semi-Riemannian manifolds with a second fundamental form. Let (L, g) and (S, gs) be Riemannian manifolds and f (L, g) → (S, gs) is a smooth map. Then the second fundamental form h satisfies the relation

h*E1,E2=E1hh*E2h*E1E2

for E1, E2 ∈ Γ(TL), where h is the pull back connection and , the Riemannian connection of the metrics g and gS, respectively. Furthermore, if (∇h*)(E1, E2) = 0 for all E1, E2 ∈ Γ(TL) (see [40], page 119), h is said to be totally geodesic and if trace(∇h*) = 0 for all E1, E2 ∈ Γ(TL), h is termed as harmonic map (see [40], page 73).

4 Anti-Invariant Lorentzian and Lagrangian Lorentzian Submersions From (LCS)n-Manifolds

We first recall the definition of an anti-invariant Lorentzian submersion whose total manifold is an (LCS)n-manifold.

Definition 4.1. ([32, 33]). Let L be an (LCS)n-manifold (dim(L) = 2m + 1) with (LCS)n-structure (φ, ζ, η, g, α) and S be a semi-Riemannian manifold with gS as its semi-Riemannian metric. If there is a Lorentzian submersion γ: L → S such that the vertical distribution Kerγ* is anti-invariant with respect to φ, i.e., φKerγ*Kerγ*, then the semi-Riemannian submersion γ is known as an anti-invariant Lorentzian submersion.

In this instance, the horizontal distribution Kerγ* is decomposed as

Kerγ*=φKerγ*μ,

where μ is an orthogonal complementary distribution of φKerγ* in Kerγ* and it is invariant with respect to φ.

For an anti-invariant submersion γ: L → S, if the Reeb vector field ζ is tangential (or normal) to ker γ*, then ζ is said to be vertical Reeb vector field (VRVF) (or horizontal Reeb vector field (HRVF)).

More information on anti-invariant Lorentzian submersions from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gN) may be found in [32, 33, 35, 36].

Remark 4.2. Throughout this paper, We consider a (LCS)n-manifold (L, φ, ζ, η, g, α) as the total manifold of the anti-invariant Lorentzian submersion.

The notion of Lagrangian submersion is a particular case of the anti-invariant submersion. Next, we review the definition of an LLS from (LCS)n-manifold onto a semi-Riemannian manifold.

Definition 4.3. [34] Let γ be an anti-invariant Lorentzian submersion from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gN). If μ = {0} or μ = Span{ζ}, i.e., Kerγ*=φ(Kerγ*) or Kerγ*=φ(Kerγ*)<ζ>, correspondingly, then we say that the submersion γ is a Lagrangian Lorentzian submersion (an LLS).

Remark 4.4. This situation has been investigated as a particular example of an anti-invariant Lorentzian submersion; for additional information, see [3236].

5 Anti-Invariant Lorentzian Submersions With Vertical Reeb Vector Field

In the present segment, we begin with the anti-invariant Lorentzian submersions admitting VRVF from (LCS)n-manifolds (L, φ, ζ, η, g, α). Let γ be an anti-invariant Lorentzian submersion from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gS). For any EΓ(Kerγ*), we write

φE=BE+CE,

where BEΓ(Kerγ*) and CEΓ(Kerγ*). We now calculate the impact of the (LCS)n-structure on tensor fields on L. T and A of the submersion γ.

Lemma 5.1. Let γ be an anti-invariant Lorentzian submersion from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gS) with VRVF. Then, we have

TUφVαgU,Vζ+2ηUηVζ=BTUV+ηVU,
HUφV=CTUV+φ̂UV,
̂VBE+TVCE=BHVE,
TVBE+HVCE=CHVE+φTVE,
AEφV=BAEV,
HEφV+αηVE=φVEV+CAEV,
VEBF+AECF=BHEF+αgE,Fζ+2ηEηFζ,
AEBF+HECF=CHEF+φAEF,

where U, V ∈ Γ(Kerγ*) and E,FΓ(Kerγ*).

Proof For any U, V ∈ Γ(Kerγ*), from (2.6), we infer

UφV=φUV+αgU,Vζ+2ηUηVζ+ηVU.

Using (3.5), 3.6 and 5.1 in the above equation, we obtain

HUφV+TUφV=BTUV+CTUV+φ̂VU+αgV,Uζ+2ηVηUζ+ηUV.

In light of the fact that ζ is vertical, equating the vertical and horizontal components of (5.8), we get (5.2) and (5.3), correspondingly. By Equation 2.6, we have

EφF=φEF+αgE,Fζ+2ηEηFζ+ηFE,

for any E, F ∈ Γ(Kerγ*).

On using Eqs 3.7, 3.8, 5.1, we get

AEBF+VEBF+HECF+AECF=BHEF+CHEF+φAEF+αgE,Fζ+2ηEηFζ+ηFE.

If we compare the vertical and horizontal components of (5.9) and using the fact that ζ is vertical, we get (5.6) and (5.7), respectively. The rest of the claims may be derived in the same way

Now, we discuss anti-invariant Lorentzian submersions from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold such that the Reeb vector field ζ is vertical. Let us consider that γ is an anti-invariant Lorentzian submersion admitting VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gS). Then, using (5.1) and the condition (S2), we come up with

gγ*φV,γ*CE=0,

for every EΓ(Kerγ*) and V ∈ Γ(Kerγ*), this suggests that

TN=γ*φKerγ*γ*μ.

As a result, we demonstrate:

Theorem 5.2. Let (L, φ, ζ, η, g, α) is an (LCS)n-manifold of dimension (2L + 1) and (S, gS) is a semi-Riemannian manifold of dimension s. Let γ (L, φ, ζ, η, g) → (S, gS) be an anti-invariant such that φ(Kerγ*)=Kerγ*. Then the Reeb vector field ζ is vertical and l = s.

Proof By the assumption φΓ(Kerγ*)=Γ(Kerγ*), we have

gζ,φU=gφξ,U=0,

for any U ∈ Γ(Kerγ*), which shows that the Reeb vector field is vertical. Now, we assume that U1,,Uk1,ζ=Uk is an orthonormal frame of Γ(Kerγ*), where k = 2L − s + 1.

Since φΓ(Kerγ*)=Γ(Kerγ*), then ϕU1, …, ϕUk−1 form an orthonormal frame of Γ(Kerγ*). Therefore, in view of (5.1) we get k = s + 1, which implies that l = n.

Theorem 5.3. Let (L, φ, ζ, η, g, α) be an (LCS)n-manifold of dimension (2L + 1) and (S, gS) is a semi-Riemannian manifold of dimension s. If γ (L, φ, ζ, η, g) → (S, gS) is an anti-invariant Lorentzian submersion with VRVF, then the fibers are not totally umbilical.

Proof Using (2.2) and 3.5, we have

TUζ=αU

for any U ∈ Γ(Kerγ*). We suppose that the fibers are totally umbilical, then we have

TUV=gU,VH

for any vertical vector fields U and V, where H is the mean curvature vector field of the fiber. Since Tζζ=0, we have H = 0, which prove that the fibers are minimal. Hence the fibers are totally geodesic, which is a contradiction to the fact that TUζ=αU0, which proves the theorem.

From 2.4 and 5.1, we have following Lemmas.

Lemma 5.4. Let γ be an anti-invariant Lorentzian submersion with VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) to a semi-Riemannian manifold (S, gS). Then we have

BCE=0,φBE+C2E=E,

for any EΓ(Kerγ*).

Lemma 5.5. Let γ be an anti-invariant Lorentzian submersion with VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) to a semi-Riemannian manifold (S, gS). Then we have

CE=AEζ,
gAEζ,φU=0,
gFAEξ,φU=gAEζ,ϕAFU+αηUgAEζ,F,
gE,AFζ=gF,AEζ,

for E,FΓ(Kerγ*) and U ∈ Γ(Kerγ*).

Proof In the light of Equations 3.7, 2.5, we get (5.11). For EΓ(Kerγ*) and U ∈ Γ(Kerγ*), Equations 3.2, 5.1, and .5.11 give

gAEζ,φU=gφEBE,φU=gE,UηEηUgφBE,U.

Since φBEΓ(Kerγ*) and ζ ∈ Γ(Kerγ*), Equation 5.15 implies (5.12). Now, from (5.12) we get

gFAEζ,φU=gAEζ,FφU,

for E,FΓ(Kerγ*) and U ∈ Γ(Kerγ*). The geodesic condition together with Equation 5.15 yield

gFAEζ,φU=gAEζ,φAFUgAEζ,φVFU+αηUgAEζ,F.

Since φ(VFU)Γ(φKerγ*)=Γ(Kerγ*), we obtain (5.13). Using the skew-symmetry of Aand (3.4), we directly get (5.14).

6 Anti-Invariant Lorentzian Submersions With Horizontal Reeb Vector Field

Example. Let R9 be a 9-dimensional semi-Riemannian space given by.

R9={(ū1,,ūn,v̄1,.,v̄n,w̄)|ūi,v̄i,w̄R,i=1,,9}.

Then we choose an (LCS)9-structure (φ, ζ, η, g) on R9 such as

ξ=3w̄,η=13dw̄+inv̄idūi,g=ηη+19indūidūidv̄idv̄i,

φ(ū1)=v̄1,φ(ū2)=v̄2,φ(ū3)=ū3,φ(ū4)=v̄4,φ(v̄1)=ū1,

φ(v̄2)=ū1,φ(v̄3)=v̄3,φ(v̄4)=v̄4,φ(w̄)=0,where ūi,v̄i=EiT(R9), 1 ≤ i ≤ 4 are vector fields. Indeed (R9,φ,ζ,η,g) is an (LCS)9 manifold [6].

Now, we consider the map γ:(LCS)9=(R9,φ,ζ,η,g)(R5,g5) defined by

γū1,ū2,ū3,ū4,v̄1,v̄2,v̄3,v̄4,w̄ū1+ū2,v̄1+v̄2,ū3v̄33,ū4v̄43,3z̄,

where g5 is the semi-Riemannian metric of R5. Then the Jacobian matrix of γ is

1100000000000110000013000130000013000130000000003.

Since rank of the Jacobian matrix is equal to 5, the map γ is a submersion. On the other hand, we can easily see that γ satisfies the condition (C2). Therefore, γ is a Lorentzian submersion. Now, after some computation, we turn up

Kerγ*=SpanV1=E5+E6,V2=E1+E2,V3=13E3+E7,V4=13E4+E8,

and

Kerγ*=SpanH1=E1+E2,H2=E5+E6,H3=13E3E7,H4=13E4E8,H5=ζ.

In addition, we notice that φ(Vi) = Hi for 1 ≤ i ≤ 4, which implies that φ(Kerγ*)(Kerγ*). Thus γ is an anti-invariant Lorentzian submersion and ζ is a HRVF.

Let γ be an anti-invariant Lorentzian submersion from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gN). For any EΓ(Kerγ*), we write

φE=BE+CE,

where BEΓ(Kerγ*) and CEΓ(Kerγ*). At first, we examine the behaviour of the tensor fields T and A for the (LCS)n-manifold submersion γ.

Lemma 6.1. Let γ be an anti-invariant Lorentzian submersion from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS) with HRVF, then we have

TUφV=BTUV,
HUφVαgU,Vζ+2ηUηVζ=CTUV+φ̂UV,
̂VBE+TVCE=BHVEαηEV,
TVBE+HVCE=CHVE+φTVE,
AEφV=BAEV,
HEφV=φVEV+CAEV,
VEBF+AECF=BHEF,
AEBF+HECF=CHEF+φAEF+αgE,Fζ+2ηEηFζ+ηFE,

where U, V ∈ Γ(Kerγ*) and E,FΓ(Kerγ*).

Proof. The proof is quite similar to proof of Lemma 5.1. As a result, we leave it out.

Next, we study the properties of anti-invariant Lorentzian submersions from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto a semi-Riemannian manifold (S, gS) if the Reeb vector field ζ is horizontal. Using (6.1), we have μ=φμζ.

Now, let V and E denote the vertical and horizontal vector fields, respectively. In the light of the previous relationship and (2.6), we arrive at

gφV,CE=0gγ*φV,γ*CE=0
TN=γ*φKerγ*γ*μ.

From Eqs 2.6, 6.1, we conclude the following Lemma.

Lemma 6.2. Let γ be an anti-invariant Lorentzian submersion with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) to (S, gS). Then

BCE=0,φ2E=φBE+C2E,

for any EΓ(Kerγ*).

Lemma 6.3. Let γ be an anti-invariant Lorentzian submersion with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) to (S, gS). Then

BE=AEζ,
TUζ=αU,
gAEζ,φU=0,
gFAEζ,φU=gAEζ,φAFU,
gECF,φU=gCF,φAEU

for E,FΓ(Kerγ*) and U ∈ Γ(Kerγ*).

Proof On using eequations (2.5) (3.8), and (5.1), we obtain (6.6). Using (3.6) and 2.5, we obtain (6.7). Since AEζ is vertical and φU is horizontal for EΓ(Kerγ*) and U ∈ Γ(Kerγ*), we have (6.8). Also (6.8) gives

gFAEζ,φU=gAEζ,FφU,

for E,FΓ(Kerγ*) and U ∈ Γ(Kerγ*). Then using (3.7) and 2.6 we have

gFAEζ,φU=gAEζ,φAFUgAEζ,φFU.

Since φ(FU)Γ(Kerγ*), we obtain (6.9). From (4.1) we get

gCF,φU=0,
0=gECF,φU+gCF,EφU=gECF,φU+gCF,φEU,
gECF,φU=gCF,φAEU.

Hence, we obtain (6.10).

7 Lagrangian Lorentzian Submersions With Vertical Reeb Vector Field From (LCS)n-Manifold

In this section, the integrability and totally geodesicness of the horizontal distribution of LLS admitting VRVF from (LCS)n-manifolds will be determined. The behavior of the O’Neill’s tensor T of such a submersion is first investigated. From Lemma 6.1, we obtain the following:

Lemma 7.1. Let γ be an LLS with a VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS), then we have

TUφVαgU,Vζ+2ηUηVζ=φTUVαηVU,
TVφE=φTVE,
TVζ=αV,
TζE=αE,

for U, V ∈ Γ(Kerγ*) and E,FΓ(Kerγ*).

Proof. For a Lagrangian submersion, we have CE=0,EΓ(Kerγ*). Thus, assertions (7.1) and (7.2) follow from 5.2 and 5.4, respectively. Eqs 7.3 follows from 3.5 and 5.13.

Remark 7.2. It is known from [41] that the fibers of a semi-Riemannian submersion are totally geodesic if the O’Neill’s tensor T vanishes ie., T=0.

From Lemma 7.1, we can notice that the O’Neill’s tensor T0. Therefore, in view of Remark 7.2, we immediately get the next result.

Theorem 7.3. Let γ be an LLS with a VRVF from an (LCS)n-manifold (L, φ, ζ, η, g) onto (S, gS). Then the fibers of γ cannot be totally geodesic.

Next, we give some results about the characteristic of the O’Neill’s tensor A of γ.

Corollary 7.4. Let γ be an LLS with a VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS), then we have

AEφV=φAEV,
AEφF=φAEF,
AEζ=αE

for V ∈ Γ(Kerγ*) and E,FΓ(Kerγ*).

Proof. The assertions (7.4) and (7.5) follow from 5.5 and 5.8, respectively. The last assertion follows from 3.3 and 3.7.

Remark 7.5. In fact in a semi-Riemannian submersion, the integrability and totally geodesicness of the horizontal distribution are comparable to each other. This situation can be noticed from 3.4 and 3.8. In this case, the O’Neill’s tensor A vanishes.

From Eq. 7.6, we can observe that the O’Neill’s tensor A can not vanish for γ. Thus, we state the following result.

Theorem 7.6. Let γ be an LLS with a VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then the totally geodesicness of horizontal distribution of γ can not be integrable.

Remark 7.7. A smooth map γ (M, g) → (N, gN) between semi-Riemannian manifolds is said to be a totally geodesic map if γ* preserves parallel translation. Moreover, Vilms [41] classified totally geodesic Lorentzian submersions and verified that a Lorentzian submersion γ (L, g) → (S, gS) is totally geodesic if and only if both O’Neill’s tensors T and A vanish.

Thus, in view of Remark 7.7 and from Theorem 7.3 or Theorem 7.6, we turn up the following theorem.

Theorem 7.8. Let γ be an LLS admitting a VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then the submersion γ can not be a totally geodesic map.

Finally, we exhibit a necessary and sufficient condition for submersion γ to be harmonic.

Theorem 7.9. Let γ be an LLS with a VRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then γ is harmonic if and only if traceφTV|Kerγ*=0 for V ∈ Γ(Kerγ*), where φTV|Kerγ* is the restriction of φTV to Kerγ*.

Proof. From [42], we know that γ is harmonic if and only if γ has minimal fiber. Let {e1, …, ek, ζ} be an orthonormal frame of Kerγ*. Thus γ is harmonic if and only if i=1kTeiei+Tζζ=0. Since Tζζ=0, it follows that γ is harmonic if and only if i=1kTeiei=0. Now, we calculate i=1kTeiei. By orthonormal expansion, we can write

i=1kTeiei=i=1kj=1kgTeiei,φejφej,

where {φe1, …, φek} is an orthonormal frame of φKerγ*. Since Tei is skew-symmetric, we obtain

i=1kTeiei=i,j=1kgTeiφej,eiφej.

Here, from (7.1), we know

Teiφej=φTeiej+αgei,ejζ+ηejei+2ηeiηejζ.

Thus, we get

i=1kTeiei=i,j=1kgφTeiej,eiφej,

since both η(ej) = 0 and η(ei) = 0. Using (3.3), we arrive

i=1kTeiei=i,j=1kgφTejei,eiφej.

Since φe1, …, φek are linearly independent, from (7.7), we see that

i=1kTeiei=0i,j=1kgφTeiej,ei=0.

It is clear to observe that,

i,j=1kgφTejei,ei=0i=1kgφTVei,ei=0

for any V ∈ Γ(Kerγ*). On the other hand,

TraceφTV|Kerγ*=i=1kgφTVei,ei+gTVζ,ζ

and by (7.3),

TraceφTV|Kerγ*=i=1kgφTVei,ei.

Thus Eqs 7.8.10.–.Eqs 7.7.10 complete the proof.

Remark 7.10. Since an LLS is a specific case of an anti-invariant Lorentzian submersion. Then, in the view of Remark 7.7, Theorem 7.3, Theorem 7.6 and Theorem 7.8 also hold for anti-invariant Lorentzian submersions with a VRVF.

Example.

Let R5={(ū1,ū2,v̄1,v̄2,w̄)|(ū1,ū2,v̄1,v̄2,w̄)(0,0,0,0,0,0)}, where (ū1,ū2,v̄1,v̄2,w̄) be the standard coordinates in R5 and R2 be (LCS)n-manifolds as in previous Example.

Now, let us consider the mapping π:(LCS)5=(R5,φ,ζ,η,g)(R2,g2) defined by the following:

γū1,ū2,v̄1,v̄2,w̄ū1v̄23,ū2v̄13,

where g2 is the semi-Riemannian metric of R2. Then the Jacobian matrix of γ is as follows:

13013000131300.

Since the rank of the matrix is equal to 2, the map γ is a submersion. On the other hand we can easily see that γ holds the condition (C2). Then, by a direct computation, we turn up

Kerγ*=SpanV1=13E1+E4,V2=13E2+E3,V3=ζ,

and

Kerγ*=SpanH1=13E1E4,H2=13E2E3.

It is obvious to recognize that φ(V1) = H1, φ(V2) = H2 and φ(V3) = 0, which mean

φKerγ*=Kerγ*,

As a result γ is an LLS such that ζ is a VRVF.

8 Lagrangian Lorentzian Submersions With Horizontal Reeb Vector Field From an (LCS)n-Manifold

In this section, we examine the LLS with a HRVF from (LCS)n-manifolds (M, φ, ζ, η, g, α) onto a semi-Riemannian manifold.

Theorem 8.1. Let the dimension of (LCS)n-manifold (L, φ, ζ, η, g, α) be (2m + 1) and (S, gS) be a semi-Riemannian manifold of dimension n. If γ (L, φ, ζ, η, g) → (S, gS) is an LLS with HRVF, then m + 1 = n.

Proof. Let us consider that U1, U2, …, Uk is an orthonormal frame of (Kerγ*), where k = 2m − n + 1. Since φ(Kerγ*)=Kerγ*{ζ}, {φU1, …, φUk, ζ} forms an orthonormal frame of Γ(Kerγ*). So, from (5.10) we get k = n − 1 which implies that m + 1 = n.

Note that the proof of Theorem 8.1 has also been given in [32], but we gave it here for clarity.

From Lemma 5.1, we deduce the next corollary.

Corollary 8.2. Let γ be an LLS with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then, we have

TUφV=φTUV,
TVφE=φTVE,
TVζ=αV.

for U, V ∈ Γ(Kerγ*) and EΓ(Kerγ*).

Proof. Assertions (8.1) and (8.2) follow from (6.2) and 6.3, respectively. The last assertion (8.3) follows from (5.13) and 3.6 or directly from (6.7).

From (8.3), we see that the tensor T can not be zero, so we have the following results.

Theorem 8.3. Let γ be an LLS from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then, the fibers of γ can not be totally geodesic.

Corollary 8.4. Let γ be an LLS with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then, we have

AEφV=φAEV,
AEBF=φAEF+αgE,FHζ+2ηEηFHζ+ηFE,
AζV=αφV,
AζE=αφE

for V ∈ Γ(Kerγ*) and E,FΓ(Kerγ*).

Proof. Assertions (8.4) and (8.5) follow from (6.4) and (6.5), respectively. The third assertion (8.6) follows from (3.3) and (3.7). The last one comes from (8.7).

From (8.4) and (8.5), it can be easily seen that the tensor A can not be zero. Thus, by Remark 7.5, we have the following result.

Theorem 8.5. Let γ be an LLS with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then, the horizontal distribution of γ can not be integrable.

In view of Remark 7.7 and Theorem 8.3 or Theorem 8.5, we get the following result.

Corollary 8.6. Let γ be an LLS with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then, the submersion γ can not be a totally geodesic map.

Finally, we give a result concerning the harmonicity of such submersions.

Theorem 8.7. Let γ is an LLS with a HRVF from an (LCS)n-manifold (L, φ, ζ, η, g, α) onto (S, gS). Then γ can not be harmonic.

Proof. Let {e1, …, ek} be an orthonormal frame of Kerγ*. Then {φe1, …, φek, ζ} forms an orthonormal frame of Kerγ*. Hence, we have

i=1kTeiei=i,j=1kgTeiei,φejφej+gTeiei,ζζ.

Using the skew-symmetricness of Tei and (8.1), we obtain

i=1kTeiei=i,j=1kgφTeiej,eiφej+gTeiζ,eiζ.

By (3.3) and (8.3), we get

i=1kTeiei=i,j=1kgφTejei,eiφej.

Now, we assume that γ is harmonic. Then i=1kTeiei=0. From (8.8), it follows that i,j=1kg(φTejei,ei)φej=0. This implies that the set {φe1, …, φek, ζ} is linearly independent.

Remark 8.8. In view of Remark 7.7, Theorem 8.3, Theorem 8.5 and Corollary 8.6 also hold for anti-invariant Lorentzian submersions with a HRVF.

Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling editor declared a past co-authorship with one of the authors AI.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors are highly thankful to referees and the handling editor for their valuable suggestions and comments which have improved the contents of the paper.

References

1. Shaikh AA On Lorentzian Almost Paracontact Manifolds with a Structure of the Concircular Type. Kyungpook Math J (2003) 43:305–14.

Google Scholar

2. Matsumoto K On Lorentzian Paracontact Manifolds. Bull Yamagata Univ Natur Sci (1989) 12:151–6.

Google Scholar

3. Mihai I, Rosca R On Lorentzian P-Sasakian Manifolds. In: Classical Analysis (Kazimierz Dolny, 1991). River Edge, NJ: World Sci. Publ. (1992). p. 155–69.

Google Scholar

4. Chaubey SK, Shaikh AA On 3-dimensional Lorentzian Concircular Structure Manifolds. Commun Korean Math Soc (2019) 34(1):303–19. doi:10.4134/CKMS.c180044

CrossRef Full Text | Google Scholar

5. Shaikh AA, Baishya KK On Concircular Structure Spacetimes. J Math Stat (2005) 1:129–32. doi:10.3844/jmssp.2005.129.132

CrossRef Full Text | Google Scholar

6. Shaikh AA, Matsuyama Y, Hui SK On Invariant Submanifolds of ( LCS ) N -manifolds. J Egypt Math Soc (2016) 24:263–9. doi:10.1016/j.joems.2015.05.008

CrossRef Full Text | Google Scholar

7. Siddiqi MD, Chaubey SK Almost η-conformal Ricci Solitons in (LCS)3-manifolds. Sarajevo J Math (2020) 16:245–60. doi:10.5644/SJM.16.02

CrossRef Full Text | Google Scholar

8. Yadav SK, Chaubey SK, Suthar DL Some Geometric Properties of η Ricci Solitons and Gradient Ricci Solitons on (Lcs) N -manifolds. Cubo (2017) 19(2):33–48. doi:10.4067/s0719-06462017000200033

CrossRef Full Text | Google Scholar

9. Mantica CA, Molinari LG A Notes on Concircular Structure Spacetimes. Commun Korean Math Soc (2019) 34(2):633–5. doi:10.4134/CKMS.c180138

CrossRef Full Text | Google Scholar

10. Alías LJ, Romero A, Sánchez M Uniqueness of Complete Spacelike Hypersurfaces of Constant Mean Curvature in Generalized Robertson-Walker Spacetimes. Gen Relat Gravit (1995) 27(1):71–84. doi:10.1007/bf02105675

CrossRef Full Text | Google Scholar

11. Nash JN The Imbedding Problem for Riemannian Manifolds. Ann Math (1956) 63(2):20–63. doi:10.2307/1969989

CrossRef Full Text | Google Scholar

12. Falcitelli M, Ianus S, Pastore AM Riemannian Submersions and Related Topics. River Edge, NJ: World Scientific (2004).

Google Scholar

13. Ṣahin B Riemannian Submersions, Riemannian Maps in Hermitian Geometry and Their Applications. London, United Kingdom: Academic Press (2017).

Google Scholar

14. Allison D Lorentzian Clairaut Submersions. Geometriae Dedicata (1996) 63:309–19. doi:10.1007/bf00181419

CrossRef Full Text | Google Scholar

15. O’Neill B The Fundamental Equations of a Submersion. Mich Math J (1966) 13:458–69. doi:10.1307/MMJ/1028999604

CrossRef Full Text | Google Scholar

16. O’Neill B Semi Riemannian Geometry with Application to Relativity. New York: Academic Press (1983).

Google Scholar

17. Gray A Pseudo-Riemannian Almost Product Manifolds and Submersion. J Math Mech (1967) 16:715–37.

Google Scholar

18. Magid MA Submersions from Anti-de Sitter Space with Totally Geodesic Fibers. J Diff Geom (1981) 16:323–31. doi:10.4310/jdg/1214436107

CrossRef Full Text | Google Scholar

19. Watson B Almost Hermitian Submersions. J Diff Geom (1976) 11(1):147–65. doi:10.4310/jdg/1214433303

CrossRef Full Text | Google Scholar

20. Falcitelli M, Pastore AM A Note on Almost Kähler and Nearly Kähler Submersions. J Geom (2000) 69:79–87. doi:10.1007/bf01237477

CrossRef Full Text | Google Scholar

21. Ṣahin B Invariant and Anti-invariant Riemannian Maps to Kähler Manifolds. Int J Geom Methods Mod Phy (2010) 7(3):1–19. doi:10.1142/SO219887810004324

CrossRef Full Text | Google Scholar

22. Shahid A, Tanveer F Anti-invariant Riemannian Submersions from Nearly Kählerian Manifolds. Filomat (2013) 27(7):1219–35. doi:10.2298/FIL1307219A

CrossRef Full Text | Google Scholar

23. Akyol MA, Gündüzalp Y Semi-invariant Semi-riemannian Submersions. Commun Fak Sci Univ Ser A1 Math Stat (2018) 67(1):80–92. doi:10.1501/Commua1_0000000832

CrossRef Full Text | Google Scholar

24. Gündüzalp Y Paracontact Semi-riemannian Submersion. Turk J Math (2013) 37:114–28. doi:10.3906/mat-1103-10

CrossRef Full Text | Google Scholar

25. Faghfouri M, Mashmouli S On Anti-invariant Semi-riemannian Submersions from Lorentzian Para-Sasakian Manifolds. Filomat (2018) 32(10):3465–78. doi:10.2298/fil1810465f

CrossRef Full Text | Google Scholar

26. Siddiqi MD, Chaubey SK, Siddiqui AN Clairaut Anti-invariant Submersions from Lorentzian Trans-sasakian Manifolds. Ajms (2021). ahead-of-print. doi:10.1108/AJMS-05-2021-0106

CrossRef Full Text | Google Scholar

27. Siddiqi MD, Ahmad M Anti-invariant Semi-riemannian Submersions from Indefinite Almost Contact Metric Manifolds. Konuralp J Math (2020) 8(1):38–9.

Google Scholar

28. Ṣahin B Anti-invariant Riemannian Submersions from Almost Hermitian Manifolds. Cent Eur J Math (2010) 8(3):437–47. doi:10.2478/s11533-010-0023-6

CrossRef Full Text | Google Scholar

29. Taştan HM On Lagrangian Submersions Hacettepe. J Math Stat (2014) 43(6):993–1000. doi:10.15672/HJMS.2014437529

CrossRef Full Text | Google Scholar

30. Gündüzalp Y Anti-invariant Riemannian Submersions from Almost Product Riemannian Manifolds. Math Sci Appl E-notes (2013) 1:58–66.

Google Scholar

31. Taştan HM, Özdemir F, Sayar C On Anti-invariant Riemannian Submersions Whose Total Manifolds Are Locally Product Riemannian. J Geom (2017) 108:411–22. doi:10.1007/s00022-016-0347-x

CrossRef Full Text | Google Scholar

32. Küpeli Erken İ, Murathan C Anti-invariant Riemannian Submersions from Sasakian Manifolds (2013). New York, NY: Cornell University, Press. arxiv: 1302.4906v1 [math.DG].

Google Scholar

33. Lee JW Anti-invariant ξ-Riemannian Submersions from Almost Contact Manifolds. Hacettepe J Math Stat (2013) 42(3):231–41.

Google Scholar

34. Taştan HM Lagrangian Submersions from normal Almost Contact Manifolds. Filomat (2017) 31(12):3885–95. doi:10.2298/FIL1712885T

CrossRef Full Text | Google Scholar

35. Beri̇ A, Küpeli̇ Erken İ, Murathan C Anti-invariant Riemannian Submersions from Kenmotsu Manifolds onto Riemannian Manifolds. Turk J Math (2016) 40(3):540–52. doi:10.3906/mat-1504-47

CrossRef Full Text | Google Scholar

36. Murathan C, Küpeli E Anti-invariant Riemannian Submersions from Cosymplectic Manifolds onto Riemannian Manifolds. Filomat (2015) 29(7):1429–44. doi:10.2298/fil1507429m

CrossRef Full Text | Google Scholar

37. Siddiqi MD, Akyol MA Anti-invariant ξ ⊥ -Riemannian Submersions from Hyperbolic β-Kenmotsu Manifolds. Cubo (2018) 20(1):79–94. doi:10.4067/s0719-06462018000100079

CrossRef Full Text | Google Scholar

38. Siddiqi MD, Akyol MA Anti-invariant ξ-Riemannian Submersions from Almost Hyperbolic Contact Manifolds. Int Elect J Geom (2019) 12(1):32–42.

Google Scholar

39. Taştan HM, Siddiqi MD Anti-invariant and Lagrangian Submersions from Trans-sasakian. Balkan J Geom App (2020) 25(2):106–23.

Google Scholar

40. Akyol MA, Sari R, Aksoy E Semi-invariant-ξ-Riemannian Submersions from Almost Contact Metric Manifolds. Int J Geom Methods Mod Phys (2017) 14(5):170074. 17. doi:10.1142/s0219887817500748

CrossRef Full Text | Google Scholar

41. Vilms J Totally Geodesic Maps. J Diff Geom (1970) 4:73–9. doi:10.4310/jdg/1214429276

CrossRef Full Text | Google Scholar

42. Eells J, Sampson JH Harmonic Mappings of Riemannian Manifolds. Am J Mathematics (1964) 86(1):109–60. doi:10.2307/2373037

CrossRef Full Text | Google Scholar

Keywords: (LCS)n-manifolds, Lorentzian submersion, anti-invariant Lorentzian submersion, Lagrangian Lorentzian submersion, horizontal distribution

Citation: Siddiqi MD, Khan MA, Ishan AA and Chaubey SK (2022) Anti-Invariant Lorentzian Submersions From Lorentzian Concircular Structure Manifolds. Front. Phys. 10:812190. doi: 10.3389/fphy.2022.812190

Received: 09 November 2021; Accepted: 11 January 2022;
Published: 28 February 2022.

Edited by:

Josef Mikes, Palacký University, Czechia

Reviewed by:

Yılmaz Gündüzalp, Dicle University, Turkey
Mehraj Lone, National Institute of Technology, Srinagar, India

Copyright © 2022 Siddiqi, Khan, Ishan and Chaubey. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Meraj A. Khan, meraj79@gmail.com; S. K. Chaubey, sudhakar.chaubey@shct.edu.om

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.