An investigation on the existence of warped product irrotational screen-real lightlike submanifolds of metallic semi-Riemannian manifolds

Abstract

Purpose

The purpose of this paper is to study the geometry of screen real lightlike submanifolds of metallic semi-Riemannian manifolds. Also, the authors investigate whether these submanifolds are warped product lightlike submanifolds or not.

Design/methodology/approach

The paper is design as follows: In Section 3, the authors introduce screen-real lightlike submanifold of metallic semi Riemannian manifold. In Section 4, the sufficient conditions for the radical and screen distribution of screen-real lightlike submanifolds, to be integrable and to be have totally geodesic foliation, have been established. Furthermore, the authors investigate whether these submanifolds can be written in the form of warped product lightlike submanifolds or not.

Findings

The geometry of the screen-real lightlike submanifolds has been studied. Also various results have been established. It has been proved that there does not exist any class of irrotational screen-real r-lightlike submanifold such that it can be written in the form of warped product lightlike submanifolds.

Originality/value

All results are novel and contribute to further study on lightlike submanifolds of metallic semi-Riemannian manifolds.

Citation

Shanker, G. and Yadav, A. (2022), "An investigation on the existence of warped product irrotational screen-real lightlike submanifolds of metallic semi-Riemannian manifolds", Arab Journal of Mathematical Sciences, Vol. 28 No. 2, pp. 142-151. https://doi.org/10.1108/AJMS-09-2020-0060

1. Introduction

It is well known that the study of semi-Riemannian manifolds and its submanifolds is more complicated as compared to Riemannian manifolds and its submanifolds. It is observed that the induced metric on submanifolds of semi-Riemannian manifolds has two cases, either non-degenerate or degenerate. In case of non-degenerate, there is no complications to do calculus on these submanifolds. On the other hand, if submanifold has a degenerate metric then there is non-trivial intersection of tangent bundle and normal bundle. Due to this, it is not possible to induce many structures uniquely and with same character as structures of ambient space. The study of degenerate submanifolds is known as lightlike geometry. Due to extensive use as a tool to understand theory of relativity, it becomes a topic of interest for mathematicians and physicists.

In 1996, Duggal and Bejancu gave a detail explanation of lightlike geometry in [1]. Later, many research articles have been published on lightlike geometry. In physics, various spacetime models have been studied with the help of lightlike geometry.

Crasmareanu and Hretcanu [2] introduced golden Riemannian manifolds by using golden ratio. Later, Spinadel introduced generalization of golden means known as metallic means [3–5]. For any positive integers p and q, the positive solutions of the equation y2−py−q=0, are known as metallic means and

σp,q=p+p2+4q2.

is known as (p,q) metallic number. A metallic semi-Riemannian manifold (N~, g~) is a semi-Riemannian manifold with metallic structure P~ such that g~ is P~-compatible metric. Different types of submanifolds of metallic and golden Riemannian manifolds have been studied in [1, 3–9]. Apart from this, the geometry of various submanifolds of metallic and golden semi-Riemannian manifolds have been studied in [10–12]. This paper is categorized as follows:

In Section 1, we give brief description of lightlike geometry and metallic semi-Riemannian manifolds. In Section 2, the necessary definitions and theorems required for the current work have been mentioned. In Section 3, we introduce geometry of screen-real lightlike submanifolds of metallic semi-Riemannian manifolds. The necessary and sufficient conditions for integrability and to be totally geodesic foliations of Rad(TNˇ) and S(TNˇ) have been established. In Section 4, we prove that “there does not exist any class of irrotational screen-real r-lightlike submanifolds that can be written in the form of warped product lightlike submanifold.”

2. Preliminaries

A submanifold (Nˇm, gˇ) of a semi-Riemannian manifold (N~m+n, g~) with constant index q (1≤q≤m+n−1, m, n≥1) is known as degenerate (lightlike) submanifold, if the induced metric gˇ is degenerate [7].

Due to generate induced metric on TNˇ, for any u∈Nˇ, there exist non zero intersection of TuNˇ (m-dimensional) and TuNˇ⊥ (n-dimensional), which is called Rad(TNˇ). A lightlike submanifold is known as r-lightlike, if there exists a smooth distribution Rad(TNˇ) of rank r>0, such that every member u of Nˇ goes to an r-dimensional subspace Rad(TuNˇ) of TuNˇ. Let S(TNˇ) (screen distribution) and S(TNˇ⊥) (screen transversal distribution) are non-degenerate complementary sub-bundles of Rad(TNˇ) in TNˇ and TNˇ⊥ respectively. Let ltr(TNˇ)(lightlike transversal bundle) and tr(TNˇ)(transversal bundle) be complementary but not orthogonal vector bundles to Rad(TNˇ) and TNˇ in S(TN⊥)⊥ and TN~|Nˇ respectively.

Then, the orthogonal decomposition of tr(TNˇ) and TNˇ|Nˇ are given by (for detail see [7])

(2.1)tr(TNˇ)=ltr(TNˇ)⊥S(TNˇ⊥)

and

(2.2)TN~|Nˇ=TNˇ⊕tr(TNˇ)=[Rad(TNˇ)⊕ltr(TNˇ)]⊥S(TNˇ)⊥S(TNˇ⊥)

respectively.

Theorem 2.1.

[7] Let (N~, g~) be a semi-Riemannian manifold, (Nˇ, gˇ, S(TNˇ), S(TNˇ⊥)) be its r-lightlike submanifold. Then there exists a vector bundle ltr(TNˇ) and a basis of Γ(ltr(TNˇ)|u) containing a smooth section {Ni} of S(TNˇ⊥)⊥|u, for a coordinate neighborhood u of Nˇ, such that

(2.3)g∼ij(Ni, ξj)=δij, g∼ij(Ni, Nj)=0,

for any i, j ∈{1, 2, …, r}, where {ξi} is a lightlike basis of Γ(Rad(TNˇ)).

For any U, V∈Γ(TNˇ) and W∈Γ(tr(TNˇ)), the Gauss and Weingarten formulae are

(2.4)∇∼UV=∇UV+h(U, V),

(2.5)∇∼UW=−AWU+∇U⊥W,

where {∇UV, AWU} and {h(U, V), ∇Y⊥W} belong to Γ(TNˇ) and Γ(tr(TNˇ)) respectively, and ∇ is a induced connection on Nˇ. Further, from (2.4) and (2.5), we deduce that

(2.6)∇∼UV=∇UV+hl(U, V)+hs(U, V),

(2.7)∇∼UN=−ANU+∇Ul(N)+Ds(U, N), N∈Γ(ltr(TNˇ)),

(2.8)∇∼UW1=−AW1U+∇Us(W1)+Dl(U, W1), W1∈Γ(S(TNˇ⊥)).

Eqns (2.4), (2.6) are known as Gauss equations and (2.5), (2.7), (2.8) are known as Weingarten equations respectively, for the lightlike submanifold Nˇ of N~.

Using metric connection ∇~ and (2.4)-(2.8), we get the following equations:

(2.9)g∼(hs(U, V), W′)+g∼(U, Dl(V, W′))=gˇ(AW′U, V),

(2.10)g∼(hl(U, V), ξ)+g~(U, hl(V, ξ))=−gˇ(U, ∇Vξ),

for any ξ∈Γ(Rad(TNˇ)) U, V∈Γ(TNˇ), and W′∈Γ(S(TNˇ⊥)).

Since the induced connection is not necessarily Levi- Civita connection, for any U1, U2, U3∈Γ(TNˇ) and U, U′∈Γ(tr(TNˇ)), we have following formula

(2.11)(∇U1gˇ)(U2, U3)=g∼(hl(U1, U2), U3)+g∼(hl(U1, U3), U2).

Let S denote projection map on S(TNˇ) from TNˇ. Then, for any U, V∈Γ(TNˇ) and ξ∈Γ(Rad(TNˇ)), we have the following equations:

(2.12)∇USV=∇U*SV+h*(U,SV).

(2.13)∇Vξ=Aξ*V+∇V*t(ξ),

where {h*(U, PV), ∇V*t(ξ)} and {∇U∗SV, Aξ∗V} belong to Γ(Rad(TNˇ)) and Γ(S(TNˇ)) respectively.

For detail understanding of Eqns (2.4)–(2.13), see [7] (pp. 196–198).

Definition 2.1.

A metallic semi-Riemannian manifold is a smooth manifold with (1,1) tensor field P∼ on N∼ such that

(2.14)P∼2=pP∼+qI,

and g∼ is P∼-compatible, i.e.

(2.15)g∼(P∼U, V)=g∼(U, P∼V).

Using (2.14) in (2.15), we obtain

(2.16)g∼(P∼U, P∼V)=pg∼(P∼U, V)+qg∼(U, V),

for any U, V∈Γ(TNˇ) [2, 8].

If (∇∼UP∼)V=0, then P∼ is called locally metallic structure. Throughout the paper, we assume that P∼ is a locally metallic structure.

3. Screen-real lightlike submanifolds

Definition 3.1.

A lightlike submanifold (Nˇ, gˇ, S(TNˇ)) of a metallic semi-Riemannian manifold (N~, g~, P~) is said to be a screen-real lightlike submanifold if it satisfies the following:

P~(Rad(TNˇ)=Rad(TNˇ)  &  P~(S(TNˇ))⊆S(TNˇ⊥).

Clearly,

P∼(ltr(TNˇ)=ltr(TNˇ)  &  P∼(μ)=μ.

From above decomposition of distributions, we get

(3.1)TN~|Nˇ=[Rad(TNˇ)⊕ltr(TNˇ)]⊕orth.S(TNˇ)⊕orth.P~(S(TNˇ))⊕orth.μ.

For any U∈Γ(TNˇ), using (3.1), we obtain

U=RU+SU,

where R and S are projection maps on Rad(TNˇ) and S(TNˇ) respectively. Applying P~ on above equation and using (3.1), we obtain

(3.2)P∼U=RU+S′U,

where P∼RU=RU, P∼SU=S′U and R, S′ are projection maps on Rad(TNˇ) and S(TNˇ⊥) respectively.

For any w∈tr(TNˇ), we have

(3.3)P∼(w)=Bw+C1w+C2w+C3w,

where B, C1, C2 and C3 are projection maps on S(TNˇ), ltr(TNˇ), P∼S(TNˇ) and μ respectively.

For any w1∈Γ(ltr(TNˇ)), w2∈Γ(P∼S(TNˇ)) and w3∈Γ(μ), (3.3) takes following different forms, respectively

(3.4)P∼(w1)=C1w1,

(3.5)P∼(w2)=Bw2+C2w2,

(3.6)P∼(w3)=C3w3.

Example 3.1. Let (N∼=ℝ26, g∼, P∼) be a six dimensional semi-Euclidean space, where gˇ is a semi-Euclidean metric with signature (+ – + + + –). Let us define

P~(x1, x2, x3, x4, x5, x6)=(σx1, σx2, (p−σ)x3, σx4, σx5, (p−σ)x6),

where (x1, x2, x3, x4, x5, x6) is the standard coordinate system of ℝ26. Then, it can be easily verified that P~ is a metallic structure.

Let us define a submanifold Nˇ of N~ such that

x1=sinh σu1,  x2=cosh σu1,

x3=σqu4, x4=u4,

x5=u1, x6=0.

Then we can find following tangent vectors of the above submanifold

U1=sinhσ∂∂x1+coshσ∂∂x2+∂∂x5, U2=σq∂∂x3+∂∂x4,

such that TNˇ=span{U1, U2}. Clearly, Nˇ is a lightlike submanifold with

Rad(TNˇ)=span{U1=ξ},

S(TNˇ)=span{U2},

ltr(TNˇ)=span{N=12(−sinhσ∂∂x1−coshσ∂∂x2+∂∂x5)},

S(TNˇ⊥)=span{W=−q∂∂x3+σ∂∂x4},

where P∼ξ=σξ, P∼(N)=σN, P∼(U2)=W. Since P∼ satisfies P~(Rad(TNˇ))=(Rad(TNˇ)) and P~(S(TNˇ))=S(TNˇ⊥), Nˇ is a screen real lightlike submanifold.

Theorem 3.2.

Let (Nˇ,gˇ,S(TNˇ)) be a screen-real lightlike submanifold of a metallic semi-Riemannian manifold, then following equations hold:

R∇UV+B(hs(U,V))=∇URV−AS′VU,C1hl(U, V)=hl(U, RV)+Dl(U, S′V)

and

hs(U, RV)+∇UsS′V=S′∇UV+C2hs(U, V)+C3hs(U, V).

Proof. Using (3.2)-(3.6) in (∇~UP~)V=0, for any U, V∈Γ(TNˇ), we obtain

(3.7)∇URV+hl(U, RV)+hs(U, RV)−AS′VU+∇UsS′V+Dl(U, S′V)=R∇UV+S′∇UV+C1hl(U, V)+B(hs(U, V))+C2hs(U, V)+C3hs(U, V).

By equating tangential, ltr(TNˇ) and S(TNˇ⊥) components in the above equation, we get required results. □

Theorem 3.3.

Let (Nˇ,gˇ,S(TNˇ)) be a screen-real lightlike submanifold of a metallic semi-Riemannian manifold, then

1. Rad(TNˇ) is integrable if and only if

   hs(P~ξ2, ξ1)=hs(P~ξ1, ξ2).

2. S(TNˇ) is integrable if and only if

   R(−AP~VU+AP~UV)=p(h∗(U, V)−h∗(V, U)).

Proof. (1) For any ξ1, ξ2∈Γ(Rad(TNˇ)) and U∈Γ(S(TNˇ)), Rad(TNˇ) is integrable if and only if,

g~([ξ1, ξ2], U)=0.

Expanding g~([ξ1, ξ2], U) and using (2.16), we get

(3.8)g~([ξ1, ξ2], U)=1qg~(P~(∇~ξ1ξ2−∇~ξ2ξ1), P~U)−pqg~(∇~ξ1ξ2−∇~ξ2ξ1, P~U).

Using (2.6), (2.8) and (2.12) in (3.8), we get

(3.9)g~([ξ1, ξ2], U)=1qg~(hs(P~ξ2, ξ1)−hs(P~ξ1, ξ2), P~U).

From (3.11), we obtain g~([ξ1, ξ2]U)=0 if and only if

hs(P~ξ2, ξ1)=hs(P~ξ1, ξ2)

(2) For any U, V∈Γ(S(TNˇ)) and N∈Γ(ltr(TNˇ)), S(TNˇ) is integrable if and only if,

g~([U, V], N)=0.

Expanding g~([U, V], N) and using (2.16), we get

(3.10)g~([U, V], N)=1qg~(P~(∇~UV−∇~VU), P~N)−pqg~(∇~UV−∇~VU, P~N).

Using (2.6), (2.8) and (2.12) in (3.10), we obtain

(3.11)g~([U, V], N)=1qg~(−AP~VU+AP~UV, P~N)−pqg~(h∗(U, V)−h∗(V, U), P~N).

From (3.11), we obtain g~([U,V],N)=0 if and only if

g~(R(−AP~VU+AP~UV), P~N)=pg~(h∗(U, V)−h∗(V, U), P~N),

i.e.

R(−AP~VU+AP~UV)=p(h*(U, V)−h*(V, U)).      □

Theorem 3.4.

Let (Nˇ, gˇ, S(TNˇ)) be a screen-real lightlike submanifold of a metallic semi-Riemannian manifold, then

1. Rad(TNˇ) defines a totally geodesic foliation if and only if

   hs(ξ1, P~ξ2)=phs(ξ1, ξ2).

2. S(TNˇ) defines a totally geodesic foliation if and only if

   R(AP~VU)=−ph*(V,U).

Proof. (1) For any ξ1,ξ2∈Γ(Rad(TNˇ)) and U∈Γ(S(TNˇ)), Rad(TNˇ) defines a totally geodesic foliation if and only if g~(∇ξ1ξ2,U)=0.

Using (2.16), we get

(3.12)g~(∇ξ1ξ2, U)=g~(∇~ξ1ξ2, U)=1qg~(P~∇~ξ1ξ2, P~U)−pqg~(∇~ξ1ξ2, P~U).

Using (2.6), (2.8) and (2.12) in (3.12), we obtain

(3.13)g~(∇~ξ1ξ2, U)=1qg~(hs(ξ1, P~ξ2), P~U)−pqg~(hs(ξ1, ξ2), P~U).

From (3.13), we get g~(∇~ξ1ξ2, U)=0 if and only if

hs(ξ1, P~ξ2)=phs(ξ1, ξ2).

(2) For any U, V∈S(TNˇ) and N∈ltr(TNˇ), S(TNˇ) defines a totally geodesic foliation if and only if, g~(∇UV, N)=0.

Using (2.16), we get

(3.14)g~(∇UV, N)=g~(∇~UV, N)=1qg~(P~∇~UV, P~N)−pqg~(∇~UV, P~N).

Using (2.6), (2.8) and (2.12) in (3.14), we obtain

(3.15)g~(∇~UV, N)=1qg~(−AP~VU, P~N)−pqg~(−AV∗U+h∗(V, U), P~N).

From (3.15), we get g~(∇UV, N)=0 if and only if

R(AP~VU)=−ph∗(V, U).              □

4. Warped product lightlike submanifolds

Definition 4.1.

[7] A product manifold Nˇ=N1×λN2, where (N1,g1) is an r-dimensional totally lightlike submanifold and (N2,g2) is an m-dimensional semi-Riemannian submanifold of a semi-Riemannian manifold N~, is known as a warped product lightlike submanifold with induced degenerate metric gˇ defined as

(4.1)gˇ(U, V)=gˇ1(π1*U, π1*V)+(λ o π1)2gˇ2(π2*U, π2*V),

where U, V∈Γ(TNˇ), λ(non constant)∈C∞(N1, ℝ), π1 and π2 are projection maps from N1×N2 to N1 and N2 respectively and * denotes tangent map.

Theorem 4.1.

Let Nˇ=N1×λN2 be a warped product lightlike submanifold. Then, for any ξ∈Γ(Rad(TNˇ)) and U∈Γ(S(TM)), we have ∇ξU∈Γ(S(TNˇ)).

Proof. For any ξ∈Γ(Rad(TNˇ)) and U, V∈Γ(S(TM)), the Koszul formula is

(4.2)2g~(∇~ξU, V)=ξg~(U, V)+Ug~(ξ, V)−Vg~(ξ, U)+g~([ξ, U], V)+g~([V, ξ], U)−g~([U, V], ξ).

In the present situation, this reduces to

2g~(∇~ξU, V)=ξg~(U, V)−g~([U, V], ξ).

If ∇ξU∈Γ(Rad(TNˇ)), then above equation reduces to

ξg~(U, V)=ξ(λoπ1)2gˇ2(U, V)=0.

Since gˇ2 is constant on Nˇ1, we get

ξ(λ)λgˇ2(U ,V)=0.

Since λ is non-constant and gˇ2 is a positive definite metric, this contradicts our assumption.

Hence, we must have ∇ξU∈Γ(S(TNˇ)). □

Definition 4.2.

[7] An r-lightlike submanifold is said to be a irrotational lightlike submanifold if and only if

(4.3)hl(ξ, U)=0, hs(ξ, U)=0,

for any U∈S(TNˇ) and ξ∈Rad(TNˇ).

Theorem 4.2.

Let (Nˇ, g, S(TNˇ)) be an irrotational screen-real r-lightlike submanifold of a metallic semi-Riemannian manifold, then the induced connection is a metric connection.

Proof. Let ∇ be a connection induced from the ambient connection ∇~. Then, for any U,V∈Γ(TNˇ), ∇ is said to be a metric connection if and only if hl(U,V)=0.

From (4.2), hl(U,ξ)=0. Now, it is enough to show that h(U,V)=0, if U,V∈S(TNˇ).

Using (2.16), we get

(4.4)g~(∇~UV, ξ)=1qg~(P~∇~UV, P~ξ)−pqg~(∇~UV, P~ξ).

Since ∇~ is a metric connection, equation (4.4) reduces to

(4.5)g~(hl(U ,V), ξ)=−1qg~(P~V, ∇~UP~ξ)+pqg~(P~V, ∇~Uξ).

Using (2.6) in (4.5), we get

(4.6)g~(hl(U, V), ξ)=−1qg~(P~V, hs(U, P~ξ))+pqg~(P~V, hs(U, ξ)).

Since hs(U, P~ξ)=0 and hs(U, ξ)=0, (4.6) becomes

g~(hl(U, V), ξ)=0

This implies hl(U,V)=0. This completes the proof. □

Theorem 4.3.

There does not exist any class of irrotational screen-real r-lightlike submanifolds that can be written in the form of warped product lightlike submanifolds.

Proof. If possible, let there exist a class of irrotational screen-real r- lightlike submanifolds such that any Nˇ in this class can be written as warped product lightlike submanifolds i.e. Nˇ=N1×λN2.

Using Theorem (4.1) in (4.1), we obtain

(4.7)g~(∇~ξU, V)=ξ(λ)λgˇ2(U, V).

Since Nˇ is irrotational, using Theorem (4.2), for any U,V∈Γ(TNˇ), we get hl(U,V)=0,

From (2.4), we obtain

(4.8)g~(ξ, hl(U, V))=−g~(∇~ξU, V)=0.

From (4.7) and (4.8), we get

ξ(λ)λgˇ2(U, V)=0.

This implies that either λ is constant or gˇ2 is a degenerate metric. In either case, it is a contradiction. This completes the proof. □

5. Conclusion

Our aim in this paper is to investigate whether it is possible to write lightlike submanifolds of metallic semi-Riemannian manifolds in the form of warped product lightlike submanifolds or not. In this context, we introduce the screen real lightlike submanifolds and find that, it is difficult to say that screen real lightlike submanifolds are warped product lightlike submanifolds or not. We find a special class of screen real lightlike submanifolds namely, “irrotational screen real lightlike submanifolds”that can never be written in the form of warped product lightlike submanifolds.