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

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

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.”

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])

and

respectively.

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

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

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:

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

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:

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).

and $g∼$ is $P∼$-compatible, i.e.

Using (2.14) in (2.15), we obtain

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.

Clearly,

From above decomposition of distributions, we get

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

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

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

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

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

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

Then we can find following tangent vectors of the above submanifold

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

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

and

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

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

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,

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

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

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

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

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

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

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

i.e.

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

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

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

(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

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

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

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

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

In the present situation, this reduces to

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

Since $gˇ2$ is constant on $Nˇ1,$ we get

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

Hence, we must have $∇ξU∈Γ(S(TNˇ)).$ □

for any $U∈S(TNˇ)$ and $ξ∈Rad(TNˇ).$

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

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

Using (2.6) in (4.5), we get

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

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

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

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

From (4.7) and (4.8), we get

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

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.

The second author is thankful to CSIR (Govt. of India) for providing financial assistance in terms of JRF scholarship vide letter no. (09/1051/(0022)/2018-EMR-I).