Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds

Gauree Shanker (Department of Mathematics and Statistics, Central University of Punjab, Bathinda, India)
Ankit Yadav (Department of Mathematics and Statistics, Central University of Punjab, Bathinda, India) (Department of Mathematics, VIT-AP University, Amarvati, India)
Ramandeep Kaur (Department of Mathematics and Statistics, Central University of Punjab, Bathinda, India)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 30 September 2024

189

Abstract

Purpose

The screen Cauchy–Riemann (SCR)-lightlike submanifold is an important class of submanifolds of semi-Riemannian manifolds. It contains various other classes of submanifolds as its sub-cases. It has been studied under various ambient space. The purpose of this research is to study the geometry of SCR-lightlike submanifolds of metallic semi-Riemannian manifolds.

Design/methodology/approach

The article is divided into five sections. The first section is introductory section which represents brief overview of the conducted research of this article. The second section outlines the key results that are utilized throughout the paper. In section three, the definition of SCR-lightlike submanifold is constructed with one non-trivial example. In section four and five, the important results on integrability, totally geodesic foliations and warped product are given.

Findings

The SCR-lightlike submanifold is introduced. One non-trivial example is constructed which helps to understand the given structure. The necessary and sufficient conditions for the integrability and to be totally geodesic for various distributions are obtained. The necessary and sufficient conditions for induced connection on totally umbilical SCR-lightlike submanifolds to be a metric connection are discussed. Various results are found on totally umbilical SCR-lightlike submanifolds. Finally, the existence of the warped product lightlike submanifold of the type N×λNT is studied.

Originality/value

SCR-lightlike submanifolds have been explored within ambient manifolds possessing various structures, such as Kaehler, Sasakian and Kenmotsu structures. In this article, we investigate this structure on submanifolds of metallic semi-Riemannian manifolds. This original and authentic research will aid researchers in advancing the study of semi-Riemannian manifolds.

Keywords

Citation

Shanker, G., Yadav, A. and Kaur, R. (2024), "Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-11-2021-0286

Publisher

:

Emerald Publishing Limited

Copyright © 2024, Gauree Shanker, Ankit Yadav and Ramandeep Kaur

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) license. 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 license may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

The study of lightlike submanifolds in semi-Riemannian manifolds presents a significantly greater level of complexity compared to study of submanifolds of Riemannian manifolds. In case of semi-Riemannian manifolds, the induced metric on submanifold has two cases, either degenerate or non-degenerate. If metric is non-degenerate, various structures of the ambient space can be induced uniquely and with same characteristic to ambient space. But this is not true for the case of degenerate metric. Also, if the submanifold has a degenerate metric then the intersection of the normal bundle and the tangent bundle is non-trivial. Due to this, we cannot study the geometry of degenerate or lightlike submanifolds by using tools of non-degenerate submanifold. The whole study of how to do calculus on lightlike submanifolds is known as lightlike geometry. The lightlike geometry is known as mathematical language of general theory of relativity. Due to this reason, it becomes a developing research topic in mathematics and physics.

In 1996, Duggal and Bejancu accumulated research on lightlike submanifolds in Ref. [1]. Later, various category of lightlike submanifolds, on the basis of action of (1,1) tensor on tangent bundle, of semi-Riemannian manifolds are studied.

The root x=1+52 of the equation x2x − 1 = 0 is known as a golden ratio [4]. This ratio has many applications in paintings, pictures, temples, fractals etc. Crasmeranue and Hretcanu [2] introduced golden manifold by defining the tensor ϕ on Riemannian manifold, such that ϕ2 = ϕ + 1. The metallic means, the generalization of golden means, was introduced by Spinadel [3–5]. Later, on the basis of metallic structure, geometry of various lightlike submanifolds of golden and metallic semi-Riemannian manifold are introduced in [6–23] and [32–34]. In Ref. [24], the author introduced Cauchy–Riemann (CR)- submanifolds of Kaehler manifolds which contains invariant and totally-real submanifolds as its subcases. But it was observed that CR-lightlike submanifolds does not contain invariant and screen real lightlike submanifolds as its sub-cases. Therefore, a new class SCR-lighlike submanifolds has been introduced by Duggal and Sahin [25] which contains invariant lightlike submanifolds and screen-lightlike submanifolds as its sub-cases. Later, the concept of SCR-lightlike submanifolds has been studied in indefinite Sasakian manifolds with the name of contact SCR-lightlike submanifolds [26]. In Ref. [17], author introduced SCR-lightlike submanifolds and investigated the existence of warped product SCR-lightlike submanifolds of indefinite nearly Kahler. The metallic semi-Riemannian manifolds is an important class of semi-Riemannian manifolds, therefore, we introduce SCR-lightlike submanifolds of metallic semi-Riemannian manifolds which contains invariant lightlike submanifolds [18] and screen real lightlike submanifolds [27] of metallic semi-Riemannian manifolds as its special cases. The sections of this article are as follows:

In section 2, some basics of lightlike geometry and definition of metallic semi-Riemannian manifolds are recalled. In section 3, we introduce SCR-lightlike submanifold of metallic semi-Riemannian manifold. One example is also constructed. In section 4, we establish the necessary and sufficient conditions for the integrability of the distributions Rad(TN), D′ and D, and the foliations determined by the distribution Rad(TN), D′ and D to be totally geodesic, where Rad(TN), D′ and D represents radical distribution, invariant distribution and anti-invariant distribution of the tangent bundle TN. In section 5, the necessary and sufficient conditions for induced connection on totally umbilical SCR-lightlike submanifolds to be a metric connection are derived. Also, it is studied that the totally umbilical SCR-lightlike submanifolds with smooth vector fields Hs ∈ Γ(μ) ⊆Γ(S(TM)) cannot be the warped product lightlike submanifold of the type N×λNT.

2. Preliminaries

Suppose (Ñm+n,g̃) be a semi-Riemannian manifold with constant index q (1 ≤ q ≤ m + n − 1, m, n ≥ 1), then the submanifold (Nm, g) is known as degenerate(lightlike) submanifold if the induced metric g is degenerate [28].

If there exists a smooth distribution N(known as null distribution) such that Np=TNpTNp for every p ∈ N and with rank r then N is called the r-lightlike submanifold. The null distribution N is also denoted by Rad(TN). Also, on the basis of non-degenerate complementary subbundles S(TN)(screen distribution) and S(TN)(screen transversal distribution) of Rad(TN) in TN and TN respectively,

we have the following decomposition (for detail see Ref. [28])

(2.1)TÑ|N=TNtr(TN)=[Rad(TN)ltr(TN)]S(TN)S(TN)
where
(2.2)tr(TN)=ltr(TN)S(TN)
Theorem 2.1.

[28] Let (Ñ,g̃) be a semi-Riemannian manifold and (N, g, S(TN), S(TN)) be its r-lightlike submanifold. Then, for a coordinate neighborhood u of N, there exists a vector bundle ltr(TN) and a basis of Γ(ltr(TN)|u) contains a smooth section {Ni} of S(TN)|u such that, for any i, j ∈ {1, 2, …, r}, where {ξi}(1 ≤ i ≤ r,)

(2.3) g̃ij(Ni,ξj)=δij,g̃ij(Ni,Nj)=0,
is a lightlike basis of Γ(Rad(TN)).

For any X1, X2 ∈ Γ(TN) and W ∈ Γ(tr(TN)), the Gauss and Weingarten formula are

(2.4)̃X1X2=X1X2+h(X1,X2),
(2.5)̃X1W=AWX1+X1W,
where {X1X2,AUX1} and {h(X1,X2),X1W} belong to Γ(TN) and Γ(tr(TN)) respectively, and ∇ is induced connection on N. Further (2.4) and (2.5) reduce to
(2.6)̃X1X2=X1X2+hl(X1,X2)+hs(X1,X2),
(2.7)̃X1N=ANX1+X1l(N)+Ds(X1,N),NΓ(ltr(TN)),
(2.8)̃X1W=AWX1+X1s(W)+Dl(X1,W),WΓ(S(TN)).
For the lightlike submanifold N of Ñ, (2.4), (2.6) and (2.5), (2.7), (2.8) are known as Gauss equations and Weingarten equations respectively.

Also, we have the following equations [28]:

(2.9)g̃(hs(X1,X2),W)+g̃(X1,Dl(X2,W))=g(AWX1,X2),
(2.10)g̃(hl(X1,X2),ξ)+g̃(X1,hl(X2,ξ))=g(X1,X2ξ),
for any ξ ∈ Γ(Rad(TN)), X1, X2 ∈ Γ(TN) and W ∈ Γ(S(TN)).

Since the induced connection from the ambient connection is not always a Levi-Civita connection. Then, for any X1, X2, Z ∈ Γ(TN) and U, V ∈ Γ(tr(TN)), we have following formulas

(2.11)(X1g)(X2,Z)=g̃(hl(X1,X2),Z)+g̃(hl(X1,Z),X2)
Let us consider S a projection map on S(TN) from TN. Then, we have the following equations, for any X1, X2 ∈ Γ(TN) and ξ ∈ Γ(Rad(TN)):
(2.12)X1SY=X1*SX2+h*(X1,SX2)
(2.13)X1ξ=Aξ*X1+X1*t(ξ),
where {h*(X1,PY),X1*t(ξ)} and {X1*SY,Aξ*X1} belong to Γ(Rad(TN)) and Γ(S(TN))), respectively.

See pg. 196–198 [28], for detail explanation of equations (2.4)-(2.13).

Definition 2.1.

[18] A smooth semi-Riemannian manifold (Ñ,g̃) with (1,1) tensor field P̃ such that

(2.14) P̃2=pP̃+qI,
and g̃ is P̃-compatible, i.e.,
(2.15) g̃(P̃X1,X2)=g̃(X1,P̃X2)
is said to be a metallic semi-Riemannian manifold.

From (2.14) in (2.15), we get

(2.16)g̃(P̃X1,P̃X2)=pg̃(P̃X1,X2)+qg̃(X1,X2),
for any X1, X2 ∈ Γ(TN) [2,8].

Also, if (̃X1P̃)X2=0 then P̃ is called locally metallic structure. Throughout the paper, we assume that (̃X1P̃)X2=0.

3. SCR-lightlike submanifolds

Definition 3.1.

A screen CauchyRiemann (SCR)-lightlike submanifold of a metallic semi-Riemannian manifold is a lightlike submanifold (N, g, S(TN)) which satisfies the following conditions:

  • (i)

    P̃(Rad(TN))=Rad(TN),

  •      i.e., Rad(TN) is invariant with respect to P̃.

  • (ii)

    There exist non-null distribution D and D of S(TN) such that

  •      P̃(D)=D,P̃(D)S(TN) and DD = {0}, where D is orthogonal complementary to D in S(TN).

Here, N is proper if, D ≠ {0} and D ≠ {0}.

From above definition, ltr(TN) is invariant w.r.t P̃ and

TN=DD,
where D′ = Rad(TN) D.

Suppose μ be the orthogonal complementary part to P̃D in S(TN), then

tr(TN)=ltr(TN)μP̃(D)
Example 3.1.

Suppose (Ñ=R17,g̃,P̃) be a 7-dimensional metallic semi-Euclidean space with semi-Euclidean metric g̃ and sign (−+ + + + + −).

Let us define, for the standard coordinate system (y1, y2, y3, y4, y5, y6, y7) of R17,

P̃(y1,y2,y3,y4,y5,y6,y7)=(σy1,σy2,(pσ)y3,σy4,σy5,σy6,(pσ)y7).
Then, P̃ is a metallic structure.

Suppose N is a submanifold of Ñ such that

y1=y2=u1,y3=σqu2,
y4=u2,y5=sinσu3,
y6=cosσu3.
from above system, we have following tangent vectors:
Z1=y1+y2,Z2=σqy3+y4,
Z3=sinσy5+cosσy6.
thus,
Rad(TN)=span{Z1=ξ},S(TN)=span{Z2,Z3},
ltr(TN)=spanN=12y1y2,
W=qy3+σy4S(TN),
where P̃ξ=σξ, P̃(N)=σN, P̃(Z2)W and P̃(Z3)=σZ3. Here, D′ = D Rad(TN) = span{Z1, Z3} and D = span{Z2}. Hence, N is SCR-lightlike submanifold.

Let Q1, Q2 and R be the projection maps on Rad(TN), D and D, respectively.

For any X ∈ Γ(TN), we have

(3.1)X=Q1X+Q2X+RX
or
(3.2)X=QX+RX,
where Q is projection map on D′ such that QX = Q1X + Q2X.

Applying P̃ on (3.2), we obtain

(3.3)P̃X=P̃Q1X+P̃Q2X+P̃RX.
We write above equation as
(3.4)P̃X=TX+wX,
where TX and wX are the tangential and transversal components of P̃X, respectively.

For any U ∈ tr(TN), we assume

(3.5)P̃U=BU+CU,
where BU and CU are tangential and transversal components of P̃U, respectively.

For any X, Y ∈ Γ(TN), expanding the expression (̃XP̃)Y=0, by using (3.3)-(3.5), we obtain

(3.6)ATQ1Y*X+X*TQ1Y+hl(X,TQ1Y)+hl(X,TQ1Y)+XTQ2Y+hs(X,TQ2Y)+hs(X,TQ2Y)AwRYX+Dl(X,wRY)+XswRY=P̃Q1XY+P̃Q2XY+P̃RXY+Chl(X,Y)+Bhs(X,Y)+Chs(X,Y).
By comparing components of Rad(TN), D, D, ltr(TN), and S(TN), we get the following equations
(3.7)X*TQ1Y+Q1XTQ2Y=Q1AwRYX+TQ1XY,
(3.8)Q2XTQ2Y=Q2ATQ1Y*X+Q2AwRYX+TQ2XY,
(3.9)RXTQ2Y=RATQ1Y*X+RAwRYX+Bhs(X,Y),
(3.10)hl(X,TQ1Y)+hl(X,TQ2Y)+Dl(X,wRY)=Chl(X,Y),
and
(3.11)hs(X,TQ1Y)+hs(X,TQ2Y)+XswRY=wRXY+Chs(X,Y).
Lemma 3.1.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of a metallic semi-Riemannian manifold (Ñ,P̃). Then we have

(3.12) (XT)Y=AwYX+Bhs(X,Y)
and
(3.13) (Xtw)Y=h(X,TY)+Dl(X,wY)+Chl(X,Y)+Chs(X,Y),
where (∇XT)Y = ∇XTY − TXY and (Xtw)Y=XtYwXY.

Proof. For any X, Y ∈ Γ(TN), we have

̃XP̃Y=P̃̃XY.
Expanding above equation by using (3.4), (3.5) and Gauss and Weingarten formula, we obtain
̃X(TY+wY)=P̃(XY+hl(X,Y)+hs(X,Y)),
this reduces to
XTY+h(X,TY)AwYX+XsY+Dl(X,wY)=TXY+wXY+Chl(X,Y)+Bhs(X,Y)+Chs(X,Y)
By comparing tangential and transversal components in above equation, we obtain
(3.15) (XT)Y=AwYX+Bhs(X,Y)
and
(3.16) (Xtw)Y=h(X,TY)+Dl(X,wY)+Chl(X,Y)+Chs(X,Y).

4. Integrability and totally geodesic foliations

In this section, we find necessary and sufficient conditions for the distributions Rad(TN), D and D′ to be integrable and to define totally geodesic foliations.

Theorem 4.1.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of a metallic semi-Riemannian manifold (Ñ,P̃). Then,

  • (i)

    the distribution Rad(TN) is integrable if and only if,

  •      Q2ATQ1ξ2*ξ1=Q2ATQ1ξ1*ξ2 and hs(ξ1, TQ1ξ2) = hs(ξ2, TQ1ξ1).

  • (ii)

    the distribution Rad(TN) defines totally geodesic foliations if and only if,

phl(ξ1,Y)=hl(ξ1,TY)+Dl(ξ1,wY).

Proof. (i) Let ξ1, ξ2 ∈ Rad(TN), from (3.7), we obtain

(4.1) Q2ATQ1ξ1*ξ2=TQ2ξ1ξ2
Interchanging role of ξ1 and ξ2 in (4.1), we obtain
(4.2) Q2ATQ1ξ2*ξ1=TQ2ξ2ξ1
Subtracting (4.2) from (4.1), we obtain
(4.3) Q2ATQ1ξ2*ξ1Q2ATQ1ξ1*ξ2=TQ2[ξ1,ξ2].
Similarly, from (3.11), we obtain
(4.4) hs(ξ1,TQ1ξ2)=wRξ1ξ2+Chs(ξ1,ξ2)
Interchanging role of ξ1 and ξ2 in (4.4) and subtract resulting equation from (4.4), we obtain
(4.5) hs(ξ1,TQ1ξ2)hs(ξ2,TQ1ξ1)=wR[ξ1,ξ2]
From (4.3) and (4.5) we obtain, Rad(TN) is integrable if and only if,

Q2ATQ1ξ2*ξ1=Q2ATQ1ξ1*ξ2 and hs(ξ1, TQ1ξ2) = hs(ξ2, TQ1ξ1).

(ii) For any ξ1, ξ2 ∈ Rad(TN) and Y ∈ Γ(S(TN)), from (2.4), we have

(4.6) g̃(ξ1ξ2,Y)=g̃(̃ξ1ξ2,Y)
Since ̃ is metric connection, using (2.16) in above equation, we obtain
(4.7) g̃(ξ2,̃ξ1Y)=1qg̃(P̃ξ2,P̃̃ξ1Y)+pqg̃(P̃ξ2,̃ξ1Y).
Using P̃̃ξ1Y=̃ξ1P̃Y in (4.7), (4.7) reduces to
(4.8) g̃(ξ2,̃ξ1Y)=1qg̃(P̃ξ2,̃ξ1(TY+wY))+pqg̃(P̃ξ2,̃ξ1Y).
Using (2.6) and (2.8), we get □
(4.9) qg̃(ξ2,̃ξ1Y)=g̃(P̃ξ2,hl(ξ1,TY)+Dl(ξ1,wY)phl(ξ1,Y)).
Since Rad(TN) defines totally geodesic foliation if and only if ξ1ξ2Rad(TN), from (4.9) and (4.6), we obtain ξ1ξ2Rad(TN) if and only if
phl(ξ1,Y)=hl(ξ1,TY)+Dl(ξ1,wY).

Theorem 4.2.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of metallic semi-Riemannian manifold (Ñ,P̃). Then,

  • (i)

    the distribution Dis integrable if and only if,

hs(X,P̃Y)=hs(Y,P̃X)
  • (ii)

    the distribution Ddefines totally geodesic foliations if and only if,

hs(X,P̃Y)phs(X,Y)Γ(μ).

Proof.

  • (i)

    For any X, Y ∈ Γ(D′), from (3.11), we obtain

(4.10) hs(X,TQ1Y)+hs(X,TQ2Y)=wRXY+Chs(X,Y).
Interchanging role of X and Y in (4.10), we get
(4.11) hs(Y,TQ1X)+hs(Y,TQ2X)=wRYX+Chs(Y,X).
Subtracting (4.10) from (4.11), we obtain
(4.12) hs(X,TQ1Y)+hs(X,TQ2Y)hs(Y,TQ1X)hs(Y,TQ2X)=wR[X,Y]
Above equation can be written as
(4.13) hs(X,P̃Y)hs(Y,P̃X)=wR[X,Y].
From (4.12), D′ is integrable if and only if
hs(X,P̃Y)=hs(Y,P̃X)
  • (ii)

    For any X, Y ∈ Γ(D′), the distribution D′ defines totally geodesic foliation if and only if ∇XY ∈ Γ(D′).

 For any Z ∈ Γ(D), from (2.6), we have

g(XY,Z)=g(̃XY,Z).
Using (2.16) in above equation, we get
(4.14) qg(XY,Z)=qg̃(̃XY,Z)=g̃(P̃̃XY,P̃Z)pg̃(̃XY,P̃Z).
Using (2.6) in (4.14), we get
(4.15) qg(XY,Z)=g̃(hs(X,P̃Y),P̃Z)pg̃(hs(X,Y),P̃Z).
Above equation reduces to
(4.16) qg(XY,Z)=g̃(hs(X,P̃Y)phs(X,Y),P̃Z)
From (4.16), we obtain ∇XY ∈ Γ(TN) if and only if hs(X,P̃Y)phs(X,Y)Γ(μ).

Theorem 4.3.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of metallic semi-Riemannian manifold Ñ. Then,

  • (i)

    the distribution D is integrable if and only if,

TQAwRZ1Z2=TQAwRZ2Z1.
  • (ii)

    the distribution D defines totally geodesic foliation if and only if, BZ1sP̃Z2=AP̃Z2Z1. and TQ1AP̃Z2Z1=pQ1AP̃Z2Z1.

Proof.

(4.17) QAwRYZ1=TQZ1Z2,
Interchanging Z1 and Z2, we obtain
(4.18) QAwRXZ2=TQZ2Z1,
Subtracting (4.18) from (4.17), we obtain
(4.19) QAwRXZ2+QAwRYZ1=TQ[Z1,Z2]

From (4.19), we obtain D is integrable if and only if

QAwRZ1Z2=QAwRZ2Z1.
  • (ii)

    For any Z1, Z2 ∈ Γ(D) and X ∈ Γ(D), from (2.6), we have

(4.20) g(Z1Z2,X)=g̃(̃Z1Z2,X).
Using (2.16) in (4.20), we obtain
(4.21) qg(Z1Z2,X)=g̃(P̃̃Z1Z2,P̃X)pg̃(̃Z1Z2,P̃X).
Since ̃XP̃Y=P̃̃XY, above equation reduces to
(4.22) qg(Z1Z2,X)=g̃(̃Z1P̃Z2,P̃X)pg̃(̃Z1P̃Z2,X).
Expanding above equation using (2.6) and (2.8) in (4.22), we obtain
(4.23) qg(Z1Z2,X)=g̃(BZ1sP̃Z2,X)+pg̃(AP̃Z2Z1,X).
On the other hand, for any N ∈ ltr(TN), we have
(4.24) g̃(Z1Z2,N)=g̃(̃Z1Z2,N)=1qg̃(P̃̃Z1Z2,P̃N)pqg̃(P̃̃Z1Z2,N).
Using (2.16) in (4.24), we obtain
(4.25) g̃(Z1Z2,N)=1qg̃(P̃̃Z1Z2,P̃N)pqg̃(P̃̃Z1Z2,N).

Further, expanding (4.25) by using (2.8), we obtain

(4.26) g̃(Z1Z2,N)=1qg̃(AP̃Z2Z1,P̃N)+pqg̃(AP̃Z2Z1,N).

Since the distribution D defines totally geodesic foliation if and only if Z1Z2Γ(D), from (4.23) and (4.26), we get, Z1Z2Γ(D) if and only if BZ1sP̃Z2=AP̃Z2Z1. and TQ1AP̃Z2Z1=pQ1AP̃Z2Z1.

5. Totally umbilical SCR-lightlike submanifolds

Definition 5.1.

[1] A lightlike submanifolds (N, g, S(TN)) of a semi-Riemannian manifold (Ñ,g̃) is said to be totally umbilical in Ñ if there is smooth transversal vector field HΓ(tr(TÑ)) on Ñ called the transversal curvature vector field of Ñ, such that, for all X1, X2 ∈ Γ(TN)

(5.1) h(X1,X2)=Hg̃(X1,X2).

Alternatively, N is totally umbilical, if and only if on each coordinate neighborhood U there exist a smooth vector fields HlΓ(ltr(TN)) and HsΓ(S(TN)) such that

(5.2)hl(X1,X2)=Hlg̃(X1,X2),
(5.3)hs(X1,X2)=Hsg̃(X1,X2),
(5.4)Dl(X1,W)=0,
for any X1, X2 ∈ Γ(TN) and W ∈ S(TN).
Theorem 5.1.

Let (N, g, S(TN)) be a totally umbilical r-lightlike SCR submanifold of metallic semi-Riemannian manifold Ñ, then the induced connectionis metric connection if and only if, g̃(X,TQ2Y)=pg̃(X,Y).

Proof. The induced connection ∇ is metric connection if and only if, for any X ∈ Γ(TN) and ξ ∈ Γ(Rad(TN)), ∇Xξ ∈ Rad(TN).

Suppose N is totally umbilical lightlike submanifold and Y ∈ S(TN), from (2.16), we get

(5.5) g(Xξ,Y)=1qg(P̃̃Xξ,P̃Y)pqg(P̃̃Xξ,Y).
Since ̃ is metric connection, (5.5) reduces to
(5.6) g(Xξ,Y)=1qg̃(P̃ξ,̃X(TY+wY))+pqg̃(P̃ξ,̃XY).
Using (5.2) and (5.3) in (5.6), we get
(5.7) g(Xξ,Y)=g̃(Hl,P̃ξ)1qg̃(X,TQ2Y)+pqg̃(X,Y)1qg̃(ξ̃,Dl(X,wY)).
Since Dl(XP̃Q2Y)=0 and g̃(Hl,P̃ξ) is not identically zero, from (5.7) g(∇Xξ, Y) = 0 if and only if, g̃(X,TQ2Y)=pg̃(X,Y).

This implies, the induced connection ∇ is metric connection if and only if, g̃(X,TQ2Y)=pg̃(X,Y). □

Theorem 5.2.

Let N be a totally umbilical lightlike submanifold of metallic semi-Riemannian manifold. Then

  • (i)

    Rad(TN) is always integrable.

  • (ii)

    Rad(TN) always defines totally geodesic foliation.

Proof. (i) Let ξ1, ξ2 ∈ Rad(TN) and X ∈ Γ(S(TN)), from (2.16), we get

(5.8) g̃([ξ1,ξ2],X)=1qg̃(̃ξ1P̃ξ2,P̃X)pqg̃(̃ξ1ξ2,P̃X)1qg̃(̃ξ2P̃ξ1,P̃X)+pqg̃(̃ξ2ξ1,P̃X).
Above equation reduces to
(5.9) g̃([ξ1,ξ2],X)=1qg̃(P̃ξ2,̃ξ1P̃X)+pqg̃(ξ2,̃ξ1P̃X)+1qg̃(P̃ξ2,̃ξ1P̃X)pqg̃(ξ2,̃ξ1P̃X).
Using (2.6) and (2.8) in above equation, we get
(5.10) g̃([ξ1,ξ2],X)=1qg̃(P̃ξ2,Dl(ξ1,wX)+pqg̃(ξ2,Dl(ξ1,wX))+1qg̃(P̃ξ2,Dl(ξ1,wX)pqg̃(ξ2,Dl(ξ1,wX))1qg̃(P̃ξ2,hl(ξ1,TX)+pqg̃(ξ2,hl(ξ1,TX))+1qg̃(P̃ξ2,hl(ξ1,TX)pqg̃(ξ2,hl(ξ1,TX)).
Since N is totally umbilical, Dl(ξ, wX) = 0 and hl(ξ, wX) = 0. From (5.10), we obtain
(5.11) g̃([ξ1,ξ2],X)=0.
Therefore, [ξ1, ξ2] ∈ Rad(TN) implies Rad(TN) is integrable.

(ii) Let ξ1, ξ2 ∈ Rad(TN) and X ∈ Γ(S(TN)), from (2.16), we obtain

(5.12) g̃(ξ1ξ2,X)=1qg̃(̃ξ1P̃ξ2,TX)+1qg̃(̃ξ1P̃ξ2,wX)pqg̃(̃ξ1ξ2,TX)pqg̃(̃ξ1ξ2,TX),
this implies
(5.13) g̃(ξ1ξ2,X)=1qg̃(hl(ξ1,TX),P̃ξ2)+1qg̃(Dl(ξ1,wX),P̃ξ2)pqg̃(hl(ξ1,TX),ξ2)pqg̃(Dl(ξ1,wX),ξ2).
Since N is totally umbilical, h(X,P̃ξ1)=Hg̃(X,P̃ξ1)=0, h(X,P̃ξ1)=Hg̃(X,ξ1)=0 and Dl(ξ, wX) = 0. From (5.13), we get
g̃(ξ1ξ2,X)=0.
Therefore ξ1ξ2Rad(TN). □

Theorem 5.3.

Let N be a totally umbilical lightlike submanifold of metallic semi-Riemannian manifold. Then

  • (i)

    Dis integrable.

  • (ii)

    Ddefines totally geodesic foliation if and only if, Hs ∈ Γ(μ)).

Proof. (i) From Theorem 4.2 (i), D′ is integrable if and only if,

hs(X,P̃Y)=hs(P̃X,Y).
Since N is totally umbilical,
(5.14) hs(X,P̃Y)=Hsg̃(X,P̃Y)
and
(5.15) hs(P̃X,Y)=Hsg̃(P̃X,Y).
Since g̃(X,P̃Y)=g̃(P̃X,Y), from (5.14) and (5.15), we obtain
(5.16) hs(X,P̃Y)=hs(P̃X,Y)=Hsg̃(P̃X,Y)=hs(P̃X,Y).
Therefore, D′ is always integrable.

(ii) From Theorem 4.2 (ii), D′ defines totally geodesic foliation if and only if, hs(X,P̃Y)phs(X,Y)Γ(μ).

Since N is totally umbilical, D′ defines totally geodesic foliation if and only if,

Hs(g̃(X,P̃Y)pg̃(X,Y)Γ(μ)).
This implies, D′ defines totally geodesic foliation if and only if, Hs ∈ Γ(μ)). □

Theorem 5.4.

Let N be a totally umbilical lightlike submanifold of metallic semi-Riemannian manifold. Then, for any Z1 and Z2 ∈ D

  • (i)

    D is integrable if and only if,

TQAwRZ1Z2=TQAwRZ2Z1.
  • (ii)

    D defines totally geodesic foliation if and only, if, Hl = 0.

Proof. The Proof is same as Theorem 4.3. □

The notion of warped product was defined by Bishop and O′ Niell in Ref. [29] as a generalization of the notion of the Cartesian product. Further, various types of warped product submanifolds have been studied in Riemannian manifolds with certain metric structures. Later, the warped product lightlike submanifolds of a semi-Riemannian manifold have been studied by Sahin [22]. In the Theorem 5.6, we investigate the existence of warped product lightlike submanifolds of the type N×λNT in metallic semi-Riemannian manifold Ñ when Hs ∈ Γ(μ).

Definition 5.2.

[29] Let B × F be a product manifold of Riemannain manifolds B and F with Riemannian metric gB and gF, respectively, and π: B × FB and η: F × BF be projection maps. The manifold B × F equipped with Riemannian metric g such that

g=gB+λ2gF
that is, for any X ∈ TN at (p, q),
X=π*(X)+λ2η*(X)2,
is known as warped product N = B ×λF.

If λ (Known as warping function) is constant then the warped product manifold is trivial. For differentiable function λ on N, the gradient ∇λ is defined by

g(λ,Z)=Zλ,ZΓ(TN).
Theorem 5.5.

[29] Let N = B ×λF be a warped product manifolds. If X1, X2 ∈ T(B) and Y1, Y2 ∈ T(F), then

(5.17) X1X2T(B),
(5.18) X1Y1=Y1X1=X1λλY1,
(5.19) Y1Y2=g(Y1,Y2)λλ.

Theorem 5.6.

Let (N, g, S(TN)) be totally umbilical SCR-lightlike submanifold of metallic semi-Riemannian manifold Ñ with Hs ∈ Γ(μ) Then, there does not exist any non-trivial lightlike warped product of the type N×λNT

Proof. Let X ∈ Γ(DT) and Z ∈ Γ(D). Then, from (5.18), we obtain

(5.20) XZ=ZX=(Zlnλ)X
this implies
(5.21) g(XZ,X)=Zlnλg(X,X).
Using (2.16) in (5.21), we get
(5.22) Zlnλg(X,X)=Zlnλ{1qg(PX,PX)pqg(PX,X)}.
Above equation can be re-written as
(5.23) Zlnλg(X,X)=1qg(ZlnλPX,PX)pqg(ZlnλPX,X).
Again, using (5.18) in (5.23), we get
(5.24) Zlnλg(X,X)=1qg(PXZ,PX)pqg(PXZ,X).
Using (2.6) in equation in (5.24), we obtain
(5.25) Zlnλg(X,X)=1qg̃(̃PXZhl(PX,Z)hs(PX,Z),PX)
(5.26) pqg̃(̃PXZhl(PX,Z)hs(PX,Z),X)
Since N is totally umbilical, hlg(PX, Z) = Hlg(PX, Z) and hs(PX, Z) = Hsg(PX, Z). Therefore, equation (5.25) reduces to
(5.27) Zlnλg(X,X)=1qg̃(̃PXZ,PX)pqg̃(̃PXZ,X).
Since the ̃ is Levi-Civita connection, (5.27) can be re-written as
(5.28) Zlnλg(X,X)=1qg̃(̃PXPX,Z)+pqg̃(̃PXX,Z).
Above equation reduces to
(5.29) Zlnλg(X,X)=1qg(PXPX,Z)+pqg(PXX,Z).
From theorem 5.3 if Hs ∈ Γ(μ) then ∇PXX and ∇PXX belong to D′. Therefore, g(∇PXPX, Z) = 0 and g(∇PXX, Z) = 0. From (5.29), we get (Z ln λ)g(X, X) = 0.

For Y ∈ DD′, we have

(5.30) (Zlnλ)g(Y,Y)=0.
Since D is non-degenerate distribution, we get Z ln λ = 0 this leads to λ = 0 which is a contradiction.

Hence, there does not exist any non-trivial lightlike warped product of type N×λNT.□

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Further reading

30Bahadır O, Siddiqui AN, Gülbahar M, Alkhaldi AH. Main curvatures identities on lightlike hypersurfaces of statistical manifolds and their characterizations. Mathematics. 2022; 10(13): 2290. doi: 10.3390/math10132290.

31Chen BY. Differential geometry of warped product manifolds and submanifolds. Hackensack: World Scientific; 2017.

32Crasmareanu M, Hretcanu CE. Metallic structures on Riemannian manifolds. Revista de la union matematica argentina. 2013; 54: 15-27.

33Zaidi A, Shanker G, Yadav A. Conformal Anti-Invariant Riemannian Maps from or To Sasakian Manifolds. Lobachevskii Journal of Mathematics. 2023; 44(4): 1518-1527. doi: 10.1134/S1995080223040297.

34Shanker G, Yadav A. A study on the geometry of totally umbilical (TU) screen-transversal (ST) lightlike submanifolds of metallic semi-Riemannian manifolds. AIP Conference Proceedings. 2022; 2435(1). doi: 10.1063/5.0085626.

Acknowledgements

The first author is thankful to DST Government of India for providing financial support in terms of DST-FST label-I grant vide sanction number SR/FST/MS-I/2021/104(C).

Corresponding author

Ankit Yadav can be contacted at: ankityadav93156@gmail.com

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