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On CR-lightlike submanifolds in a golden semi-Riemannian manifold

  • CR-lightlike submanifolds of a golden semi-Riemannian manifold are the focus of the research presented in this work, which aims to define and investigate these structures. Under the context of a golden semi-Riemannian manifold, we study the properties of geodesic CR-lightlike submanifolds as well as umbilical CR-lightlike submanifolds. In addition, on a golden semi-Riemannian manifold, we find numerous intriguing findings for entirely geodesic and totally umbilical CR-lightlike subamnifolds. Also, we construct an example of a CR-lightlike submanifold of a golden semi-Riemannian manifold.

    Citation: Mohammad Aamir Qayyoom, Rawan Bossly, Mobin Ahmad. On CR-lightlike submanifolds in a golden semi-Riemannian manifold[J]. AIMS Mathematics, 2024, 9(5): 13043-13057. doi: 10.3934/math.2024636

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  • CR-lightlike submanifolds of a golden semi-Riemannian manifold are the focus of the research presented in this work, which aims to define and investigate these structures. Under the context of a golden semi-Riemannian manifold, we study the properties of geodesic CR-lightlike submanifolds as well as umbilical CR-lightlike submanifolds. In addition, on a golden semi-Riemannian manifold, we find numerous intriguing findings for entirely geodesic and totally umbilical CR-lightlike subamnifolds. Also, we construct an example of a CR-lightlike submanifold of a golden semi-Riemannian manifold.



    The concept of lightlike submanifolds in geometry was initially established and expounded upon in a work produced by Duggal and Bejancu [1]. A nondegenerate screen distribution was employed in order to produce a nonintersecting lightlike transversal vector bundle of the tangent bundle. They defined the CR-lightlike submanifold as a generalization of lightlike real hypersurfaces of indefinite Kaehler manifolds and showed that CR-lightlike submanifolds do not contain invariant and totally real lightlike submanifolds. Further, they defined and studied GCR-lightlike submanifolds of Kaehler manifolds as an umbrella of invariant submanifolds, screen real submanifolds, and CR-lightlike and SCR-lightlike submanifolds in [2,3], respectively. Subsequently, B. Sahin and R. Gunes investigated geodesic property of CR-lightlike submanifolds [4] and the integrability of distributions in CR-lightlike submanifolds [5]. In the year 2010, Duggal and Sahin published a book [6]pertaining to the field of differential geometry, specifically focusing on the study of lightlike submanifolds. This book provides a comprehensive examination of recent advancements in lightlike geometry, encompassing novel geometric findings, accompanied by rigorous proofs, and exploring their practical implications in the field of mathematical physics. The investigation of the geometric properties of lightlike hypersurfaces and lightlike submanifolds has been the subject of research in several studies (see [7,8,9,10,11,12,13,14]).

    Crasmareanu and Hretcanu[15] created a special example of polynomial structure [16] on a differentiable manifold, and it is known as the golden structure (¯M,g). Hretcanu C. E. [17] explored Riemannian submanifolds with the golden structure. M. Ahmad and M. A. Qayyoom studied geometrical properties of Riemannian submanifolds with golden structure [18,19,20,21] and metallic structure [22,23]. The integrability of golden structures was examined by A. Gizer et al. [24]. Lightlike hypersurfaces of a golden semi-Riemannian manifold was investigated by N. Poyraz and E. Yasar [25]. The golden structure was also explored in the studies [26,27,28,29].

    In this research, we investigate the CR-lightlike submanifolds of a golden semi-Riemannian manifold, drawing inspiration from the aforementioned studies. This paper has the following outlines: Some preliminaries of CR-lightlike submanifolds are defined in Section 2. We establish a number of properties of CR-lightlike submanifolds on golden semi-Riemannian manifolds in Section 3. In Section 4, we look into several CR-lightlike submanifolds characteristics that are totally umbilical. We provide a complex illustration of CR-lightlike submanifolds of a golden semi-Riemannian manifold in the final section.

    Assume that (¯,g) is a semi-Riemannian manifold with (k+j)-dimension, k,j1, and g as a semi-Riemannian metric on ¯. We suppose that ¯ is not a Riemannian manifold and the symbol q stands for the constant index of g.

    [15] Let ¯ be endowed with a tensor field ψ of type (1,1) such that

    ψ2=ψ+I, (2.1)

    where I represents the identity transformation on Γ(Υ¯). The structure ψ is referred to as a golden structure. A metric g is considered ψ-compatible if

    g(ψγ,ζ)=g(γ,ψζ) (2.2)

    for all γ, ζ vector fields on Γ(Υ¯), then (¯,g,ψ) is called a golden Riemannian manifold. If we substitute ψγ into γ in (2.2), then from (2.1) we have

    g(ψγ,ψζ)=g(ψγ,ζ)+g(γ,ζ). (2.3)

    for any γ,ζΓ(Υ¯).

    If (¯,g,ψ) is a golden Riemannian manifold and ψ is parallel with regard to the Levi-Civita connection ¯ on ¯:

    ¯ψ=0, (2.4)

    then (¯,g,ψ) is referred to as a semi-Riemannian manifold with locally golden properties.

    The golden structure is the particular case of metallic structure [22,23] with p=1, q=1 defined by

    ψ2=pψ+qI,

    where p and q are positive integers.

    [1] Consider the case where is a lightlike submanifold of k of ¯. There is the radical distribution, or Rad(Υ), on that applies to this situation such that Rad(Υ)=ΥΥ, p. Since RadΥ has rank r0, is referred to as an r-lightlike submanifold of ¯. Assume that is a submanifold of that is r-lightlike. A screen distribution is what we refer to as the complementary distribution of a Rad distribution on Υ, then

    Υ=RadΥS(Υ).

    As S(Υ) is a nondegenerate vector sub-bundle of Υ¯|, we have

    Υ¯|=S(Υ)S(Υ),

    where S(Υ) consists of the orthogonal vector sub-bundle that is complementary to S(Υ) in Υ¯|. S(Υ),S(Υ) is an orthogonal direct decomposition, and they are nondegenerate.

    S(Υ)=S(Υ)S(Υ).

    Let the vector bundle

    tr(Υ)=ltr(Υ)S(Υ).

    Thus,

    Υ¯=Υtr(Υ)=S(Υ)S(Υ)(Rad(Υ)ltr(Υ).

    Assume that the Levi-Civita connection is ¯ on ¯. We have

    ¯γζ=γζ+h(γ,ζ),γ,ζΓ(Υ) (2.5)

    and

    ¯γζ=Ahζ+γh,γΓ(Υ)andhΓ(tr(Υ)), (2.6)

    where {γζ,Ahγ} and {h(γ,ζ),γh} belongs to Γ(Υ) and Γ(tr(Υ)), respectively.

    Using projection L:tr(Υ)ltr(Υ), and S:tr(Υ)S(Υ), we have

    ¯γζ=γζ+hl(γ,ζ)+hs(γ,ζ), (2.7)
    ¯γ=Aγ+lγ+λs(γ,), (2.8)

    and

    ¯γχ=Aχγ+sγ+λl(γ,χ) (2.9)

    for any γ,ζΓ(Υ),Γ(ltr(Υ)), and χΓ(S(Υ)), where hl(γ,ζ)=Lh(γ,ζ),hs(γ,ζ)=Sh(γ,ζ),lγ,λl(γ,χ)Γ(ltr(T)),sγλs(γ,)Γ(S(Υ)), and γζ,Aγ,AχγΓ(Υ).

    The projection morphism of Υ on the screen is represented by P, and we take the distribution into consideration.

    γPζ=γPζ+h(γ,Pζ),γξ=Aξγ+tγξ, (2.10)

    where γ,ζΓ(Υ),ξΓ(Rad(Υ)).

    Thus, we have the subsequent equation.

    g(h(γ,Pζ),)=g(Aγ,Pζ), (2.11)

    Consider that ¯ is a metric connection. We get

    (γg)(ζ,η)=g(hl(γ,ζ),η)+g(hl(γ,ζη),ζ). (2.12)

    Using the characteristics of a linear connection, we can obtain

    (γhl)(ζ,η)=lγ(hl(ζ,η))hl(¯γζ,η)hl(ζ,¯γη), (2.13)
    (γhs)(ζ,η)=sγ(hs(ζ,η))hs(¯γζ,η)hs(ζ,¯γη). (2.14)

    Based on the description of a CR-lightlike submanifold in [4], we have

    Υ=λλ,

    where λ=Rad(Υ)ψRad(Υ)λ0.

    S and Q stand for the projection on λ and λ, respectively, then

    ψγ=fγ+wγ

    for γ,ζΓ(Υ), where fγ=ψSγ and wγ=ψQγ.

    On the other hand, we have

    ψζ=Bζ+Cζ

    for any ζΓ(tr(Υ)), BζΓ(Υ) and CζΓ(tr(Υ)), unless 1 and 2 are denoted as ψL1 and ψL2, respectively.

    Lemma 2.1. Assume that the screen distribution is totally geodesic and that is a CR-lightlike submanifold of the golden semi-Riemannian manifold, then γζΓ(S(ΥN)), where γ,ζΓ(S(Υ)).

    Proof. For γ,ζΓ(S(Υ)),

    g(γζ,)=g(¯γζh(γ,ζ),)=g(ζ,¯γ).

    Using (2.8),

    g(γζ,)=g(ζ,Aγ+γ)=g(ζ,Aγ).

    Using (2.11),

    g(γζ,)=g(h(γ,ζ),).

    Since screen distribution is totally geodesic, h(γ,ζ)=0,

    g(¯γζ,)=0.

    Using Lemma 1.2 in [1] p.g. 142, we have

    γζΓ(S(Υ)),

    where γ,ζΓ(S(Υ)).

    Theorem 2.2. Assume that is a locally golden semi-Riemannian manifold ¯ with CR-lightlike properties, then γψγ=ψγγ for γΓ(λ0).

    Proof. Assume that γ,ζΓ(λ0). Using (2.5), we have

    g(γψγ,ζ)=g(¯γψγh(γ,ψγ),ζ)g(γψγ,ζ)=g(ψ(¯γγ),ζ)g(γψγ,ζ)=g(ψ(γγ),ζ),g(γψγψ(γγ),ζ)=0.

    Nondegeneracy of λ0 implies

    γψγ=ψ(γγ),

    where γΓ(λ0).

    Definition 3.1. [4] A CR-lightlike submanifold of a golden semi-Riemannian manifold is mixed geodesic if h satisfies

    h(γ,α)=0,

    where h stands for second fundamental form, γΓ(λ), and αΓ(λ).

    For γ,ζΓ(λ) and α,βΓ(λ) if

    h(γ,ζ)=0

    and

    h(α,β)=0,

    then it is known as λ-geodesic and λ-geodesic, respectively.

    Theorem 3.2. Assume is a CR-lightlike submanifold of ¯, which is a golden semi-Riemannian manifold. is totally geodesic if

    (Lg)(γ,ζ)=0

    and

    (Lχg)(γ,ζ)=0

    for α,βΓ(Υ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. Since is totally geodesic, then

    h(γ,ζ)=0

    for γ,ζΓ(Υ).

    We know that h(γ,ζ)=0 if

    g(h(γ,ζ),ξ)=0

    and

    g(h(γ,ζ),χ)=0.
    g(h(γ,ζ),ξ)=g(¯γζγζ,ξ)=g(ζ,[γ,ξ]+¯ξγ=g(ζ,[γ,ξ])+g(γ,[ξ,ζ])+g(¯ζξ,γ)=(Lξg)(γ,ζ)+g(¯ζξ,γ)=(Lξg)(γ,ζ)g(ξ,h(γ,ζ)))2g(h(γ,ζ)=(Lξg)(γ,ζ).

    Since g(h(γ,ζ),ξ)=0, we have

    (Lξg)(γ,ζ)=0.

    Similarly,

     g(h(γ,ζ),χ)=g(¯γζγζ,χ)=g(ζ,[γ,χ])+g(γ,[χ,ζ])+g(¯ζχ,γ)=(Lχg)(γ,ζ)+g(¯ζχ,γ)2g(h(γ,ζ),χ)=(Lχg)(γ,ζ).

    Since g(h(γ,ζ),χ)=0, we get

    (Lχg)(γ,ζ)=0

    for χΓ(S(Υ)).

    Lemma 3.3. Assume that ¯ is a golden semi-Riemannian manifold whose submanifold is CR-lightlike, then

    g(h(γ,ζ),χ)=g(Aχγ,ζ)

    for γΓ(λ),ζΓ(λ) and χΓ(S(Υ)).

    Proof. Using (2.5), we get

    g(h(γ,ζ),χ)=g(¯γζγζ,χ)=g(ζ,¯γχ).

    From (2.9), it follows that

    g(h(γ,ζ),χ)=g(ζ,Aχγ+sγχ+λs(γ,χ))=g(ζ,Aχγ)g(ζ,sγχ)g(ζ,λs(γ,χ))g(h(γ,ζ),χ)=g(ζ,Aχγ),

    where γΓ(λ),ζΓ(λ),χΓ(S(Υ)).

    Theorem 3.4. Assume that is a CR-lightlike submanifold of the golden semi-Riemannian manifold and ¯ is mixed geodesic if

    AξγΓ(λ0ψL1)

    and

    AχγΓ(λ0Rad(Υ)ψL1)

    for γΓ(λ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. For γΓ(λ),ζΓ(λ), and χΓ(S(Υ)), we get

    Using (2.5),

    g(h(γ,ζ),ξ)=g(¯γζγζ,ξ)=g(ζ,¯γξ).

    Again using (2.5), we obtain

    g(h(γ,ζ),ξ)=g(ζ,γξ+h(γ,ξ))=g(ζ,γξ).

    Using (2.10), we have

    g(h(γ,ζ),ξ)=g(ζ,Aξγ+tγξ)g(ζ,Aξγ)=0.

    Since the CR-lightlike submanifold is mixed geodesic, we have

    g(h(γ,ζ),ξ)=0
    g(ζ,Aξγ)=0
    AξγΓ(λ0ψL1),

    where γΓ(λ),ζΓ(λ).

    From (2.5), we get

    g(h(γ,ζ),χ)=g(¯γζγζ,χ)=g(ζ,¯γχ).

    From (2.9), we get

    g(h(γ,ζ),χ)=g(ζ,Aχγ+sγχ+λl(γ,χ))g(h(γ,ζ),χ)=g(ζ,Aχγ).

    Since, is mixed geodesic, then g(h(γ,ζ),χ)=0

    g(ζ,Aχγ)=0.
    AχγΓ(λ0Rad(Υ)ψ1).

    Theorem 3.5. Suppose that is a CR-lightlike submanifold of a golden semi-Riemannian manifold ¯, then is λ-geodesic if Aχη and Aξη have no component in 2ψRad(Υ) for ηΓ(λ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. From (2.5), we obtain

    g(h(η,β),χ)=g(¯ηβγζ,χ)=¯g(γζ,χ),

    where χ,βΓ(λ).

    Using (2.9), we have

    g(h(η,β),χ)=g(β,Aχη+sη+λl(η,χ))g(h(η,β),χ)=g(β,Aχη). (3.1)

    Since is λ-geodesic, then g(h(η,β),χ)=0.

    From (3.1), we get

    g(β,Aχη)=0.

    Now,

    g(h(η,β),ξ)=g(¯ηβηβ,ξ)=g(¯ηβ,ξ)=g(β,¯ηξ).

    From (2.10), we get

    g(h(η,β),ξ)=g(η,Aξη+tηξ)g(h(η,β),ξ)=g(Aξβ,η).

    Since is λ- geodesic, then

    g(h(η,β),ξ)=0
    g(Aξβ,η)=0.

    Thus, Aχη and Aξη have no component in M2ψRad(Υ).

    Lemma 3.6. Assume that ¯ is a golden semi-Riemannian manifold that has a CR-lightlike submanifold . Due to the distribution's integrability, the following criteria hold.

    (ⅰ) ψg(λl(ψγ,χ),ζ)g(λl(γ,χ),ψζ)=g(Aχψγ,ζ)g(Aχγ,ψζ),

    (ⅱ) g(λl(ψγ),ξ)=g(Aχγ,ψξ),

    (ⅲ) g(λl(γ,χ),ξ)=g(Aχψγ,ψξ)g(Aχγ,ψξ),

    where γ,ζΓ(Υ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. From Eq (2.9), we obtain

    g(λl(ψγ,χ),ζ)=g(¯ψγχ+Aχψγsψγχ,ζ)=g(χ,¯ψγζ)+g(Aχψγ,ζ).

    Using (2.5), we get

    g(λl(ψγ,χ),ζ)=g(χ,ψγζ+h(ψγ,ζ))+g(Aχψγ,ζ)=g(χ,h(γ,ψζ))+g(Aχψγ,ζ).

    Again, using (2.5), we get

    g(λl(ψγ,χ),ζ)=g(χ,¯γψζγψζ)+g(Aχψγ,ζ)=g(¯γχ,ψζ)+g(Aχψγ,ζ).

    Using (2.9), we have

    g(λl(ψγ,χ),ζ)=g(Aχγ+sγχ+λl(γ,χ),ψζ)+g(λl(ψγ,χ),ζ)g(λl(γ,χ),ψζ)=g(Aχψγ,ζ)g(Aχγ,ψζ).

    (ⅱ) Using (2.9), we have

    g(λl(ψγ,χ),ξ)=g(Aχψγsψγχ+ψγχ,ξ)=g(Aχψγ,ξ)g(χ,¯ψγξ).

    Using (2.10), we get

    g(λl(ψγ,χ),ξ)=g(Aχψγ,ξ)+g(χ,Aξψγ)g(χ,tψγ,ξ)g(λl(ψγ),ξ)=g(Aχγ,ψξ).

    (ⅲ) Replacing ζ by ψξ in (ⅰ), we have

    ψg(λl(ψγ,χ),ψξ)g(λl(γ,χ),ψ2ξ)=g(Aχψγ,ψξ)g(Aχγ,ψ2ξ).

    Using Definition 2.1 in [18] p.g. 9, we get

    ψg(λl(ψγ,χ),ψξ)g(λl(γ,χ),(ψ+I)ξ)=g(Aχψγ,ψξ)g(Aχγ,(ψ+I)ξ)ψg(λl(ψγ,χ),ψξ)g(λl(γ,χ),ψξ)g(λl(γ,χ),ξ)=g(Aχψγ,ψξ)g(Aχγ,ψξ)g(Aχγ,ξ).g(λl(γ,χ),ξ)=g(Aχψγ,ψξ)g(Aχγ,ψξ).

    Definition 4.1. [12] A CR-lightlike submanifold of a golden semi-Riemannian manifold is totally umbilical if there is a smooth transversal vector field Htr Γ(Υ) that satisfies

    h(χ,η)=Hg(χ,η),

    where h is stands for second fundamental form and χ, η Γ(Υ).

    Theorem 4.2. Assume that the screen distribution is totally geodesic and that is a totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold ¯, then

    Aψηχ=Aψχη,χ,ηΓλ.

    Proof. Given that ¯ is a golden semi-Riemannian manifold,

    ψ¯ηχ=¯ηψχ.

    Using (2.5) and (2.6), we have

    ψ(ηχ)+ψ(h(η,χ))=Aψχη+tηψχ. (4.1)

    Interchanging η and χ, we obtain

    ψ(χη)+ψ(h(χ,η))=Aψηχ+tχψη. (4.2)

    Subtracting Eqs (4.1) and (4.2), we get

    ψ(ηχχη)tηψχ+tχψη=AψηχAψχη. (4.3)

    Taking the inner product with γΓ(λ0) in (4.3), we have

    g(ψ(χη,γ)g(ψ(χη,γ)=g(Aψηχ,γ)g(Aψχη,γ).g(AψηχAψχη,γ)=g(χη,ψγ)g(χη,ψγ). (4.4)

    Now,

    g(χη,ψγ)=g(¯χηh(χ,η),ψγ)g(χη,ψγ)=g(η,(¯χψ)γψ(¯χγ)).

    Since ψ is parallel to ¯, i.e., ¯γψ=0,

    g(χη,ψγ)=ψ(¯χγ)).

    Using (2.7), we have

    g(χη,ψγ)=g(ψη,χγ+hs(χ,γ)+hl(χ,γ))g(χη,ψγ)=g(ψη,χγ)g(ψη,hs(χ,γ))g(ψη,hl(χ,γ)). (4.5)

    Since is a totally umbilical CR-lightlike submanifold and the screen distribution is totally geodesic,

    hs(χ,γ)=Hsg(χ,γ)=0

    and

    hl(χ,γ)=Hlg(χ,γ)=0,

    where χΓ(λ) and γΓ(λ0).

    From (4.5), we have

    g(χη,ψγ)=g(ψη,χγ).

    From Lemma 2.1, we get

    g(χη,ψγ)=0.

    Similarly,

    g(ηχ,ψγ)=0

    Using (4.4), we have

    g(AψηχAψχη,γ)=0.

    Since λ0 is nondegenerate,

    AψηχAψχη=0
    Aψηχ=Aψχη.

    Theorem 4.3. Let be the totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold ¯. Consequently, 's sectional curvature, which is CR-lightlike, vanishes, resulting in ¯K(π)=0, for the entire CR-lightlike section π.

    Proof. We know that is a totally umbilical CR-lightlike submanifold of ¯, then from (2.13) and (2.14),

    (γhl)(ζ,ω)=g(ζ,ω)lγHlHl{(γg)(ζ,ω)}, (4.6)
    (γhs)(ζ,ω)=g(ζ,ω)sγHsHs{(γg)(ζ,ω)} (4.7)

    for a CR-lightlike section π=γω,γΓ(λ0),ωΓ(λ).

    From (2.12), we have (Ug)(ζ,ω)=0. Therefore, from (4.6) and (4.7), we get

    (γhl)(ζ,ω)=g(ζ,ω)lγHl, (4.8)
    (γhs)(ζ,ω)=g(ζ,ω)sγHs. (4.9)

    Now, from (4.8) and (4.9), we get

    {¯R(γ,ζ)ω}tr=g(ζ,ω)lγHlg(γ,ω)lζHl+g(ζ,ω)λl(γ,Hs)g(γ,ω)λl(ζ,Hs)+g(ζ,ω)sγHsg(γ,ω)sζHs+g(ζ,ω)λs(γ,Hl)g(γ,ω)λs(ζ,Hl). (4.10)

    For any βΓ(tr(Υ)), from Equation (4.10), we get

    ¯R(γ,ζ,ω,β)=g(ζ,ω)g(lγHl,β)g(γ,ω)g(lζHl,β)+g(ζ,ω)g(λl(γ,Hs),ζ)g(γ,ω)g(λl(ζ,Hs),β)+g(ζ,ω)g(sγHs,β)g(γ,ω)g(sζHs,β)+g(ζ,ω)g(λs(γ,Hl),β)g(γ,ω)g(λs(ζ,Hl,β).
    R(γ,ω,ψγ,ψω)=g(ω,ψγ)g(lγHl,ψω)g(γ,ψγ)g(lωHl,ψω)+g(ω,ψγ)g(λl(γ,Hs),ψω)g(γ,ψγ)g(λl(ω,Hs),ψω)+g(ω,ψγ)g(sγHs,ψω)g(γ,ψγ)g(sωHs,ψω)+g(ω,ψγ)g(λs(γ,Hl),ψω)g(γ,ψγ)g(λs(ω,Hl,ψU).

    For any unit vectors γΓ(λ) and ωΓ(λ), we have

    ¯R(γ,ω,ψγ,ψω)=¯R(γ,ω,γ,ω)=0.

    We have

    K(γ)=KN(γζ)=g(¯R(γ,ζ)ζ,γ),

    where

    ¯R(γ,ω,γ,ω)=g(¯R(γ,ω)γ,ω)

    or

    ¯R(γ,ω,ψγ,ψω)=g(¯R(γ,ω)ψγ,ψω)

    i.e.,

    ¯K(π)=0

    for all CR-sections π.

    Example 5.1. We consider a semi-Riemannian manifold R62 and a submanifold of co-dimension 2 in R62, given by equations

    υ5=υ1cosαυ2sinαυ3z4tanα,
    υ6=υ1sinαυ2cosαυ3υ4,

    where αR{π2+kπ; kz}. The structure on R62 is defined by

    ψ(υ1,υ2,υ3,υ4,υ5,υ6)=(¯ϕ υ1,¯ϕυ2,ϕυ3,ϕυ4,ϕυ5,ϕυ6).

    Now,

    ψ2(υ1,υ2,υ3,υ4,υ5,υ6)=((¯ϕ+1) υ1,(¯ϕ+1)υ2,(ϕ+1)υ3,(ϕ+1)υ4,
    (ϕ+1)υ5,(ϕ+1)υ6)
    ψ2=ψ+I.

    It follows that (R62,ψ) is a golden semi-Reimannian manifold.

    The tangent bundle Υ is spanned by

    Z0=sinα υ5cosα υ6ϕ υ2,
    Z1=ϕ sinα υ5ϕ cosα υ6+ υ2,
    Z2=υ5¯ϕ sinα υ2+υ1,
    Z3=¯ϕ cosα υ2+υ4+iυ6.

    Thus, is a 1-lightlike submanifold of R62 with RadΥ=Span{X0}. Using golden structure of R62, we obtain that X1=ψ(X0). Thus, ψ(RadΥ) is a distribution on . Hence, the is a CR-lightlike submanifold.

    In general relativity, particularly in the context of the black hole theory, lightlike geometry finds its uses. An investigation is made into the geometry of the golden semi-Riemannian manifolds that are CR-lightlike in nature. There are many intriguing findings on completely umbilical and completely geodesic CR-lightlike submanifolds that are examined. We present a required condition for a CR-lightlike submanifold to be completely geodesic. Moreover, it is demonstrated that the sectional curvature K of an entirely umbilical CR-lightlike submanifold of a golden semi-Riemannian manifold ¯ disappears.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The present work (manuscript number IU/R&D/2022-MCN0001708) received financial assistance from Integral University in Lucknow, India as a part of the seed money project IUL/IIRC/SMP/2021/010. All of the authors would like to express their gratitude to the university for this support. The authors are highly grateful to editors and referees for their valuable comments and suggestions for improving the paper. The present manuscript represents the corrected version of preprint 10.48550/arXiv.2210.10445. The revised version incorporates the identities of all those who have made contributions, taking into account their respective skills and understanding.

    Authors have no conflict of interests.



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