The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

Ahmad Alalyani; M. A. Abdou; M. Basseem; Ahmad Alalyani; M. A. Abdou; M. Basseem

doi:10.3934/math.2024937

AIMS Mathematics

2024, Volume 9, Issue 7: 19240-19260. doi: 10.3934/math.2024937

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Research article Special Issues

The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

1.
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65526, Saudi Arabia
2.
Department of Mathematics, Faculty of Education, Alexandria University, Egypt
3.
Department of Mathematics, Faculty of Engineering, Sinai University, Egypt

Received: 04 March 2024 Revised: 20 May 2024 Accepted: 27 May 2024 Published: 11 June 2024
MSC : 65R20, 45B05, 65H10

The orthogonal polynomials approach with Gegenbauer polynomials is an effective tool for analyzing mixed integral equations (MIEs) due to their orthogonality qualities. This article reviewed recent breakthroughs in the use of Gegenbauer polynomials to solve mixed integral problems. Previous authors studied the problem with a continuous kernel that combined both Volterra (V) and Fredholm (F) components; however, in this paper, we focused on a singular Carleman kernel. The kernel of FI was measured with respect to position in the space ${L}_{2}[-\mathrm{1, 1}],$ while the kernel of Ⅵ was considered as a function of time in the space $C[0, T], T < 1$ . The existence of a unique solution was discussed in ${L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right]$ space. The solution and its error stability were both investigated and commented on. Finally, numerical examples were reviewed, and their estimated errors were assessed using Maple (2022) software.

Keywords:

Citation: Ahmad Alalyani, M. A. Abdou, M. Basseem. The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel[J]. AIMS Mathematics, 2024, 9(7): 19240-19260. doi: 10.3934/math.2024937

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Abstract

1. Introduction

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 $(\overline{M}, g)$ . 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.

2. Opening statements

Assume that $(\overline{\aleph}, g)$ is a semi-Riemannian manifold with $(k + j)$ -dimension, $k, j \ge 1$ , and $g$ as a semi-Riemannian metric on $\overline{\aleph}.$ We suppose that $\overline{\aleph}$ is not a Riemannian manifold and the symbol $q$ stands for the constant index of $g$ .

^[15] Let $\overline{\aleph}$ be endowed with a tensor field $\psi$ of type $(1, 1)$ such that

$\begin{equation} \psi^{2} = \psi+I, \end{equation}$

(2.1)

where $I$ represents the identity transformation on $\Gamma (\Upsilon \overline{\aleph})$ . The structure $\psi$ is referred to as a golden structure. A metric $g$ is considered $\psi$ -compatible if

$\begin{equation} g(\psi \gamma, \zeta) = g(\gamma, \psi \zeta) \end{equation}$

(2.2)

for all $\gamma$ , $\zeta$ vector fields on $\Gamma (\Upsilon \overline{\aleph})$ , then $(\overline{\aleph}, g, \psi)$ is called a golden Riemannian manifold. If we substitute $\psi \gamma$ into $\gamma$ in (2.2), then from (2.1) we have

$\begin{equation} g(\psi \gamma, \psi \zeta) = g(\psi \gamma, \zeta)+g(\gamma, \zeta). \end{equation}$

(2.3)

for any $\gamma, \zeta \in \Gamma (\Upsilon \overline{\aleph}).$

If $(\overline{\aleph}, g, \psi)$ is a golden Riemannian manifold and $\psi$ is parallel with regard to the Levi-Civita connection $\overline{\nabla}$ on $\overline{\aleph}:$

$\begin{equation} \overline{\nabla}\psi = 0, \end{equation}$

(2.4)

then $(\overline{\aleph}, g, \psi)$ 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

$\psi^{2} = p\psi+ qI,$

where $p$ and $q$ are positive integers.

^[1] Consider the case where $\aleph$ is a lightlike submanifold of $k$ of $\overline{\aleph}.$ There is the radical distribution, or $Rad(\Upsilon\aleph)$ , on $\aleph$ that applies to this situation such that $Rad(\Upsilon\aleph) = \Upsilon\aleph \cap \Upsilon\aleph^{\bot},$ $\forall$ $p \in \aleph.$ Since $Rad\Upsilon\aleph$ has rank $r \ge 0,$ $\aleph$ is referred to as an $r$ -lightlike submanifold of $\overline{\aleph}.$ Assume that $\aleph$ is a submanifold of $\aleph$ that is $r$ -lightlike. A screen distribution is what we refer to as the complementary distribution of a Rad distribution on $\Upsilon \aleph$ , then

$\begin{equation} \label{1e1} \nonumber \Upsilon \aleph = Rad\Upsilon\aleph \bot S(\Upsilon\aleph). \end{equation}$

As $S(\Upsilon\aleph)$ is a nondegenerate vector sub-bundle of $\Upsilon\overline{\aleph}|_{\aleph},$ we have

$\begin{equation} \label{1e2} \nonumber \Upsilon\overline{\aleph}|_{\aleph} = S(\Upsilon\aleph) \bot S(\Upsilon\aleph)^{\bot}, \end{equation}$

where $S(\Upsilon\aleph)^{\bot}$ consists of the orthogonal vector sub-bundle that is complementary to $S(\Upsilon\aleph)$ in $\Upsilon\overline{\aleph}|_{\aleph}.$ $S(\Upsilon\aleph), S(\Upsilon\aleph^{\bot})$ is an orthogonal direct decomposition, and they are nondegenerate.

$\begin{equation} \label{1e3} \nonumber S(\Upsilon\aleph)^{\bot} = S(\Upsilon\aleph^{\bot}) \bot S(\Upsilon\aleph^{\bot})^{\bot}. \end{equation}$

Let the vector bundle

$\begin{equation} \label{1e4} \nonumber tr(\Upsilon\aleph) = ltr(\Upsilon\aleph) \bot S(\Upsilon\aleph^{\bot}). \end{equation}$

Thus,

$\begin{equation} \label{1e5} \nonumber \Upsilon\overline{\aleph} = \Upsilon\aleph \oplus tr(\Upsilon\aleph) = S(\Upsilon\aleph) \bot S(\Upsilon\aleph^{\bot}) \bot (Rad(\Upsilon\aleph)\oplus ltr(\Upsilon\aleph). \end{equation}$

Assume that the Levi-Civita connection is $\overline{\nabla}$ on $\overline{\aleph}.$ We have

$\begin{equation} \overline{\nabla}_{\gamma}\zeta = \nabla_{\gamma}\zeta + h(\gamma, \zeta), \; \forall \, \gamma, \zeta \in \Gamma (\Upsilon\aleph) \end{equation}$

(2.5)

and

$\begin{equation} \overline{\nabla}_{\gamma}\zeta = - A_{h}\zeta + \nabla_{\gamma}^{\bot}h, \; \forall \gamma \in \Gamma (\Upsilon\aleph)\; and \; h \in \Gamma (tr(\Upsilon\aleph)), \end{equation}$

(2.6)

where $\{ \nabla_{\gamma}\zeta, A_{h}\gamma \}$ and $\{ h(\gamma, \zeta), \nabla_{\gamma}^{\bot}h \}$ belongs to $\Gamma (\Upsilon\aleph)$ and $\Gamma (tr(\Upsilon\aleph)),$ respectively.

Using projection $L : tr(\Upsilon\aleph) \rightarrow ltr(\Upsilon\aleph),$ and $S: tr(\Upsilon\aleph) \rightarrow S(\Upsilon\aleph^{\bot}),$ we have

$\begin{equation} \overline{\nabla}_{\gamma}\zeta = \nabla_{\gamma}\zeta + h^{l}(\gamma, \zeta) + h^{s}(\gamma, \zeta), \end{equation}$

(2.7)

$\begin{equation} \overline{\nabla}_{\gamma}\aleph = - A_{\aleph}\gamma + \nabla_{\gamma}^{l}\aleph + \lambda^{s}(\gamma, \aleph), \end{equation}$

(2.8)

and

$\begin{equation} \overline{\nabla}_{\gamma} \chi = - A_{\chi}\gamma + \nabla_{\gamma}^{s} + \lambda^{l}(\gamma, \chi) \end{equation}$

(2.9)

for any $\gamma, \zeta \in \Gamma (\Upsilon\aleph), \; \aleph \in \Gamma (ltr(\Upsilon\aleph)),$ and $\chi \in \Gamma (S(\Upsilon\aleph^{\bot})),$ where $h^{l}(\gamma, \zeta) = Lh(\gamma, \zeta), \; h^{s}(\gamma, \zeta) = Sh(\gamma, \zeta), \; \nabla_{\gamma}^{l}\aleph, \; \lambda^{l}(\gamma, \chi) \in \Gamma (ltr(\mathcal{T}\aleph)), \; \nabla_{\gamma}^{s}\; \lambda^{s}(\gamma, \aleph) \in \Gamma (S(\Upsilon\aleph^{\bot})),$ and $\nabla_{\gamma}\zeta, \; A_{\aleph}\gamma, \; A_{\chi}\gamma \in \Gamma (\Upsilon\aleph).$

The projection morphism of $\Upsilon\aleph$ on the screen is represented by $P$ , and we take the distribution into consideration.

$\begin{array}{c} \nabla_{\gamma}P\zeta = \nabla_{\gamma}^{*}P\zeta + h^{*}(\gamma, P\zeta), \\ \nabla_{\gamma}\xi = - A_{\xi}^{*}\gamma + \nabla_{\gamma}^{*t}\xi, \end{array}$

(2.10)

where $\gamma, \zeta \in \Gamma(\Upsilon\aleph), \; \xi \in \Gamma(Rad(\Upsilon\aleph)).$

Thus, we have the subsequent equation.

$\begin{equation} {g}(h^{*}(\gamma, P\zeta), \aleph) = g(A_{\aleph}\gamma, P\zeta), \end{equation}$

(2.11)

Consider that $\overline{\nabla}$ is a metric connection. We get

$\begin{equation} (\nabla_{\gamma}{g})(\zeta, \eta) = {g}(h^{l}(\gamma, \zeta), \eta)+ {g}(h^{l}(\gamma, \zeta \eta), \zeta). \end{equation}$

(2.12)

Using the characteristics of a linear connection, we can obtain

$\begin{equation} (\nabla_{\gamma}h^{l})(\zeta, \eta) = \nabla_{\gamma}^{l}(h^{l}(\zeta, \eta))- h^{l}(\overline{\nabla}_{\gamma}\zeta, \eta)-h^{l}(\zeta, \overline{\nabla}_{\gamma}\eta), \end{equation}$

(2.13)

$\begin{equation} (\nabla_{\gamma}h^{s})(\zeta, \eta) = \nabla_{\gamma}^{s}(h^{s}(\zeta, \eta))- h^{s}(\overline{\nabla}_{\gamma}\zeta, \eta)-h^{s}(\zeta, \overline{\nabla}_{\gamma}\eta). \end{equation}$

(2.14)

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

$\Upsilon\aleph = \lambda \oplus \lambda',$

where $\lambda = Rad(\Upsilon\aleph) \bot {\psi} Rad(\Upsilon\aleph) \bot \lambda_{0}.$

$S$ and $Q$ stand for the projection on $\lambda$ and $\lambda'$ , respectively, then

$\begin{equation} \label{1e11} \nonumber {\psi}\gamma = f\gamma + w\gamma \end{equation}$

for $\gamma, \zeta \in \Gamma (\Upsilon\aleph),$ where $f\gamma = {\psi}S\gamma$ and $w\gamma = {\psi}Q\gamma.$

On the other hand, we have

$\begin{equation} \label{1e12} \nonumber {\psi}\zeta = B\zeta + C\zeta \end{equation}$

for any $\zeta \in \Gamma (tr(\Upsilon\aleph)),$ $B\zeta \in \Gamma (\Upsilon\aleph)$ and $C\zeta \in \Gamma (tr(\Upsilon\aleph)),$ unless $\aleph_{1}$ and $\aleph_{2}$ are denoted as ${\psi}L_{1}$ and ${\psi}L_{2},$ respectively.

Lemma 2.1. Assume that the screen distribution is totally geodesic and that $\aleph$ is a CR-lightlike submanifold of the golden semi-Riemannian manifold, then $\nabla_{\gamma}\zeta \in \Gamma (S(\Upsilon N)),$ where $\gamma, \zeta \in \Gamma (S(\Upsilon \aleph)).$

Proof. For $\gamma, \zeta \in \Gamma (S(\Upsilon\aleph)),$

$\begin{eqnarray} {g}(\nabla_{\gamma}\zeta, \aleph) & = & {g} (\overline{\nabla}_{\gamma}\zeta - h(\gamma, \zeta), \aleph)\\ & = & - {g} (\zeta, \overline{\nabla}_{\gamma} \aleph). \end{eqnarray}$

Using (2.8),

$\begin{eqnarray} {g}(\nabla_{\gamma}\zeta, \aleph) & = & - {g}(\zeta, -A_{\aleph}\gamma + \nabla_{\gamma}^{\bot}\aleph) \\ & = & {g}(\zeta, A_{\aleph}\gamma). \end{eqnarray}$

Using (2.11),

${g}(\nabla_{\gamma}\zeta, \aleph) = {g}(h^{*}(\gamma, \zeta), \aleph).$

Since screen distribution is totally geodesic, $h^{*}(\gamma, \zeta) = 0,$

${g} (\overline{\nabla}_{\gamma}\zeta, \aleph) = 0.$

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

$\nabla_{\gamma}\zeta \in \Gamma (S(\Upsilon \aleph)),$

where $\gamma, \zeta \in \Gamma (S(\Upsilon \aleph)).$

Theorem 2.2. Assume that $\aleph$ is a locally golden semi-Riemannian manifold $\overline{\aleph}$ with CR-lightlike properties, then $\nabla_{\gamma}\psi\gamma = \psi\nabla_{\gamma}\gamma$ for $\gamma \in \Gamma (\lambda_{0}).$

Proof. Assume that $\gamma, \zeta \in \Gamma (\lambda_{0}).$ Using (2.5), we have

$\begin{eqnarray} {g}(\nabla_{\gamma}\psi\gamma, \zeta) & = & {g}(\overline{\nabla}_{\gamma}\psi\gamma-h(\gamma, \psi\gamma), \zeta)\\ {g}(\nabla_{\gamma}\psi\gamma, \zeta) & = & {g}(\psi(\overline{\nabla}_{\gamma}\gamma), \zeta) \\ {g}(\nabla_{\gamma}\psi\gamma, \zeta) & = & {g}(\psi(\nabla_{\gamma}\gamma), \zeta), \\ {g}(\nabla_{\gamma}\psi\gamma - \psi(\nabla_{\gamma}\gamma), \zeta) & = & 0. \end{eqnarray}$

Nondegeneracy of $\lambda_{0}$ implies

$\nabla_{\gamma}\psi\gamma = \psi(\nabla_{\gamma}\gamma),$

where $\gamma \in \Gamma (\lambda_{0}).$

3. Geodesic CR-lightlike submanifolds

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

$h(\gamma, \alpha) = 0,$

where $h$ stands for second fundamental form, $\gamma \in \Gamma (\lambda),$ and $\alpha \in \Gamma (\lambda').$

For $\gamma, \zeta \in \Gamma (\lambda)$ and $\alpha, \beta \in \Gamma (\lambda')$ if

$h(\gamma, \zeta) = 0$

and

$h(\alpha, \beta) = 0,$

then it is known as $\lambda$ -geodesic and $\lambda'$ -geodesic, respectively.

Theorem 3.2. Assume $\aleph$ is a CR-lightlike submanifold of $\overline{\aleph}$ , which is a golden semi-Riemannian manifold. $\aleph$ is totally geodesic if

$(L_{g})(\gamma, \zeta) = 0$

and

$(L_{\chi}{g})(\gamma, \zeta) = 0$

for $\alpha, \beta \in \Gamma (\Upsilon\aleph), \; \xi \in \Gamma (Rad(\Upsilon\aleph)) \;$ , and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. Since $\aleph$ is totally geodesic, then

$h(\gamma, \zeta) = 0$

for $\gamma, \zeta \in \Gamma (\Upsilon\aleph).$

We know that $h(\gamma, \zeta) = 0$ if

${g}(h(\gamma, \zeta), \xi) = 0$

and

${g}(h(\gamma, \zeta), \chi) = 0.$

$\begin{eqnarray} {g}(h(\gamma, \zeta), \xi) & = & {g} (\overline{\nabla}_{\gamma} \zeta - \nabla_{\gamma}\zeta, \xi) \\ {} & = & - {g} (\zeta, [\gamma, \xi] + \overline{\nabla}_{\xi}\gamma \\ {} & = & - {g} (\zeta, [\gamma, \xi]) + {g}(\gamma, [\xi, \zeta]) + {g}(\overline{\nabla}_{\zeta}\xi, \gamma) \\ {} & = & - (L_{\xi}{g})(\gamma, \zeta) + {g}(\overline{\nabla}_{\zeta}\xi, \gamma) \\ {} & = & - (L_{\xi}{g})(\gamma, \zeta) - {g}(\xi, h(\gamma, \zeta))) \\ 2{g}(h(\gamma, \zeta) & = & - (L_{\xi}{g})(\gamma, \zeta). \end{eqnarray}$

Since ${g}(h(\gamma, \zeta), \xi) = 0,$ we have

$\begin{equation} \label{2e1} \nonumber (L_{\xi}{g})(\gamma, \zeta) = 0. \end{equation}$

Similarly,

$\begin{eqnarray} \ {g}(h(\gamma, \zeta), \chi) & = & {g}(\overline{\nabla}_{\gamma}\zeta - \nabla_{\gamma}\zeta, \chi) \\ {} & = & - {g} (\zeta, [\gamma, \chi]) + {g}(\gamma, [\chi, \zeta]) + {g}(\overline{\nabla}_{\zeta}\chi, \gamma) \\ {} & = & - (L_{\chi}{g})(\gamma, \zeta) + {g}(\overline{\nabla}_{\zeta}\chi, \gamma) \\ 2{g}(h(\gamma, \zeta), \chi) & = & - (L_{\chi}{g})(\gamma, \zeta). \end{eqnarray}$

Since ${g}(h(\gamma, \zeta), \chi) = 0,$ we get

$\begin{equation} \label{2e2} \nonumber (L_{\chi}{g})(\gamma, \zeta) = 0 \end{equation}$

for $\chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Lemma 3.3. Assume that $\overline{\aleph}$ is a golden semi-Riemannian manifold whose submanifold $\aleph$ is CR-lightlike, then

${g}(h(\gamma, \zeta), \chi) = {g}(A_{\chi}\gamma, \zeta)$

for $\gamma \in \Gamma (\lambda), \; \zeta \in \Gamma (\lambda') \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. Using (2.5), we get

$\begin{eqnarray} {g} (h(\gamma, \zeta), \chi) & = & {g}(\overline{\nabla}_{\gamma}\zeta - \nabla_{\gamma}\zeta, \chi) \\ {} & = & {g}(\zeta, \overline{\nabla}_{\gamma}\chi). \end{eqnarray}$

From (2.9), it follows that

$\begin{eqnarray} {g}(h(\gamma, \zeta), \chi) & = & - {g}(\zeta, - A_{\chi}\gamma + \nabla_{\gamma}^{s}\chi + \lambda^{s}(\gamma, \chi)) \\ {} & = & {g}(\zeta, A_{\chi}\gamma) - {g} (\zeta, \nabla_{\gamma}^{s}\chi) - {g}(\zeta, \lambda^{s}(\gamma, \chi)) \\ {g}(h(\gamma, \zeta), \chi) & = & {g}(\zeta, A_{\chi}\gamma), \end{eqnarray}$

where $\gamma \in \Gamma (\lambda), \; \zeta \in \Gamma (\lambda') \; , \; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Theorem 3.4. Assume that $\aleph$ is a CR-lightlike submanifold of the golden semi-Riemannian manifold and $\overline{\aleph}$ is mixed geodesic if

$A^{*}_{\xi}\gamma \in \Gamma (\lambda_{0} \bot {\psi}L_{1})$

and

$A_{\chi}\gamma \in \Gamma (\lambda_{0} \bot Rad(\Upsilon\aleph) \bot {\psi}L_{1})$

for $\gamma \in \Gamma (\lambda), \; \xi \in \Gamma (Rad(\Upsilon\aleph)), \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. For $\gamma \in \Gamma (\lambda), \; \zeta \in \Gamma (\lambda'), \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})),$ we get

Using (2.5),

$\begin{eqnarray} {g}(h(\gamma, \zeta), \xi) & = & {g}(\overline{\nabla}_{\gamma}\zeta - \nabla_{\gamma}\zeta, \xi) \\ {} & = & - {g}(\zeta, \overline{\nabla}_{\gamma}\xi). \end{eqnarray}$

Again using (2.5), we obtain

$\begin{eqnarray} {} {g}(h(\gamma, \zeta), \xi) & = & - {g}(\zeta, \nabla_{\gamma}\xi + h(\gamma, \xi)) \\ {} & = & - {g}(\zeta, \nabla_{\gamma}\xi). \end{eqnarray}$

Using (2.10), we have

$\begin{eqnarray} {} {g}(h(\gamma, \zeta), \xi) & = & - {g}( \zeta, - A_{\xi}^{*}\gamma + \nabla^{*t}_{\gamma}\xi) \\ {g}( \zeta, A_{\xi}^{*}\gamma)& = & 0. \end{eqnarray}$

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

${g}(h(\gamma, \zeta), \xi) = 0$

$\Rightarrow {g}( \zeta, A_{\xi}^{*}\gamma) = 0$

$\Rightarrow A_{\xi}^{*}\gamma \in \Gamma (\lambda_{0} \bot {\psi}L_{1}),$

where $\gamma \in \Gamma (\lambda), \zeta \in \Gamma (\lambda').$

From (2.5), we get

$\begin{eqnarray} {g}(h(\gamma, \zeta), \chi) & = & {g} (\overline{\nabla}_{\gamma}\zeta - \nabla_{\gamma}\zeta, \chi) \\ {} & = & - {g} (\zeta, \overline{\nabla}_{\gamma}\chi). \end{eqnarray}$

From (2.9), we get

$\begin{eqnarray} {} {g}(h(\gamma, \zeta), \chi) & = & - {g} (\zeta, - A_{\chi}\gamma + \nabla_{\gamma}^{s}\chi + \lambda^{l}(\gamma, \chi))\\ {g}(h(\gamma, \zeta), \chi) & = & {g} (\zeta, A_{\chi}\gamma). \end{eqnarray}$

Since, $\aleph$ is mixed geodesic, then ${g}(h(\gamma, \zeta), \chi) = 0$

$\Rightarrow {g} (\zeta, A_{\chi}\gamma) = 0.$

$A_{\chi}\gamma \in \Gamma (\lambda_{0} \bot Rad(\Upsilon\aleph) \bot\psi_{1}).$

Theorem 3.5. Suppose that $\aleph$ is a CR-lightlike submanifold of a golden semi-Riemannian manifold $\overline{\aleph},$ then $\aleph$ is $\lambda'$ -geodesic if $A_{\chi}\eta$ and $A_{\xi}^{*}\eta$ have no component in $\aleph_{2}\bot {\psi}Rad(\Upsilon\aleph)$ for $\eta \in \Gamma (\lambda'), \; \xi \in \Gamma (Rad(\Upsilon\aleph)), \; \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. From (2.5), we obtain

$\begin{eqnarray} {g}(h(\eta, \beta), \chi) & = & {g}(\overline{\nabla}_{\eta}\beta - \nabla_{\gamma}\zeta, \chi) \\ {} & = & - \overline{g}(\nabla_{\gamma}\zeta, \chi), \end{eqnarray}$

where $\chi, \beta \in \Gamma (\lambda').$

Using (2.9), we have

$\begin{eqnarray} {g}(h(\eta, \beta), \chi) & = & - {g}(\beta, - A_{\chi}\eta + \nabla_{\eta}^{s}+ \lambda^{l}(\eta, \chi)) \\ {g}(h(\eta, \beta), \chi) & = & {g}(\beta, A_{\chi}\eta) . \end{eqnarray}$

(3.1)

Since $\aleph$ is $\lambda'$ -geodesic, then ${g}(h(\eta, \beta), \chi) = 0.$

From (3.1), we get

${g}(\beta, A_{\chi}\eta) = 0.$

Now,

$\begin{eqnarray} {g}(h(\eta, \beta), \xi) & = & {g}(\overline{\nabla}_{\eta}\beta - \nabla_{\eta}\beta, \xi) \\ {} & = & {g}(\overline{\nabla}_{\eta}\beta, \xi) = - {g}(\beta, \overline{\nabla}_{\eta} \xi). \end{eqnarray}$

From (2.10), we get

$\begin{eqnarray} {} {g}(h(\eta, \beta), \xi) & = & - {g}(\eta, - A_{\xi}^{*}\eta + \nabla_{\eta}^{*t}\xi) \\ {g}(h(\eta, \beta), \xi) & = & {g}( A_{\xi}^{*}\beta, \eta). \label{2e4} \end{eqnarray}$

Since $\aleph$ is $\lambda'$ - geodesic, then

${g}(h(\eta, \beta), \xi) = 0$

$\Rightarrow {g}( A_{\xi}^{*}\beta, \eta) = 0.$

Thus, $A_{\chi}\eta$ and $A^{*}_{\xi}\eta$ have no component in $M_{2}\bot \psi Rad(\Upsilon\aleph).$

Lemma 3.6. Assume that $\overline{\aleph}$ is a golden semi-Riemannian manifold that has a CR-lightlike submanifold $\aleph$ . Due to the distribution's integrability, the following criteria hold.

(ⅰ) $\psi{g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) - {g}(\lambda^{l}(\gamma, \chi), {\psi}\zeta) = {g}(A_{\chi}{\psi}\gamma, \zeta) - {g}(A_{\chi}\gamma, {\psi}\zeta),$

(ⅱ) ${g}(\lambda^{l}({\psi}\gamma), \xi) = {g}(A_{\chi}\gamma, {\psi} \xi),$

(ⅲ) ${g}(\lambda^{l}(\gamma, \chi), \xi) = {g}(A_{\chi}{\psi}\gamma, \psi\xi) - {g}(A_{\chi}\gamma, \psi\xi),$

where $\gamma, \zeta \in \Gamma (\Upsilon\aleph), \; \xi \in \Gamma (Rad(\Upsilon\aleph)),$ and $\chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. From Eq (2.9), we obtain

$\begin{eqnarray} {g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) & = & {g}(\overline{\nabla}_{\psi\gamma}\chi + A_{\chi}{\psi}\gamma - \nabla_{\psi\gamma}^{s}\chi, \zeta) \\ {} & = & - {g}(\chi, \overline{\nabla}_{\psi\gamma}\zeta) + {g}(A_{\chi}{\psi}\gamma, \zeta). \end{eqnarray}$

Using (2.5), we get

$\begin{eqnarray} {} {g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) & = & - {g}(\chi, {\nabla}_{\psi\gamma}\zeta + h({\psi}\gamma, \zeta)) + {g}(A_{\chi}{\psi}\gamma, \zeta)\\ {} & = & - {g}(\chi, h(\gamma, {\psi}\zeta)) + {g}(A_{\chi}{\psi}\gamma, \zeta). \end{eqnarray}$

Again, using (2.5), we get

$\begin{eqnarray} {} {g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) & = & - {g}(\chi, \overline{\nabla}_{\gamma}{\psi}\zeta - \nabla_{\gamma}{\psi}\zeta) + {g}(A_{\chi}{\psi}\gamma, \zeta) \\ {} & = & {g}(\overline{\nabla}_{\gamma}\chi, {\psi}\zeta) + {g}(A_{\chi}{\psi}\gamma, \zeta). \end{eqnarray}$

Using (2.9), we have

$\begin{eqnarray} {} {g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) & = & {g}(- A_{\chi}\gamma + \nabla_{\gamma}^{s}\chi + \lambda^{l}(\gamma, \chi), {\psi}\zeta) + \\ {} {g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) - {g}(\lambda^{l}(\gamma, \chi), {\psi}\zeta) & = & {g}(A_{\chi}{\psi}\gamma, \zeta) - {g}( A_{\chi}\gamma, {\psi}\zeta). \end{eqnarray}$

(ⅱ) Using (2.9), we have

$\begin{eqnarray} {g}(\lambda^{l}({\psi}\gamma, \chi), \xi) & = & {g}( A_{\chi}{\psi}\gamma - \nabla_{{\psi}\gamma}^{s}\chi + {\nabla}_{{\psi}\gamma}\chi, \xi ) \\ {} & = & {g}( A_{\chi}{\psi}\gamma, \xi) - {g}(\chi, \overline{\nabla}_{{\psi}\gamma}\xi ). \end{eqnarray}$

Using (2.10), we get

$\begin{eqnarray} {} {g}(\lambda^{l}({\psi}\gamma, \chi), \xi) & = & {g}( A_{\chi}{\psi}\gamma, \xi) + {g}(\chi, A^{*}_{\xi}{\psi}\gamma ) - {g}(\chi, \nabla^{*t}_{{\psi}\gamma}, \xi) \\ {g}(\lambda^{l}({\psi}\gamma), \xi) & = & {g}( A_{\chi}\gamma, {\psi} \xi). \end{eqnarray}$

(ⅲ) Replacing $\zeta$ by $\psi\xi$ in (ⅰ), we have

$\begin{eqnarray} \psi{g}(\lambda^{l}({\psi}\gamma, \chi), \psi\xi) - {g}(\lambda^{l}(\gamma, \chi), {\psi^{2}}\xi) = {g}(A_{\chi}{\psi}\gamma, \psi\xi) - {g}( A_{\chi}\gamma, {\psi^{2}}\xi). \end{eqnarray}$

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

$\begin{eqnarray} \psi{g}(\lambda^{l}({\psi}\gamma, \chi), \psi\xi) - {g}(\lambda^{l}(\gamma, \chi), (\psi+I)\xi) & = & {g}(A_{\chi}{\psi}\gamma, \psi\xi) - {g}( A_{\chi}\gamma, (\psi+I)\xi)\\ \psi{g}(\lambda^{l}({\psi}\gamma, \chi), \psi\xi) - {g}(\lambda^{l}(\gamma, \chi), \psi\xi) - {g}(\lambda^{l}(\gamma, \chi), \xi) & = & {g}(A_{\chi}{\psi}\gamma, \psi\xi) - {g}( A_{\chi}\gamma, \psi\xi) - \\ && {g}( A_{\chi}\gamma, \xi).\\ {g}(\lambda^{l}(\gamma, \chi), \xi) & = & {g}(A_{\chi}{\psi}\gamma, \psi\xi) - {g}( A_{\chi}\gamma, \psi\xi). \end{eqnarray}$

4. Totally umbilical CR-lightlike submanifolds

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 $H \in tr\ \Gamma (\Upsilon \aleph)$ that satisfies

$h(\chi, \eta) = H g(\chi, \eta),$

where $h$ is stands for second fundamental form and $\chi, \ \eta \in \ \Gamma (\Upsilon \aleph).$

Theorem 4.2. Assume that the screen distribution is totally geodesic and that $\aleph$ is a totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold $\overline{\aleph}$ , then

$A_{\psi \eta}\chi = A_{\psi\chi}\eta, \; \forall \; \chi, \eta \in \Gamma \lambda'.$

Proof. Given that $\overline{\aleph}$ is a golden semi-Riemannian manifold,

$\psi\overline{\nabla}_{\eta}\chi = \overline{\nabla}_{\eta}\psi\chi.$

Using (2.5) and (2.6), we have

$\begin{eqnarray} \psi(\nabla_{\eta}\chi) + \psi(h(\eta, \chi)) & = & - A_{\psi\chi}\eta + \nabla_{\eta}^{t}\psi\chi. \end{eqnarray}$

(4.1)

Interchanging $\eta$ and $\chi$ , we obtain

$\begin{equation} \psi(\nabla_{\chi}\eta) + \psi(h(\chi, \eta)) = - A_{\psi \eta}\chi + \nabla_{\chi}^{t}\psi \eta. \end{equation}$

(4.2)

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

$\begin{eqnarray} \psi(\nabla_{\eta}\chi - \nabla_{\chi}\eta) - \nabla_{\eta}^{t}\psi\chi + \nabla_{\chi}^{t}\psi \eta & = & A_{\psi \eta}\chi - A_{\psi\chi}\eta. \end{eqnarray}$

(4.3)

Taking the inner product with $\gamma \in \Gamma (\lambda_{0})$ in (4.3), we have

$\begin{eqnarray} {g}(\psi(\nabla_{\chi}\eta, \gamma) - {g}(\psi(\nabla_{\chi}\eta, \gamma) & = & {g}(A_{\psi \eta}\chi, \gamma) \\ && -{g}(A_{\psi\chi}\eta, \gamma). \\ {g}(A_{\psi \eta}\chi - A_{\psi\chi}\eta, \gamma) = {g}&(\nabla_{\chi}\eta,&- \psi\gamma) {g}(\nabla_{\chi}\eta, \psi\gamma). \end{eqnarray}$

(4.4)

Now,

$\begin{eqnarray} {g}(\nabla_{\chi}\eta, \psi\gamma) & = & {g}(\overline{\nabla}_{\chi}\eta - h(\chi, \eta), \psi\gamma) \\ {g}(\nabla_{\chi}\eta, \psi\gamma) & = & - {g}(\eta, (\overline{\nabla}_{\chi}\psi)\gamma - \psi(\overline{\nabla}_{\chi}\gamma)). \end{eqnarray}$

Since $\psi$ is parallel to $\overline{\nabla},$ i.e., $\overline{\nabla}_{\gamma}\psi = 0,$

${g}(\nabla_{\chi}\eta, \psi\gamma) = - \psi(\overline{\nabla}_{\chi}\gamma)).$

Using (2.7), we have

$\begin{eqnarray} {g}(\nabla_{\chi}\eta, \psi\gamma) & = &- {g} (\psi \eta, \nabla_{\chi}\gamma + h^{s}(\chi, \gamma) + h^{l}(\chi, \gamma))\\ {g}(\nabla_{\chi}\eta, \psi\gamma) & = & - {g} (\psi \eta, \nabla_{\chi}\gamma) - {g}(\psi \eta, h^{s}(\chi, \gamma)) - {g}(\psi \eta, h^{l}(\chi, \gamma)). \end{eqnarray}$

(4.5)

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

$h^{s}(\chi, \gamma) = H^{s}{g}(\chi, \gamma) = 0$

and

$h^{l}(\chi, \gamma) = H^{l}{g}(\chi, \gamma) = 0,$

where $\chi \in \Gamma (\lambda')$ and $\gamma \in \Gamma (\lambda_{0}).$

From (4.5), we have

${g}(\nabla_{\chi}\eta, \psi\gamma) = - {g} (\psi \eta, \nabla_{\chi}\gamma).$

From Lemma 2.1, we get

${g}(\nabla_{\chi}\eta, \psi\gamma) = 0.$

Similarly,

${g}(\nabla_{\eta}\chi, \psi\gamma) = 0$

Using (4.4), we have

$g(A_{\psi \eta}\chi - A_{\psi\chi}\eta, \gamma) = 0.$

Since $\lambda_{0}$ is nondegenerate,

$A_{\psi \eta}\chi - A_{\psi \chi}\eta = 0$

$\Rightarrow A_{\psi \eta}\chi = A_{\psi\chi}\eta.$

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

Proof. We know that $\aleph$ is a totally umbilical CR-lightlike submanifold of $\overline{\aleph},$ then from (2.13) and (2.14),

$\begin{eqnarray} (\nabla_{\gamma}h^{l})(\zeta, \omega) = g(\zeta, \omega)\nabla^{l}_{\gamma}H^{l}-H^{l}\{ (\nabla_{\gamma}g)(\zeta, \omega)\} , \end{eqnarray}$

(4.6)

$\begin{eqnarray} (\nabla_{\gamma}h^{s})(\zeta, \omega) = g(\zeta, \omega)\nabla^{s}_{\gamma}H^{s}-H^{s}\{ (\nabla_{\gamma}g)(\zeta, \omega)\} \end{eqnarray}$

(4.7)

for a CR-lightlike section $\pi = \gamma\wedge \omega, \; \gamma \in \Gamma (\lambda_{0}), \omega \in \Gamma (\lambda').$

From (2.12), we have $(\nabla_{U}g)(\zeta, \omega) = 0.$ Therefore, from (4.6) and (4.7), we get

$\begin{eqnarray} (\nabla_{\gamma}h^{l})(\zeta, \omega) = g(\zeta, \omega)\nabla^{l}_{\gamma}H^{l} , \end{eqnarray}$

(4.8)

$\begin{eqnarray} (\nabla_{\gamma}h^{s})(\zeta, \omega) = g(\zeta, \omega)\nabla^{s}_{\gamma}H^{s} . \end{eqnarray}$

(4.9)

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

$\begin{eqnarray} \{\overline{R}(\gamma, \zeta)\omega \}^{tr} & = & g(\zeta, \omega)\nabla^{l}_{\gamma} H^{l}- g(\gamma, \omega)\nabla^{l}_{\zeta} H^{l} + g(\zeta, \omega)\lambda^{l}(\gamma, H^{s}) \\ && - g(\gamma, \omega)\lambda^{l}(\zeta, H^{s})+g(\zeta, \omega)\nabla^{s}_{\gamma} H^{s}-g(\gamma, \omega)\nabla^{s}_{\zeta}H^{s} \\ && +g(\zeta, \omega)\lambda^{s}(\gamma, H^{l})-g(\gamma, \omega)\lambda^{s}(\zeta, H^{l}). \end{eqnarray}$

(4.10)

For any $\beta \in \Gamma (tr(\Upsilon\aleph))$ , from Equation (4.10), we get

$\begin{eqnarray} \overline{R}(\gamma, \zeta, \omega, \beta ) & = & g(\zeta, \omega){g}(\nabla^{l}_{\gamma} H^{l}, \beta)- g(\gamma, \omega){g}(\nabla^{l}_{\zeta} H^{l}, \beta) + \\ && g(\zeta, \omega){g}(\lambda^{l}(\gamma, H^{s}), \zeta) - g(\gamma, \omega){g}(\lambda^{l}(\zeta, H^{s}), \beta)+ \\ && g(\zeta, \omega) {g}(\nabla^{s}_{\gamma} H^{s}, \beta)-g(\gamma, \omega){g}(\nabla^{s}_{\zeta}H^{s}, \beta) \\ && +g(\zeta, \omega){g}(\lambda^{s}(\gamma, H^{l}), \beta)- g(\gamma, \omega){g}(\lambda^{s}(\zeta, H^{l}, \beta). \label{4e11} \end{eqnarray}$

$\begin{eqnarray} {R}(\gamma, \omega, \psi\gamma, \psi\omega ) & = & g(\omega, \psi\gamma){g}(\nabla^{l}_{\gamma} H^{l}, \psi\omega)- g(\gamma, \psi\gamma){g}(\nabla^{l}_{\omega} H^{l}, \psi\omega) + \\ && g(\omega, \psi\gamma){g}(\lambda^{l}(\gamma, H^{s}), \psi\omega) - g(\gamma, \psi\gamma){g}(\lambda^{l}(\omega, H^{s}), \psi\omega)+ \\ && g(\omega, \psi\gamma) {g}(\nabla^{s}_{\gamma} H^{s}, \psi\omega)-g(\gamma, \psi\gamma){g}(\nabla^{s}_{\omega}H^{s}, \psi\omega) + \\ && g(\omega, \psi\gamma){g}(\lambda^{s}(\gamma, H^{l}), \psi\omega)- g(\gamma, \psi\gamma){g}(\lambda^{s}(\omega, H^{l}, \psi U). \label{4e13} \end{eqnarray}$

For any unit vectors $\gamma \in \Gamma (\lambda)$ and $\omega \in \Gamma (\lambda'),$ we have

$\overline{R}(\gamma, \omega, \psi\gamma, \psi\omega) = \overline{R}(\gamma, \omega, \gamma, \omega) = 0.$

We have

$K(\gamma) = K_{N}(\gamma \wedge \zeta) = g(\overline{R}(\gamma, \zeta)\zeta, \gamma),$

where

$\overline{R}(\gamma, \omega, \gamma, \omega) = g(\overline{R}(\gamma, \omega)\gamma, \omega)$

$\overline{R}(\gamma, \omega, \psi\gamma, \psi\omega) = g(\overline{R}(\gamma, \omega)\psi\gamma, \psi\omega)$

i.e.,

$\overline{K}(\pi) = 0$

for all CR-sections $\pi.$

5. Example

Example 5.1. We consider a semi-Riemannian manifold $R_{2}^{6}$ and a submanifold $\aleph$ of co-dimension $2$ in $R_{2}^{6},$ given by equations

$\upsilon_{5} = \upsilon_{1}{\mathrm{cos}}\alpha - \upsilon_{2}{\mathrm{sin}}\alpha - \upsilon_{3}z_{4} {\mathrm{tan}}\alpha,$

$\upsilon_{6} = \upsilon_{1}{\mathrm{sin}}\alpha - \upsilon_{2}{\mathrm{cos}}\alpha - \upsilon_{3}\upsilon_{4},$

where $\alpha \in R - \{ \frac{\pi}{2} + k\pi; \ k \in z \}.$ The structure on $R^{6}_{2}$ is defined by

$\psi( \frac{\partial}{\partial \upsilon_{1}}, \frac{\partial}{\partial \upsilon_{2}}, \frac{\partial}{\partial \upsilon_{3}}, \frac{\partial}{\partial \upsilon_{4}}, \frac{\partial}{\partial \upsilon_{5}} , \frac{\partial}{\partial \upsilon_{6}}) = (\overline{\phi} \ \frac{\partial}{\partial \upsilon_{1}}, \overline{\phi} \frac{\partial}{\partial \upsilon_{2}}, \phi \frac{\partial}{\partial \upsilon_{3}}, \phi \frac{\partial}{\partial \upsilon_{4}}, \phi \frac{\partial}{\partial \upsilon_{5}}, \phi \frac{\partial}{\partial \upsilon_{6}} ).$

Now,

$\psi^{2} ( \frac{\partial}{\partial \upsilon_{1}}, \frac{\partial}{\partial \upsilon_{2}}, \frac{\partial}{\partial \upsilon_{3}}, \frac{\partial}{\partial \upsilon_{4}}, \frac{\partial}{\partial \upsilon_{5}} , \frac{\partial}{\partial \upsilon_{6}}) = ((\overline{\phi}+ 1) \ \frac{\partial}{\partial \upsilon_{1}}, (\overline{\phi} + 1) \frac{\partial}{\partial \upsilon_{2}}, (\phi + 1) \frac{\partial}{\partial \upsilon_{3}}, (\phi + 1) \frac{\partial}{\partial \upsilon_{4}},$

$( \phi + 1) \frac{\partial}{\partial \upsilon_{5}}, ( \phi + 1) \frac{\partial}{\partial \upsilon_{6}} )$

$\psi^{2} = \psi + I.$

It follows that $(R_{2}^{6}, \psi)$ is a golden semi-Reimannian manifold.

The tangent bundle $\Upsilon\aleph$ is spanned by

$Z_{0} = - {\mathrm{sin}}\alpha \ \frac{\partial}{\partial \upsilon_{5}} - \mathrm{cos}\alpha \ \frac{\partial}{\partial \upsilon_{6}} - \phi \ \frac{\partial}{\partial \upsilon_{2}},$

$Z_{1} = - \phi \ {\mathrm{sin}}\alpha \ \frac{\partial}{\partial \upsilon_{5}} - \phi \ {\mathrm{cos}}\alpha \ \frac{\partial}{\partial \upsilon_{6}} + \ \frac{\partial}{\partial \upsilon_{2}},$

$Z_{2} = \frac{\partial}{\partial \upsilon_{5}} - \overline{\phi} \ {\mathrm{sin}}\alpha \ \frac{\partial}{\partial \upsilon_{2}} + \frac{\partial}{\partial \upsilon_{1}},$

$Z_{3} = - \overline{\phi} \ {\mathrm{cos}} \alpha \ \frac{\partial}{\partial \upsilon_{2}} + \frac{\partial}{\partial \upsilon_{4}} + i \frac{\partial}{\partial \upsilon_{6}}.$

Thus, $\aleph$ is a 1-lightlike submanifold of $R_{2}^{6}$ with $Rad\Upsilon\aleph = \mathrm{Span}\{X_{0}\}$ . Using golden structure of $R_{2}^{6},$ we obtain that $X_{1} = \psi(X_{0}).$ Thus, $\psi(Rad\Upsilon\aleph)$ is a distribution on $\aleph.$ Hence, the $\aleph$ is a CR-lightlike submanifold.

6. Conclusions

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 $\aleph$ 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 $\aleph$ of a golden semi-Riemannian manifold $\overline{\aleph}$ disappears.

Use of AI tools declaration

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

Acknowledgments

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.

Conflict of interest

Authors have no conflict of interests.

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