Non-resonance with one-sided superlinear growth for indefinite planar systems via rotation numbers

1.   Introduction

The notion of lightlike submanifolds was introduced by K. L. Duggal and A. Bejancu in ^[1]. The study of lightlike submanifolds is remarkably more interesting due to non-trivial intersection of normal vector bundle and tangent bundle. Many geometers enriched the study of lightlike submanifolds (see ^[2,3]). B. Y. Chen ^[4] defined and studied slant submanifolds of an almost Hermitian manifold as generalization of complex submanifolds and totally real submanifolds for which the angle $\Psi$ between $\overline{\varrho}\zeta$ and the tangent space is constant for any tangent vector field $\zeta$. A. Lotta ^[5] investigated the concept of slant submanifolds in contact geometry. B. Sahin ^[6] studied slant submanifolds of an almost product Riemannian manifold. J. L. Cabrerizo et al. ^[7] analysed slant submanifolds of a Sasakian manifold. M. A. Khan et al. ^[8] introduced slant submanifolds in LP-contact manifolds. N. Papaghuic ^[9] introduced the notion of semi-slant submanifolds of Kaehler manifold. P. Alegre and A. Carriazo ^[10] introduced and studied bi-slant submanifolds of a para Hermitian manifold. On the other hand, Crasmareanu and Hretcanu ^[11,12] define golden structure $\overline{\varrho}$ as a (1, 1) tensor-field satisfying $\overline{\varrho}^{2} = \overline{\varrho}+ I$. A Riemannian manifold $\overline{O}$ with a golden structure $\overline{\varrho}$ is called almost golden Riemannian manifold. M. A. Qayyoom and M. Ahmad (see ^[13,14,15]) studied hypersurfaces, warped product skew semi-invariant submanifolds, skew semi-invariant submanifolds of a golden Riemannian manifold. They also studied submanifolds of locally metallic Riemannian manifolds ^[16]. B. Sahin ^[17] studied slant lightlike submanifolds of indefinite Hermitian manifolds. R. S. Gupta and A. Sharfuddin ^[18] studied slant lightlike submanifolds of indefinite Kenmotsu manifolds. J. W. Lee and D. H. Jin ^[19] studied slant lightlike submanifolds of an indefinite Sasakian manifold. Semi-slant submanifolds were defined and studied by V. Khan et al. ^[20]. Thereafter, many geometers studied semi-slant submanifolds in various spaces (see ^[21,22,23]). Moreover, many authors studied geometry of bi-slant lightlike submanifolds ^[24,25]. N. Poyraz and E. Yasar ^[3] studied lightlike submanifolds of golden semi-Riemannian manifolds. S. Kumar and A. Yadav ^[26] studied semi-slant lightlike submanifolds of golden semi-Riemannian manifolds. Johnson and Whitt studied foliations that has great importance in diffrential geometry, they considered each leaf of the foliation to be a totally geodesic submanifold of the ambient space ^[27].

In this paper, we introduce bi-slant lightlike submanifolds of golden semi-Riemannian manifolds. The paper is organized as follows: In Section 2, we define golden semi-Riemannian manifolds and present some basic concepts of lightlike submanifolds. In Section 3, we define and investigate some results on bi-slant lightlike submanifolds of golden semi-Riemannian manifolds and give two examples. We also discuss the integrability conditions of distributions on bi-slant lightlike submanifolds. In the last section, we obtain necessary and sufficient conditions for foliations determined by distributions on bi-slant lightlike submanifolds of golden semi-Riemannian manifolds to be geodesic.

2.   Basic concepts

Let $(\overline{O}, g)$ be a semi-Riemannian manifold. A golden structure on $(\overline{O}, g)$ is a non-null tensor $\overline{\varrho}$ of type (1, 1) which satisfies the equation

where $I$ is the identity transformation. We say that the metric $g$ is $\overline{\varrho}$-compatible if

for all $\zeta$, $\eta$ vector fields on $\overline{O}$. If we substitute $\overline{ \varrho}\zeta$ into $\zeta$ in (2.2), then we have

The semi-Riemannian metric is called $\overline{\varrho}$-compatible and $(\overline{O}, \overline{\varrho}, g)$ is called a golden semi-Riemannian manifold. Also, if

for all $\zeta, \eta\in\Gamma (TO)$, where $\overline{\nabla}$ is the Levi-Civita connection with respect to $g$, then $(\overline{O}, \overline{\varrho}, g)$ is called a locally golden semi-Riemannian manifold.

A submanifold $(O^{m}, g)$ immersed in a semi-Riemannian manifold $(\overline{O}^{m+n}, \overline{g})$ is called a lightlike submanifold if the metric $g$ induced from $\overline{g}$ is degenerate and the radical distribution $Rad(TO)$ is of rank $r$, where $1\leq r \leq m$. Let $S(TO)$ be a screen distribution which is a semi-Riemannian complementary distribution of $Rad(TO)$ in $TO$, i.e., $TO = Rad(TO)\perp S(TO)$.

Consider a screen transversal vector bundle $S(TO^{\bot})$, which is a semi-Riemannian complementary vector bundle of $Rad(TO)$ in $TO^{\bot}$. Since, for any local basis $\{\xi_{i}\}$ of $Rad(TO)$, there exists a local null frame $\{\varpi_{i}\}$ of sections with values in the orthogonal complement of $S(TO^{\bot})$ in $[S(TO)]^{\bot}$ such that $\overline{g}(\xi_{i}, \varpi_{j}) = \delta_{ij}$, it follows that there exists a lightlike transversal vector bundle $ltr(TO)$ locally spanned by $\{\varpi_{i}\}$. Let $tr(TO)$ be complementary (but not orthogonal) vector bundle to $TO$ in $T\overline{O}|_{O}$. Then

A submanifold $(O, g, S(TO), S(TO^{\bot}))$ of $\overline{O}$ is said to be

(i) r-lightlike if $r < min\{m, n\};$

(ii) Coisotropic if $r = n < m, S(TO^{\bot}) = \{0\};$

(iii) Isotropic if $r = m < n, S(TO) = \{0\};$

(iv) Totally lightlike if $r = m = n, S(TO) = \{0\} = S(TO^{\perp}).$

Let $\overline{\nabla}$, $\nabla$ and $\nabla^{t}$ denote the linear connections on $\overline{O}$, $O$ and vector bundle $tr(TO)$, respectively. Then the Gauss and Weingarten formulae are given by

where $\{\nabla_{\zeta}\eta, A_{\eta}\zeta\}$ and $\{h(\zeta, \eta), \nabla_{\zeta}^{t}\eta\}$ belong to $\Gamma (TO)$ and $\Gamma (ltr(TO)),$ respectively. $\nabla$ and $\nabla_{\zeta}^{t}$ are linear connections on $O$ and on the vector bundle $ltr(TO)$, respectively. The second fundamental form $h$ is a symmetric $F(O)$-bilinear form on $\Gamma (TO)$ with values in $\Gamma (tr(TO))$ and the shape operator $A_{\eta}$ is a linear endomorphism of $\Gamma (TO).$ Then we have

and $\omega\in \Gamma (S(TO^{\bot})).$ Denote the projection of $TO$ on $S(TO)$ by $\overline{P}.$ Then, by using (2.1), (2.3)–(2.5) and having the fact that $\overline{\nabla}$ is a metric connection we obtain

We set

for $\zeta, \eta\in \Gamma (TO)$ and $\xi\in \Gamma (RadTO).$ By using above equations, we obtain

In general, the induced connection $\nabla$ on $O$ is not metric connection. Since $\overline{\nabla}$ is a metric connection, by using (2.3) we get

However, it is important to note that $\nabla^{*}$ is a metric connection on $S(TO).$ From now on, we briefly denote $(O, g, S(TO), S(TO^{\bot}))$ by $O$ in this paper.

3.   Bi-slant lightlike submanifolds

To define bi-slant lightlike submanifolds, we require a lemma:

Lemma 3.1. ^[28] Let $O$ be a $q$-lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$ of index $2q.$ Suppose there exists a screen distribution $S(TO)$ such that $\overline{\varrho}Rad(TO)\subset S(TO)$ and $\overline{\varrho}ltr(TO)\subset S(TO).$ Then, $\overline{\varrho}Rad(TO)\bigcap\overline{\varrho}ltr(TO) = \{0\}$ and any complementary distribution to $\overline{\varrho}Rad(TO)\oplus\overline{\varrho}ltr(TO)$ in $S(TO)$ is Riemannian.

Definition 3.1. Let $O$ be a $q$-lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$ of index $2q$ such that $2q < dim(O).$ Then, we say that $O$ is a bi-slant lightlike submanifold of $\overline{O}$ if the following conditions are satisfied:

(i) $\overline{\varrho}Rad(TO)$ is a distribution on $O$ such that $Rad(TO)\bigcap\overline{\varrho}RadTO = \{0\}$;

(ii) there exists non-degenerate orthogonal distributions $D$, $D_{1}$ and $D_{2}$ on $O$ such that

(iii) the distribution $D$ is an invariant distribution, i.e., $\overline{\varrho}D = D;$

(iv) the distribution $D_{1}$ is slant with angle $\Psi_{1}(\neq0)$, i.e., for each $\zeta\in O$ and each non-zero vector $\zeta\in(D_{1})_{\zeta}$, the angle $\Psi_{1}$ between $\overline{\varrho}\zeta$ and the vector space $(D_{1})_{\zeta}$ is a non-zero constant, which is independent of the choice of $\zeta\in O$ and $\zeta\in(D_{1})_{\zeta}$;

(v) the distribution $D_{2}$ is slant with angle $\Psi_{2}(\neq0)$, i.e., for each $\zeta\in O$ and each non-zero vector $\zeta\in(D_{2})_{\zeta}$, the angle $\Psi_{2}$ between $\overline{\varrho}\zeta$ and the vector space $(D_{2})_{\zeta}$ is a non-zero constant, which is independent of the choice of $\zeta\in O$ and $\zeta\in(D_{2})_{\zeta}$.

These constant angle $\Psi_{1}$ and $\Psi_{2}$ are called the slant angles of distributions $D_{1}$ and $D_{2}$ respectively. A bi-slant lightlike submanifold is said to be proper if $D_{1}\neq\{0\}, D_{2}\neq\{0\}$ and $\Psi_{1}\neq \frac{\pi}{2}$, $\Psi_{2}\neq \frac{\pi}{2}$.

From the above definition, we have the following decomposition

In particular, we have

(i) If $D = 0$, and any one of $D_{1}$ and $D_{2}$ is zero, then $O$ is a slant lightlike submanifold;

(ii) If $D\neq 0$, and any one of $D_{1}$ and $D_{2}$ is zero, then $O$ is a semi-slant lightlike submanifold;

(iii) If $D\neq 0$ and $\Psi_{1} = \frac{\pi}{2}, \Psi_{2} = \frac{\pi}{2}$, then $O$ is CR-lightlike submanifold.

Thus, the above new class of lightlike submanifolds of a golden semi-Riemannian manifold includes slant, semi-slant and Cauchy-Riemann lightlike submanifolds as its subcases.

Example 3.1. Let $(R_{2}^{16}, \overline{g})$ be a semi-Euclidean space of signature $(-, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +)$ with respect to the canonical basis $\{\partial \zeta_{1}, \partial \zeta_{2}, \partial \zeta_{3}, \partial \zeta_{4}, \partial \zeta_{5}, \partial \zeta_{6}, \partial \zeta_{7}, \partial \zeta_{8}, \partial \zeta_{9}, \partial \zeta_{10}, \partial \zeta_{11}, \partial \zeta_{12}, \partial \zeta_{13}, \partial \zeta_{14}, \partial \zeta_{15}, \partial \zeta_{16}\}$ with the golden structure $\overline{\varrho}$ defined by

where $\tau = \frac{1+\sqrt{5}}{2}$ and $\overline\tau = \frac{1-\sqrt{5}}{2}$ are the roots of $\zeta^{2}-\zeta-1 = 0.$ Thus, $\overline{\varrho}^{2} = \overline{\varrho}+I$ and $\overline{\varrho}$ is a golden structure on $R_{2}^{16}$.

Let $O$ be a 9-dimensional submanifold of $(R_{2}^{16}, \overline{g})$ given by

Then, $TO$ is spanned by $\varsigma_{1}, \varsigma_{2}, \varsigma_{3}, \varsigma_{4}, \varsigma_{5}, \varsigma_{6}, \varsigma_{7}, \varsigma_{8}, \varsigma_{9}$, where

This implies that $Rad(TO) = span\{\varsigma_{1}\}$ and $S(TO) = span\{\varsigma_{2}, \varsigma_{3}, \varsigma_{4}, \varsigma_{5}, \varsigma_{6}, \varsigma_{7}, \varsigma_{8}, \varsigma_{9}\}$.

Now, $ltr(TO)$ is spannned by

and $S(TO^{\bot})$ is spanned by

Now, $\overline{\varrho}\varsigma_{1} = \varsigma_{2}, \overline{\varrho}\varpi = \varsigma_{3}$ and $\overline{\varrho}\varsigma_{4} = \overline{\tau}\varsigma_{4}, \overline{\varrho}\varsigma_{5} = \overline{\tau}\varsigma_{5},$ which means $D$ is invariant, i.e., $\overline{\varrho}D = D$ and $D = span\{\varsigma_{4}, \varsigma_{5}\}$ and $D_{1} = span\{\varsigma_{6}, \varsigma_{7}\}, D_{2} = span\{\varsigma_{8}, \varsigma_{9}\}$ are slant distributions with slant angles $\Psi_{1} = arccos(\frac{4}{\sqrt{21}})$ and $\Psi_{2} = arccos(\frac{1+\sqrt{5}}{\sqrt{2(3+\sqrt{5})}})$ respectively. Hence, $O$ is a bi-slant 1-lightlike submanifold of $R_{2}^{16}$.

Example 3.2. Let $(R_{4}^{16}, \overline{g})$ be a semi-Euclidean space of signature $(+, +, -, -, +, +, +, +, +, +, +, -, +, +, +, -)$ with respect to the canonical basis $\{\partial \zeta_{1}, \partial \zeta_{2}, \partial \zeta_{3}, \partial \zeta_{4}, \partial \zeta_{5}, \partial \zeta_{6}, \partial \zeta_{7}, \partial \zeta_{8}, \partial \zeta_{9}, \partial \zeta_{10}, \partial \zeta_{11}, \partial \zeta_{12}, \partial \zeta_{13}, \partial \zeta_{14}, \partial \zeta_{15}, \partial \zeta_{16}\}$ with the golden structure $\overline{\varrho}$ defined by

where $\tau = \frac{1+\sqrt{5}}{2}$ and $\overline\tau = \frac{1-\sqrt{5}}{2}$ are the roots of $\zeta^{2}-\zeta-1 = 0.$ Thus, $\overline{\varrho}^{2} = \overline{\varrho}+I$ and $\overline{\varrho}$ is a golden structure on $R_{4}^{16}$.

Let $O$ be a 9-dimensional submanifold of $(R_{4}^{16}, \overline{g})$ given by

Then, $TO$ is spanned by $\varsigma_{1}, \varsigma_{2}, \varsigma_{3}, \varsigma_{4}, \varsigma_{5}, \varsigma_{6}, \varsigma_{7}, \varsigma_{8}, \varsigma_{9}$, where

This implies that $Rad(TO) = span\{\varsigma_{1}\}$ and $S(TO) = span\{\varsigma_{2}, \varsigma_{3}, \varsigma_{4}, \varsigma_{5}, \varsigma_{6}, \varsigma_{7}, \varsigma_{8}, \varsigma_{9}\}$.

Now, $ltr(TO)$ is spannned by

and $S(TO^{\bot})$ is spanned by

Now, $\overline{\varrho}\varsigma_{1} = \varsigma_{2}, \overline{\varrho}\varpi = \varsigma_{3}$ and $\overline{\varrho}\varsigma_{4} = \tau\varsigma_{4}, \overline{\varrho}\varsigma_{5} = \overline{\tau}\varsigma_{5},$ which means $D$ is invariant, i.e., $\overline{\varrho}D = D$ and $D = span\{\varsigma_{4}, \varsigma_{5}\}$ and $D_{1} = span\{\varsigma_{6}, \varsigma_{8}\}, D_{2} = span\{\varsigma_{7}, \varsigma_{9}\}$ are slant distributions with slant angles $\Psi_{1} = arccos(-\frac{1}{\sqrt{6}})$ and $\Psi_{2} = arccos(\frac{1-\sqrt{5}}{\sqrt{2(3-\sqrt{5})}})$ respectively. Hence, $O$ is a bi-slant 1-lightlike submanifold of $R_{4}^{16}$.

Further, for any vector field $\zeta$ tangent to $O,$ we put

where $f\zeta$ and $F\zeta$ are tangential and transversal parts of $\overline{\varrho}\zeta$ respectively. We denote the projections on $Rad(TO), \overline{\varrho}Rad(TO), \overline{\varrho}ltr(TO), D, D_{1}$ and $D_{2}$ in $TO$ by $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ and $P_{6}$ respectively. Similarly, we denote the projections of $tr(TO)$ on $ltr(TO)$ and $S(TO^{\bot})$ by $Q_{1}$ and $Q_{2}$ respectively. Thus, for any $\zeta\in\Gamma(TO),$ we get

Now, applying $\overline{\varrho}$ to above, we get

which gives

where $\overline{\varrho}P_{2}\zeta = K_{1}\overline{\varrho}P_{2}\zeta+K_{2}\overline{\varrho}P_{2}\zeta, \overline{\varrho}P_{3}\zeta = L_{1}\overline{\varrho}P_{3}\zeta+L_{2}\overline{\varrho}P_{3}\zeta$ and $fP_{5}\zeta, fP_{6}\zeta$ (resp. $FP_{5}\zeta, FP_{6}\zeta)$ denotes the tangential (resp. transversal) component of $\overline{\varrho}P_{5}\zeta$ and $\overline{\varrho}P_{6}\zeta$. Thus, we get $\overline{\varrho}P_{1}\zeta\in\Gamma (\overline{\varrho}Rad(TO)), K_{1}\overline{\varrho}P_{2}\zeta\in\Gamma (RadTO), K_{2}\overline{\varrho}P_{2}\zeta\in\Gamma(\overline{\varrho}Rad(TO)), L_{1}\overline{\varrho}P_{3}\zeta\in\Gamma (ltr(TO)), L_{2}\overline{\varrho}P_{3}\zeta\in\Gamma(\overline{\varrho}ltr(TO)), \overline{\varrho}P_{4}\zeta\in\Gamma(\overline{\varrho}D), fP_{5}\zeta\in\Gamma (D_{1}), fP_{6}\zeta\in\Gamma (D_{2})$ and $FP_{5}\zeta, FP_{6}\zeta\in\Gamma (S(TO^{\bot})).$ Also, for any $\omega\in\Gamma (tr(TO)),$ we have

Applying $\overline{\varrho}$ to above, we obtain

which gives

where $BQ_{2}^{'}\omega, BQ_{2}^{''}\omega$(resp. $CQ_{2}^{'}\omega, CQ_{2}^{''}\omega$) denote the tangential(resp. transversal) component of $\overline{\varrho}Q_{2}\omega$. Thus, we get $\overline{\varrho}Q_{1}\omega\in\Gamma (\overline{\varrho}ltr(TO)), \; BQ_{2}^{'}\omega\in\Gamma(D_{1}), \; BQ_{2}^{''}\omega\in\Gamma(D_{2})$ and $CQ_{2}^{'}\omega, CQ_{2}^{''}\omega\in\Gamma (S(TO^{\bot}))$.

Theorem 3.2. Let $O$ be a $q$-lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$ of index $2q$. Then, $O$ is a bi-slant lightlike submanifold iff

(i) there exist a distribution $\overline{\varrho}Rad(TO)$ on $O$ such that $Rad(TO)\bigcap\overline{\varrho}Rad(TO) = \{0\}$;

(ii) there exist a screen distribution $S(TO)$ which can be splitted as

such that $D$ is an invariant distribution on $O$, i.e., $\overline{\varrho}D = D$;

(iii) there exist a constant $\kappa_{1}\in [0, 1)$ such that $f^{2}\zeta = \kappa_{1}(\overline{\varrho}+ I)\zeta$, $\forall \; \zeta\in\Gamma(D_{1})$;

(iv) there exist a constant $\kappa_{2}\in [0, 1)$ such that $f^{2}\zeta = \kappa_{2}(\overline{\varrho}+ I)\zeta$, $\forall \; \zeta\in\Gamma(D_{2})$.

In that case, $\kappa_{1} = cos^{2}\Psi_{1}$ and $\kappa_{2} = cos^{2}\Psi_{2}$, where $\Psi_{1}$ and $\Psi_{2}$ represents the slant angles of $D_{1}$ and $D_{2}$ respectively.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the distribution $D$ is invariant with respect to $\overline{\varrho}$ and $\overline{\varrho}Rad(TO)$ is a distribution on $O$ such that $Rad(TO)\bigcap\overline{\varrho}Rad(TO) = \{0\}$.

For any $\zeta\in\Gamma(D_{1})$, we have

Since, $O$ is a bi-slant lightlike submanifold, $cos^{2}\Psi_{1} = \kappa_{1}$ (constant)$\in[0, 1)$ and therefore from (3.5), we have

For all $\zeta\in\Gamma(D_{1})$,

Since, $(f^{2}-\kappa_{1}\overline{\varrho}^{2})\zeta\in\Gamma (D_{1})$ and the induced metric $g = g|_{D_{1}\times D_{1}}$ is non-degenerate (positive definite), from (3.6), we have

This proves (iii).

Suppose for any $\zeta\in\Gamma(D_{2})$, we have

Now, by using similar steps as in proof of (iii), we obtain

which proves (iv).

Conversely, suppose that conditions $(i)$–$(iv)$ are satisfied. From $(iii)$, we have $f^{2}\zeta = \kappa_{1}\overline{\varrho}^{2}\zeta$ for all $\zeta\in\Gamma(D_{1})$, where $\kappa_{1}$(constant)$\in[0, 1)$.

Now,

Using (3.7), we have

Using (3.4), we have

Further, from (iv) we have $f^{2}\zeta = \kappa_{2}\overline{\varrho}^{2}\zeta$ for all $\zeta\in\Gamma(D_{2})$, where $\kappa_{2}$(constant)$\in[0, 1)$. Now, by proceeding in the same way as above, we get $cos^{2}\Psi_{2} = \kappa_{2}(constant)$. This completes the proof. Hence, $O$ is a bi-slant lightlike submanifold. □

Theorem 3.3. Let $O$ be a $q$-lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$ of index $2q$. Then, $O$ is a bi-slant lightlike submanifold iff

(i) there exist a distribution $\overline{\varrho}Rad(TO)$ on $O$ such that $Rad(TO)\bigcap\overline{\varrho}Rad(TO) = \{0\}$;

such that $D$ is an invariant distribution on $O$, i.e., $\overline{\varrho}D = D$;

(ii) there exist a constant $\nu_{1}\in [0, 1)$ such that $BF\zeta = \nu_{1}(\overline{\varrho}+ I)\zeta$, $\forall \; \zeta\in\Gamma(D_{1})$;

(iii) there exist a constant $\nu_{2}\in [0, 1)$ such that $BF\zeta = \nu_{2}(\overline{\varrho}+ I)\zeta$, $\forall \; \zeta\in\Gamma(D_{2})$.

In that case, $\nu_{1} = \sin^{2}\Psi_{1}$ and $\nu_{2} = \sin^{2}\Psi_{2}$, where $\Psi_{1}$ and $\Psi_{2}$ represents the slant angles of $D_{1}$ and $D_{2}$, respectively.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the distribution $D$ is invariant with respect to $\overline{\varrho}$ and $\overline{\varrho}Rad(TO)$ is a distribution on $O$ such that $Rad(TO)\bigcap\overline{\varrho}Rad(TO) = \{0\}$.

Now, $\forall$ vector field $\zeta\in\Gamma(D_{1})$, we have

where $f\zeta$ and $F\zeta$ represents the tangential and transversal parts of $\overline{\varrho}\zeta$ respectively. Applying $\overline{\varrho}$ to (3.10), we get

Now, comparing the tangential components, we get

Since, $O$ is a bi-slant lightlike submanifold, $f^{2}\zeta = \kappa_{1}\overline{\varrho}^{2}\zeta, \; \; \forall\; \; \zeta\in\Gamma(D_{1})$, where $\kappa_{1}$(constant)$\in[0, 1)$. From (3.11), we get

where $1-\kappa_{1} = \nu_{1}$(constant)$\in[0, 1)$.

This proves $(iii)$.

Suppose for any $\zeta\in \Gamma(D_{2})$, we have

where $f\zeta$ and $F\zeta$ are the tangential and transversal parts of $\overline{\varrho}\zeta$ respectively.

Now, by using similar steps as in proof of (iii), we obtain

where $1-\kappa_{2} = \nu_{2}$(constant)$\in[0, 1)$.

This proves (iv).

Conversely, suppose that conditions (i)–(iv) are satisfied. From (3.11), we have

where $1-\nu_{1} = \kappa_{1}$(constant)$\in[0, 1)$.

Furthermore, from (iv) we have $BF\zeta = \nu_{2}\overline{\varrho}^{2}\zeta$, for all $\zeta\in\Gamma(D_{2})$, where $\nu_{2}$(constant)$\in[0, 1)$. Now, by proceeding in the same way as above, we get

where $1-\nu_{2} = \kappa_{2}$(constant)$\in[0, 1)$. Now, the proof follows from Theorem 3.2.

Hence, $O$ is a bi-slant lightlike submanifold. □

Corollary 3.1. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then for any slant distribution $D^{'}$ of $O$ with slant angle $\Psi$, we have

$\forall \; \zeta, \eta\in\Gamma(D^{'})$.

The proof of the above corollary follows by using similar steps as in the proof of Corollary $3.1$ of ^[17].

Theorem 3.4. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the integrability of $Rad(TO)$ holds iff

(i) $\overline{g}(h^{l}(\zeta, \overline{\varrho}\eta), \xi) = \overline{g}(h^{l}(\eta, \overline{\varrho}\zeta), \xi)$;

(ii) $\overline{g}(h^{*}(\zeta, \overline{\varrho}\eta), \varpi) = \overline{g}(h^{*}(\eta, \overline{\varrho}\zeta), \varpi)$;

(iii) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \overline{J}\varsigma) = \overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \varsigma)$;

(iv) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, f\varsigma_{1})+ \overline{g}(h^{s}(\zeta, \overline{\varrho}\eta)-h^{s}(\eta, \overline{\varrho}\zeta), F\varsigma_{1}) = \overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \varsigma_{1})$;

(v) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, f\varsigma_{2})+ \overline{g}(h^{s}(\zeta, \overline{\varrho}\eta)-h^{s}(\eta, \overline{\varrho}\zeta), F\varsigma_{2}) = \overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \varsigma_{2})$

$\forall \; \zeta, \eta, \xi \in\Gamma(RadTO), \varsigma\in\Gamma(D), \varsigma_{1}\in\Gamma(D_{1}), \varsigma_{2}\in\Gamma(D_{2})$ and $\varpi\in\Gamma(ltr(TO)).$

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. $Rad(TO)$ is integrable iff

$\forall \; \zeta, \eta, \xi \in\Gamma(RadTO), \varsigma\in\Gamma(D), \varsigma_{1}\in\Gamma(D_{1}), \varsigma_{2}\in\Gamma(D_{2})$ and $\varpi\in\Gamma(ltr(TO)). \; \overline{\nabla}$ being a metric connection and using $(2.3), (2.6), (2.9)$ and $(3.1)$, we obtain

Now, the proof follows from (3.17)–(3.21). □

Theorem 3.5. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the integrability of $\overline{\varrho}Rad(TO)$ holds iff

(i) $\overline{g}(h^{l}(\overline{\varrho}\zeta, \eta)-h^{l}(\overline{\varrho}\eta, \zeta), \overline{\varrho}\xi) = - \overline{g}(h^{l}(\overline{\varrho}\zeta, \eta)-h^{l}(\overline{\varrho}\eta, \zeta), \xi)$;

(ii) $\overline{g}(A_{\zeta}^{*}\overline{\varrho}\eta, \overline{\varrho}\varsigma) = \overline{g}(A_{\eta}^{*}\overline{\varrho}\zeta, \overline{\varrho}\varsigma)$;

(iii) $\overline{g}(A_{\zeta}^{*}\overline{\varrho}\eta-A_{\eta}^{*}\overline{\varrho}\zeta, f\varsigma_{1}) = \overline{g}(h^{s}(\overline{\varrho}\eta, \zeta)-h^{s}(\overline{\varrho}\zeta, \eta), F\varsigma_{1})$;

(iv) $\overline{g}(A_{\zeta}^{*}\overline{\varrho}\eta-A_{\eta}^{*}\overline{\varrho}\zeta, f\varsigma_{2}) = \overline{g}(h^{s}(\overline{\varrho}\eta, \zeta)-h^{s}(\overline{\varrho}\zeta, \eta), F\varsigma_{2})$;

(v) $\overline{g}(A_{\varpi}\overline{\varrho}\zeta, \overline{\varrho}\eta) = \overline{g}(A_{\varpi}\overline{\varrho}\eta, \overline{\varrho}\zeta)$

$\forall \; \zeta, \eta, \xi \in\Gamma(RadTO), \varsigma\in\Gamma(D), \varsigma_{1}\in\Gamma(D_{1}), \varsigma_{2}\in\Gamma(D_{2})$ and $\varpi\in\Gamma(ltr(TO)).$

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. $\overline{\varrho}Rad(TO)$ is integrable iff

$\forall \; \zeta, \eta, \xi \in\Gamma Rad(TO), \varsigma\in\Gamma(D), \varsigma_{1}\in\Gamma(D_{1}), \varsigma_{2}\in\Gamma(D_{2})$ and $\varpi\in\Gamma(ltr(TO)). \; \overline{\nabla}$ being a metric connection and using $(2.3), (2.6), (2.7), (2.10)$ and $(3.1)$, we obtain

Now, the proof follows from (3.22)–(3.26). □

Theorem 3.6. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the integrability of $\overline{\varrho}ltr(TO)$ holds iff

(i) $\overline{g}(A_{\varpi_{1}}\overline{\varrho}\varpi_{2}-A_{\varpi_{2}}\overline{\varrho}\varpi_{1}, \overline{\varrho}\varpi) = -\overline{g}(A_{\varpi_{1}}\overline{\varrho}\varpi_{2}-A_{\varpi_{2}}\overline{\varrho}\varpi_{1}, \varpi)$;

(ii) $\overline{g}(A_{\varpi_{1}}\overline{\varrho}\varpi_{2}, \overline{\varrho}\varsigma) = \overline{g}(A_{\varpi_{2}}\overline{\varrho}\varpi_{1}, \overline{\varrho}\varsigma)$;

(iii) $\overline{g}(A_{\varpi_{1}}\overline{\varrho}\varpi_{2}-A_{\varpi_{2}}\overline{\varrho}\varpi_{1}, f\varsigma_{1}) = \overline{g}(D^{s}(\overline{\varrho}\varpi_{2}, \varpi_{1})-D^{s}(\overline{\varrho}\varpi_{1}, \varpi_{2}), F\varsigma_{1})$;

(iv)$\overline{g}(A_{\varpi_{1}}\overline{\varrho}\varpi_{2}-A_{\varpi_{2}}\overline{\varrho}\varpi_{1}, f\varsigma_{2}) = \overline{g}(D^{s}(\overline{\varrho}\varpi_{2}, \varpi_{1})-D^{s}(\overline{\varrho}\varpi_{1}, \varpi_{2}), F\varsigma_{2})$;

(v) $\overline{g}(A_{\varpi}\overline{\varrho}\varpi_{1}, \overline{\varrho}\varpi_{2}) = \overline{g}(A_{\varpi}\overline{\varrho}\varpi_{2}, \overline{\varrho}\varpi_{1})$

$\forall \; \varpi_{1}, \varpi_{2}, \varpi\in \Gamma(ltr(TO)), \varsigma\in\Gamma(D), \varsigma_{1}\in\Gamma({D}_{1})$ and $\varsigma_{2}\in\Gamma(D_{2})$.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. $\overline{\varrho}ltr(TO)$ is integrable iff

$\forall \; \varpi_{1}, \varpi_{2}, \varpi \in\Gamma(ltr(TO)), \varsigma\in\Gamma(D), \varsigma_{1}\in\Gamma(D_{1})$ and $\varsigma_{2}\in\Gamma(D_{2}). \; \overline{\nabla}$ being a metric connection and using $(2.3), (2.6), (2.7), (2.9)$ and $(3.1)$, we obtain

The proof follows from (3.27)–(3.31). □

Theorem 3.7. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the integrability of $D$ holds iff

(i) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, f\varsigma_{1})+ \overline{g}(h^{s}(\zeta, \overline{\varrho}\eta)-h^{s}(\eta, \overline{\varrho}\zeta), F\varsigma_{1}) = \overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \varsigma_{1})$;

(ii) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, f\varsigma_{2})+ \overline{g}(h^{s}(\zeta, \overline{\varrho}\eta)-h^{s}(\eta, \overline{\varrho}\zeta), F\varsigma_{2}) = \overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \varsigma_{2})$;

(iii) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta-\nabla_{\eta}^{*}\overline{\varrho}\zeta, \overline{\varrho}\varpi) = \overline{g}(h^{*}(\zeta, \overline{\varrho}\eta)-h^{*}(\eta, \overline{\varrho}\zeta), \varpi)$;

(iii) $\overline{g}(A_{\varpi}\zeta, \overline{\varrho}\eta) = \overline{g}(A_{\varpi}\eta, \overline{\varrho}\zeta)$; $\forall \; \zeta, \eta\in \Gamma(D), \varsigma_{1}\in\Gamma(D_{1}), \varsigma_{2}\in\Gamma(D_{2})$ and $\varpi\in\Gamma(ltr(TO))$.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. $D$ is integrable iff

$\forall \; \zeta, \eta\in \Gamma(D), \varsigma_{1}\in\Gamma(D_{1}), \varsigma_{2}\in\Gamma(D_{2})$ and $\varpi\in\Gamma(ltr(TO))$. $\overline{\nabla}$ being a metric connection and using $(2.3), (2.6), (2.7), (2.9)$ and $(3.1)$, we obtain

The proof follows from (3.32)–(3.35). □

Theorem 3.8. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, any slant distribution $D^{'}$ (in particular $D_{1}, D_{2}$) is integrable iff

(i) $\overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \overline{\varrho}\varsigma)+\overline{g}(\nabla_{\eta}f\zeta-A_{F\zeta}\eta, \varsigma) = \overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \varsigma)+\overline{g}(\nabla_{\eta}f\zeta-A_{F\zeta}\eta, \overline{\varrho}\varsigma)$;

(ii) $\overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \overline{\varrho}\varpi)+ \overline{g}(\nabla_{\eta}f\zeta-A_{F\zeta}\eta, \varpi) = \overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \varpi)+\overline{g}(\nabla_{\eta}f\zeta-A_{F\zeta}\eta, \overline{\varrho}\varpi)$;

(iii) $\overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \varpi) = \overline{g}(\nabla_{\eta}f\zeta-A_{F\zeta}\eta, \varpi)$;

$\forall \; \zeta, \eta\in \Gamma(D^{'})$ (in particular $\Gamma (D_{1}), \Gamma (D_{2})$), $\varsigma\in\Gamma(D)$ and $\varpi\in\Gamma(ltr(TO))$.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. $D^{'}$ is integrable iff

$\forall \; \zeta, \eta\in \Gamma(D^{'})$ (in particular $\Gamma (D_{1}), \Gamma (D_{2})$), $\varsigma\in\Gamma(D)$ and $\varpi\in\Gamma(ltr(TO))$. Then, from $(2.3), (2.6), (2.8)$ and $(3.1)$, we obtain

The proof follows from (3.36)–(3.38). □

4.   Foliations determined by distributions

This section includes the necessary and sufficient conditions for foliations determined by distributions on a bi-slant lightlike submanifold of a golden semi-Riemannian manifold to be totally geodesic.

Definition 4.1. ^[2] A bi-slant lightlike submanifold $O$ of a golden semi-Riemannian manifold $\overline{O}$ is said to be mixed geodesic if its second fundamental form $h$ satisfies $h(\zeta, \eta) = 0$ for all $\zeta\in \Gamma(D_{1})$ and $\eta\in\Gamma(D_{2})$. Thus, $O$ is a mixed geodesic bi-slant lightlike submanifold if $h^{l}(\zeta, \eta) = 0$ and $h^{s}(\zeta, \eta) = 0$ for all $\zeta\in \Gamma(D_{1})$ and $\eta\in \Gamma(D_{2})$.

Theorem 4.1. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, $Rad(TO)$ defines a totally geodesic foliation iff

$\forall \; \zeta, \eta\in \Gamma(Rad(TO))$ and $\varsigma\in\Gamma(S(TO))$.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. The distribution $Rad(TO)$ defines a totally geodesic foliation iff $\nabla_{\zeta}\eta\in \Gamma(Rad(TO)) \; \forall \; \zeta, \eta\in \Gamma(Rad(TO)$. $\overline{\nabla}$ being a metric connection and using (2.3), (2.6), (2.7), (2.8) and (3.1) $\forall \; \zeta, \eta \in\Gamma(Rad(TO)$ and $\varsigma\in\Gamma(S(TO))$, we get

Thus, the theorem is completed. □

Theorem 4.2. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, $D$ defines a totally geodesic foliation iff

(i) $\overline{g}(\nabla_{\zeta}f\varsigma-A_{F\varsigma}\zeta, \overline{\varrho}\eta) = \overline{g}(\nabla_{\zeta}f\varsigma-A_{F\varsigma}\zeta, \eta)$;

(ii) $\overline{g}(\nabla_{\zeta}^{*}\overline{\varrho}\eta, \overline{\varrho}\varpi) = \overline{g}(h^{*}(\zeta, \overline{\varrho}\eta), \varpi)$;

(iii) $h^{*}(\zeta, \overline{\varrho}\eta)$ has no components in $\Gamma(Rad(TO))$

$\forall \; \zeta, \eta\in \Gamma(D)$, $\varsigma\in \Gamma(D^{'})$(in particular $\Gamma(D_{1}), \Gamma(D_{2})$) where $D^{'}$ is any slant distribution and $\varpi\in\Gamma(ltr(TO))$.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. The distribution $D$ defines a totally geodesic foliation iff $\nabla_{\zeta}\eta\in \Gamma(D)$, $\forall \; \zeta, \eta\in \Gamma(D)$. $\overline{\nabla}$ being a metric connection and using (2.3), (2.6), (2.8) and (3.1), $\forall \; \zeta, \eta \in\Gamma(D)$ and $\varsigma\in\Gamma(D^{'})$, we obtain

Now, from $(2.3), (2.6)$ and $(2.9)$, $\forall \; \zeta, \eta\in\Gamma(D)$ and $\varpi\in\Gamma(ltr(TO))$, we obtain

Also, from $(2.2), (2.6)$ and $(2.9)$, $\forall \; \zeta, \eta\in\Gamma(D)$ and $\varpi\in\Gamma(ltr(TO))$, we obtain

which implies

which completes the proof. □

Theorem 4.3. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the slant distribution $D^{'}$(in particular $D_{1}, D_{2}$) defines a totally geodesic foliation iff

(i) $\overline{g}(f\eta, \nabla_{\zeta}\overline{\varrho}\varsigma)+\overline{g}(F\eta, h^{s}(\zeta, \overline{\varrho}\varsigma)) = \overline{g}(\nabla_{\zeta}\overline{\varrho}\varsigma, \eta)$;

(ii) $\overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \overline{\varrho}\varpi) = \overline{g}(\nabla_{\zeta}f\eta-A_{F\eta}\zeta, \varpi)$;

(iii) $\nabla_{\zeta}f\eta-A_{F\eta}\zeta$ has no components in $\Gamma(Rad(TO))$

$\forall \; \zeta, \eta\in \Gamma(D^{'})$ (in particular $\Gamma(D_{1}), \Gamma(D_{2})$), $\varsigma\in\Gamma(D)$ and $\varpi\in\Gamma(ltr(TO))$.

Proof. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. The distribution $D^{'}$ defines a totally geodesic foliation iff $\nabla_{\zeta}\eta\in\Gamma(D^{'}) \; \forall \; \zeta, \eta\in\Gamma(D^{'})$. $\overline{\nabla}$ being a metric connection and using (2.3), (2.6) and (3.1), $\forall \; \zeta, \eta\in\Gamma(D^{'})$ and $\varsigma\in\Gamma(D)$, we obtain

which implies

Now, from (2.3), (2.6), (2.8) and (3.1), $\forall \; \zeta, \eta\in\Gamma(D^{'})$ and $\varpi\in\Gamma(ltr(TO))$, we obtain

which implies

Now, from (2.2), (2.6), (2.8) and (3.1), $\forall \; \zeta, \eta\in\Gamma(D^{'})$ and $\varpi\in\Gamma(ltr(TO))$, we obtain

which gives

This completes the proof. □

Theorem 4.4. Let $O$ be a mixed geodesic bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, the slant distribution $D^{'}$(in particular $D_{1}, D_{2}$) defines a totally geodesic foliation iff

(i) $\overline{g}(f\eta, \nabla_{\zeta}\overline{\varrho}\varsigma) = \overline{g}(\eta, \nabla_{\zeta}\overline{\varrho}\varsigma)$;

(ii) $\overline{g}(\nabla_{\zeta}\overline{\varrho}\varpi, f\eta)+\overline{g}(h^{s}(\zeta, \overline{\varrho}\varpi), F\eta) = \overline{g}(\nabla_{\zeta}\overline{\varrho}\varpi, \eta)$;

(iii) $\nabla_{\zeta}f\eta-A_{F\eta}\zeta$ has no components in $\Gamma(Rad(TO))$

$\forall \; \zeta, \eta\in \Gamma(D^{'})$ (in particular $\Gamma(D_{1}), \Gamma(D_{2})$), $\varsigma\in\Gamma(D)$ and $\varpi\in\Gamma(ltr(TO))$.

Proof. Let $O$ be a mixed geodesic bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$, we have $h^{s}(\zeta, \overline{\varrho}\varsigma) = 0$, $\forall \; \zeta\in\Gamma(D^{'})$ and $\varsigma\in\Gamma(D)$. The distribution $D^{'}$ defines a totally geodesic foliation iff $\nabla_{\zeta}\eta\in\Gamma(D^{'})$. $\overline{\nabla}$ being a metric connection and using (2.3), (2.6) and (3.1), $\forall \; \zeta, \eta\in\Gamma(D^{'})$ and $\varsigma\in\Gamma(D)$, we obtain

which implies

From (2.3), (2.6) and (3.1), $\forall \; \zeta, \eta\in\Gamma(D^{'})$ and $\varpi\in\Gamma(ltr(TO))$, we obtain

which implies

Now, from (2.2), (2.6), (2.8) and (3.1), $\forall \; \zeta, \eta\in\Gamma(D^{'})$ and $\varpi\in\Gamma(ltr(TO))$, we obtain

which gives

This completes the proof. □

Theorem 4.5. Let $O$ be a bi-slant lightlike submanifold of a golden semi-Riemannian manifold $\overline{O}$. Then, $O$ is mixed geodesic iff the following holds

(i) $F(\nabla_{\zeta}f\varsigma-A_{F\varsigma}\zeta) = -C(h^{s}(\zeta, f\varsigma)+\nabla_{\zeta}^{s}F\varsigma)$;

(ii) $h^{l}(\zeta, f\varsigma)+D^{l}(\zeta, F\varsigma) = h^{s}(\zeta, f\varsigma)+\nabla_{\zeta}^{s}F\varsigma$

$\forall \; \zeta\in\Gamma(D)$ and $\varsigma\in\Gamma(D^{'})$ (in particular $\Gamma(D_{1}), \Gamma(D_{2})$).

Proof. From (2.5), (2.6), (2.8), (3.1), (3.2) and (3.3), we obtain

Taking transversal part of above equation, we get

Hence, $h(\zeta, \varsigma) = 0$, iff $(i)$ and $(ii)$ hold, which completes the proof. □

5.   Conclusions

We investigate some interesting results on bi-slant lightlike submanifolds of golden semi-Riemannian manifolds and give two examples on such submanifolds. We also discuss the integrability of distributions on bi-slant lightlike submanifolds. Certain conditions on foliations determined by distributions on bi-slant lightlike submanifolds of golden semi-Riemannian manifolds are derived.

Use of AI tools declaration

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

Acknowledgments

The authors are grateful to the referees and the editor for their valuable suggestions and remarks that definitely improve the paper. The author, Fatemah Mofarreh, expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The authors Muqeem Ahmad and Mobin Ahmad would like to thank Integral University, Lucknow, India, for providing the manuscript number IU/R & D/2022-MCN0001737 for the present research work.

Conflict of interest

The authors declare no conflict of interest.