Analysis of Screen Generic Lightlike Submanifolds in an Indefinite Kaehler Statistical Manifold Endowed with a Quarter-Symmetric Non-Metric Connection

Abstract

This paper introduces the notion of screen generic lightlike submanifolds (SGLSs) of an indefinite Kaehler statistical manifold equipped with a quarter-symmetric non-metric (QSNM) connection, supported by suitable illustrations. Assertions for induced connection on the lightlike submanifold and integrability of the distributions are proved. The characterization theorems on parallelism and geodesicity of the SGLSs are presented. Results for the totally umbilic screen generic lightlike submanifold with a QSNM connection are also established.

1. Introduction

The study of lightlike submanifolds of semi-Riemannian manifolds was initiated by Duggal and Bejancu [1]. This field has since become a significant area of research in submanifold theory, particularly due to its profound implications in theoretical physics, including its relevance in general relativity and spacetime geometry. Lightlike submanifolds naturally arise in various physical models where the induced metric becomes degenerate, making their study both mathematically rich and physically significant. Sahin [2] and Duggal [3] introduced the concept of CR-lightlike submanifolds of an indefinite Kaehler manifold, excluding complex and totally real cases. They further explored SCR-lightlike submanifolds, which include invariant and screen real subcases but are distinct from the CR class [4]. Thereafter, GCR-lightlike submanifolds were proposed, which cover complex, SCR, and CR cases, yet excluding real lightlike curves, thereby leaving room for further refinements in the classification of lightlike submanifolds. Additionally, Refs. [5,6] introduced the notion of a generic lightlike submanifold, which is an extension of the geometry of the half lightlike submanifold of codimension 2, for an indefinite Sasakian and Kaehler manifold, respectively. The notion of generic lightlike submanifolds broadens the scope of lightlike geometry, offering a versatile framework for studying lightlike submanifolds in different geometric settings.

To further generalize this concept, Ref. [7] introduced screen generic lightlike submanifolds, which encompass SCR lightlike submanifolds and generic lightlike submanifolds. (Throughout the entire paper, we express “screen generic lightlike submanifolds” as “SGLSs” and “quarter-symmetric non-metric” as “QSNM). In this context, lightlike submanifolds provide a valuable mathematical framework for modeling and analyzing black holes, null hypersurfaces, and event horizons. Their unique geometric properties make them essential tools in understanding the causal structure of spacetime and other fundamental aspects of modern theoretical physics.

Statistical manifolds play a crucial role in modern differential geometry, as they provide a framework for examining the geometric properties of families of probability distributions. These manifolds have extensively been researched in [8,9,10,11]. Ref. [12] explored the submanifold theory of statistical manifolds, extending their applicability to various geometric and analytical problems. Due to their substantial applications in neural networks and control systems, the lightlike geometry of statistical manifolds has also become an emerging area of interest. Significant contributions have been made upon investigating the properties of lightlike statistical manifolds and integrating the concept of a statistical manifold with an indefinite Kaehler manifold, as shown in [13,14,15] and others.

A linear connection $\tilde{\mathcal{D}}$ on a semi-Riemannian manifold $(\tilde{N},\tilde{\rho })$, introduced in [16], is called a quarter-symmetric connection if its torsion tensor $\tilde{T}$ satisfies

$$\tilde{T}(E,F)=\sigma (F)\overline{\mathfrak{J}}(E)-\sigma (E)\overline{\mathfrak{J}}(F),$$

(1)

where $\overline{\mathfrak{J}}$ is a (1, 1)-tensor field, and $\sigma$ is a 1-form corresponding with a smooth vector field $\zeta$, called the torsion vector field of $\tilde{N}$, where $\sigma (E)=\tilde{\rho }(E,\zeta )$. If the linear connection $\tilde{\mathcal{D}}$ is not a metric connection, then it is called a quarter-symmetric non-metric connection (QSNM). Based on the idea of a QSNM connection for an indefinite Kaehler manifold, in this paper, the geometrical properties of the SGLS of an indefinite Kaehler statistical manifold endowed with a QSNM connection are investigated.

Beginning with a recap of some basic facts about lightlike geometry, the present study demonstrates assertions regarding the induced connection and characterization theorems concerning the integrability and parallelism of distributions in screen generic lightlike submanifolds of an indefinite Kaehler statistical manifold. The conditions for the mixed geodesicity of these submanifolds in relation to structural components are developed. Furthermore, results pertaining to totally umbilic screen generic lightlike submanifolds with QSNM connections are presented.

2. Preliminaries

In the inspection of lightlike submanifolds of semi-Riemannian geometry, several key concepts and structures are essential for understanding their geometric properties. Let us review some of these fundamental notions.

Definition 1. 

A pair $(\overline{\nabla },\tilde{\rho })$ is known as a statistical structure on a semi-Riemannian manifold $\tilde{N}$ if for all $E,F,G\in \Gamma (T\tilde{N})$, the following conditions hold:

1. 

${\overline{\nabla }}_{E}F-{\overline{\nabla }}_{F}E=[E,F]$;

2. 

$({\overline{\nabla }}_E\tilde{\rho })(F,G)=({\overline{\nabla }}_F\tilde{\rho })(E,G)$.

Then, $(\tilde{N},\tilde{\rho },\overline{\nabla })$ is called an indefinite statistical manifold. Also, there exists ${\overline{\nabla }}^{*}$ a dual connection of $\overline{\nabla }$ with respect to $\tilde{\rho }$, satisfying the condition

$$E\tilde{\rho }(F,G)=\tilde{\rho }({\overline{\nabla }}_{E}F,G)+\tilde{\rho }(F,{\overline{\nabla }}_E^{*}G).$$

Additionally, ${({\overline{\nabla }}^{*})}^{*}=\overline{\nabla }$.

Consider a semi-Riemannian manifold $\tilde{N}$ with semi-Riemannian metric $\tilde{\rho }$ of constant index q. Suppose N is a lightlike submanifold of $\tilde{N}$. Here, there $exists$ $Rad(TN)$, known as a radical distribution on N such that $Rad(T{N}_p)=T{N}_p\cap T{N}_p^{\perp },\forall p\in N$, where $T{N}_p$ is the tangent space and $T{N}_p^{\perp }$ is the orthogonal complement at p. This distribution is degenerate and not complementary to $T{N}_p$. So, N is termed as an r-lightlike submanifold of $\tilde{N}$. Now, consider $S(TN)$, known as the screen distribution, i.e.,

$$TN=(Rad(TN)\perp S(TN)).$$

and the distribution $S(T{N}^{\perp })$, called the screen transversal vector bundle, i.e.,

$$T{N}^{\perp }=Rad(TN)\perp S(T{N}^{\perp })$$

As $S(TN)$ is a non-degenerate vector sub-bundle of $T\tilde{N}{\mid }_N$, we have

$$T\tilde{N}{\mid }_N=S(TN)\perp S{(TN)}^{\perp }$$

where $S{(TN)}^{\perp }$ is the complementary orthogonal vector sub-bundle of $S(TN)$ in $T\tilde{N}{\mid }_N$. Let $tr(TN)$ and $ltr(TN)$ be complementary vector bundles to $TN$ in $T\tilde{N}{\mid }_N$ and to $Rad(TN)$ in $S{(T{N}^{\perp })}^{\perp }$. Then, we have $tr(TN)=(ltr(TN)\perp S(T{N}^{\perp }))$, $T\tilde{N}{\mid }_N=(TN\oplus tr(TN))=((Rad(TN)\oplus ltr(TN))\perp S(TN)\perp S(T{N}^{\perp }))$.

Theorem 1 

([1]). Let $(N,\rho ,S(TN),S(T{N}^{\perp }))$ be an r-lightlike submanifold $(\tilde{N},\tilde{\rho })$. Then, there exists a complementary vector bundle $ltr(TN)$ of $Rad(TN)$ in $S{(T{N}^{\perp })}^{\perp }$ and the basis of $\Gamma (ltr(TN){\mid }_U)$ consisting of smooth sections $\{N_1^{\prime },\cdots ,N_r^{\prime }\}$$S{(T{M}^{\perp })}^{\perp }{\mid }_U$ such that

$$\overline{g}(N_i^{\prime },{\xi }_j)={\delta }_{ij},\overline{g}(N_i^{\prime },N_j^{\prime })=0,i,j=0,1,\cdots ,r$$

where $\left\{{\xi }_1,\cdots ,{\xi }_r\right\}$ is a lightlike basis of $\Gamma (RadTM){\mid }_U$.

Let $(N,\rho )$ be a lightlike submanifold of $(\tilde{N},\tilde{\rho },\overline{\nabla },{\overline{\nabla }}^{*})$. The formulas derived from its structure are as follows:

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{E}F& ={\nabla }_{E}F+{\mathfrak{h}}^l(E,F)+{\mathfrak{h}}^s(E,F),{\overline{\nabla }}_E^{*}F={\nabla }_E^{*}F+{\mathfrak{h}}^{*l}(E,F)+{\mathfrak{h}}^{*s}(E,F),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{E}V& =-A_{V}E+D_E^{l}V+D_E^{s}V,{\overline{\nabla }}_E^{*}V=-A_V^{*}E+D_E^{*l}V+D_E^{*s}V,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\overline{\nabla }}_E{N}^{\prime }& =-A_{N^{\prime }}E+{\nabla }_E^l{N}^{\prime }+D^s(E,N^{\prime }),{\overline{\nabla }}_E^{*}{N}^{\prime }=-A_{N^{\prime }}^{*}E+{\nabla }_E^{*l}{N}^{\prime }+D^{*s}(E,N^{\prime }),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{E}W& =-A_{W}E+{\nabla }_E^{s}W+D^l(E,W),{\overline{\nabla }}_E^{*}W=-A_W^{*}E+{\nabla }_E^{*s}W+D^{*l}(E,W).\hfill \end{array}$$

(5)

for any $E,F\in \Gamma (TN)$, $V\in \Gamma (tr(TN))$, $N^{\prime }\in \Gamma (ltr(TN))$, and $W\in \Gamma (S(T{N}^{\perp }))$.

Equations (2)–(5) imply that

$$\tilde{\rho }({\mathfrak{h}}^s(E,F),W)+\tilde{\rho }(F,D^{*l}(E,W))=\tilde{\rho }(F,A_W^{*}E),$$

(6)

In the study of non-degenerate submanifolds, it is a well-established fact that any submanifold of a statistical manifold inherits the statistical structure and is itself a statistical manifold. However, this property does not hold in a lightlike case, and (2) implies that

$$({\nabla }_E\rho )(F,G)-({\nabla }_F\rho )(E,G)=\tilde{\rho }(F,{\mathfrak{h}}^l(E,G))-\tilde{\rho }(E,{\mathfrak{h}}^l(F,G)),$$

and

$$E\rho (F,G)-\rho ({\nabla }_{E}F,G)-\rho (F,{\nabla }_E^{*}Z)=\tilde{\rho }({\mathfrak{h}}^l(E,F),G)+\tilde{\rho }(F,{\mathfrak{h}}^{*l}(E,G)).$$

In taking into account the projection morphism P of $TN$ on $S(TN)$, the decomposition with respect to ∇ and ${\nabla }^{*}$ can be expressed as follows:

$$\begin{array}{cc}\hfill {\nabla }_{E}PF& ={\nabla }_E^{\prime }PF+{\mathfrak{h}}^{\prime }(E,PF),{\nabla }_E^{*}PF={\nabla }_E^{*\prime }PF+{{\mathfrak{h}}^{*}}^{\prime }(E,PF),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\nabla }_E\xi & =-A_{\xi }^{\prime }E+{\nabla }_E^{\prime t}\xi ,{\nabla }_E^{*}\xi =-A_{\xi }^{*\prime }E+{\nabla }_E^{*\prime t}\xi ,\hfill \end{array}$$

(8)

Using (2), (3), (6), and (8), we obtain

$$\begin{array}{cc}\hfill \tilde{\rho }({\mathfrak{h}}^l(E,PF),\xi )& =\rho (A_{\xi }^{*\prime }E,PF),\tilde{\rho }({\mathfrak{h}}^{*l}(E,PF),\xi )=\rho (A_{\xi }^{\prime }E,PF),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \tilde{\rho }({\mathfrak{h}}^{\prime }(E,PF),N^{\prime })& =\rho (A_{N^{\prime }}^{*}E,PF),\tilde{\rho }({{\mathfrak{h}}^{*}}^{\prime }(E,PF),N^{\prime })=\rho (A_{N^{\prime }}E,PF),\hfill \end{array}$$

(10)

As ${\mathfrak{h}}^l$ and ${\mathfrak{h}}^{*l}$ are symmetric, from (9), we obtain

$$\rho (A_{\xi }^{\prime }PE,PF)=\rho (PE,A_{\xi }^{\prime }PF),\rho (A_{\xi }^{*\prime }PE,PF)=\rho (PE,A_{\xi }^{*\prime }PF).$$

Let ${\overline{\nabla }}^{\circ }$ be the Levi–Civita connection with respect to $\tilde{\rho }$. It is given by ${\overline{\nabla }}^{\circ }={\displaystyle \frac{1}{2}}(\overline{\nabla }+{\overline{\nabla }}^{*})$. For $(\tilde{N},\tilde{\rho },\overline{\nabla },{\overline{\nabla }}^{*})$, the difference $(1,2)$ tensor K of $\overline{\nabla }$ and ${\overline{\nabla }}^{\circ }$ is defined as

$$K(E,F)=K_{E}F={\overline{\nabla }}_{E}F-{\overline{\nabla }}_E^{\circ }F,$$

(11)

Also, for $\overline{\nabla }$ and ${\overline{\nabla }}^{\circ }$, we have

$$K(E,F)=K(F,E),\tilde{\rho }(K_{E}F,G)=\tilde{\rho }(F,K_{E}Z),$$

(12)

for any $E,F,G\in \Gamma (TN)$.

Also, from (11), we have

$$\tilde{\rho }({\overline{\nabla }}_{E}F,G)=\tilde{\rho }(K(E,F),G)+\tilde{\rho }({\overline{\nabla }}_E^{\circ }F,G).$$

(13)

Let $(\overline{J},\tilde{\rho })$ be an indefinite almost Hermitian structure, where $\overline{J}$ is an almost complex structure and $\tilde{\rho }$ is a Hermitian metric such that for all $E,F\in \Gamma (T\tilde{N})$,

$${\overline{\mathfrak{J}}}^2=-I,\tilde{\rho }(\overline{\mathfrak{J}}E,\overline{\mathfrak{J}}F)=\tilde{\rho }(E,F).$$

(14)

An indefinite almost Hermitian structure on $\tilde{N}$ is known as an indefinite Kaehler structure if

$$({\overline{\nabla }}_E^{\circ }\overline{\mathfrak{J}})F=0{.}^{″}$$

Definition 2 

([14]). A triplet $(\overline{\nabla }={\overline{\nabla }}^{\circ }+K,\tilde{\rho },\overline{\mathfrak{J}})$ is referred to as an indefinite Kaehler statistical structure on $\tilde{N}$ if the following conditions hold:

(i) 

$(\tilde{\rho },\overline{\mathfrak{J}})$ is an indefinite Kaehler structure on $\tilde{N}$;

(ii) 

$(\overline{\nabla },\tilde{\rho })$ forms a statistical structure on $\tilde{N}$ and

$$K(E,\overline{\mathfrak{J}}F)=-\overline{\mathfrak{J}}K(E,F),$$

(15)

holds for any $E,F\in \Gamma (T\tilde{N})$.

3. Screen Generic Lightlike Submanifolds

In this section, we present and explore the concept of SGLSs within the framework of an indefinite Kaehler statistical manifold. Our investigation delves into the intricate interplay between statistical structures and indefinite Kaehler geometry, highlighting the significant role played by lightlike submanifolds in this setting. To provide a concrete understanding of the theoretical framework, we supplement our analysis with a well-structured example that demonstrates the key properties of SGLSs in an indefinite Kaehler statistical manifold.

Definition 3. 

A real lightlike submanifold N of an indefinite Kaehler statistical manifold $\tilde{N}$ is called an SGLS if it complies with the following conditions:

1. 

$Rad(TN)$ is invariant with respect to $\overline{\mathfrak{J}}$,

$$\overline{\mathfrak{J}}(Rad(TN))=Rad(TN).$$

2. 

There $exists$ $L_{\circ }$ of $S(TN)$ such that

$$L_{\circ }=\overline{\mathfrak{J}}(S(TN))\cap S(TN),$$

where $L_{\circ }$ forms a non-degenerate distribution on N.

Also, there exists a complementary non-degenerate distribution $L^{\prime }$ to $L_{\circ }$ in $S(TN)$ satisfying

$$S(TN)=L_{\circ }\oplus L^{\prime },$$

where

$$\overline{\mathfrak{J}}(L^{\prime })\neg \subseteq S(TN)and\overline{\mathfrak{J}}(L^{\prime })\neg \subseteq S(T{N}^{\perp }).$$

Let $P_{\circ }$, $P_1$, and Q represent the projection morphisms on $L_{\circ }$, $Rad(TN)$, and $L^{\prime }$, respectively.

Then, for $E\in \Gamma (TN)$,

$$\begin{array}{cc}\hfill E& =P_{\circ }E+P_{1}E+QE\hfill \\ \hfill & =PE+QE,\hfill \end{array}$$

(16)

where $L=L_{\circ }\perp Rad(TN)$, L is invariant, and $PE\in \Gamma (L)$, $QE\in \Gamma (L^{\prime })$. From (16), we have

$$\overline{\mathfrak{J}}E=\varphi E+wE,$$

(17)

where $\varphi E$ and $wE$ denote the tangential and normal parts of $\overline{\mathfrak{J}}E$. Also, $\overline{\mathfrak{J}}(L^{\prime })\ne L^{\prime }$ holds. For $F\in \Gamma (L^{\prime })$, we have

$$\overline{\mathfrak{J}}F=\varphi F+wF$$

(18)

such that $\varphi F\in \Gamma (L^{\prime })$ and $wF\in \Gamma (S(T{N}^{\perp }))$.

For $V\in \Gamma (tr(TN))$, we obtain

$$\overline{\mathfrak{J}}V=BV+CV,$$

(19)

where $BV$ represents the tangential, and $CV$ represents the normal parts of $\overline{\mathfrak{J}}V$.

Inspired by [7], we exemplified the theoretical constructs of the $SGLS$ of an indefinite Kaehler statistical manifold, clarifying their geometric and statistical implications.

Example 1. 

Let $\tilde{N}=(R_6^{12},\tilde{\rho })$ represent an indefinite Kaehler manifold, where the metric $\tilde{\rho }$ has the signature $(-,-,-,-,-,-,+,+,+,+,+,+)$ relative to the basis { $\partial s_1,\partial s_2,\partial s_3,\partial s_4,\partial s_5,\partial s_6,\partial s_7,\partial s_8,\partial s_9,\partial s_{10},\partial s_{11},\partial s_{12}$}.

With the standard coordinate system $(s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9,s_{10},s_{11},s_{12})$ of $R_6^{12}$, the map $\overline{\mathfrak{J}}$ is defined as $(s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9,s_{10},s_{11},s_{12})=(-s_2,s_1,-s_4,s_3,-s_6,s_5,-s_8,s_7,-s_{10},s_9,-s_{12},s_{11})$, and it satisfies the property ${\overline{\mathfrak{J}}}^2=-I$.

The triplet $(\overline{\nabla }={\overline{\nabla }}^{\circ }+K,\tilde{\rho },\overline{\mathfrak{J}})$, where K fulfills Equation (12) (referred to in Definition 2), defines an indefinite Kaehler statistical structure on $\tilde{N}$. Next, consider a submanifold N of $R_6^{12}$ described as $E=(0,u_{5}cos\alpha ,-u_5,-u_6,u_{1}cosh\alpha ,u_{2}cosh\alpha ,u_{1}sinh\alpha -u_2,u_1+u_{2}sinh\alpha ,u_{5}sin\alpha ,u_{6}sin\alpha ,sin{u}_{3}sinh{u}_4,cos{u}_{3}cosh{u}_4)$.

Then, N is an SGLS of an indefinite Kaehler statistical manifold $R_6^{12}$.

Example 2. 

Let $\tilde{N}=(R_2^{12},\tilde{\rho })$ represent an indefinite Kaehler manifold, where the metric $\tilde{\rho }$ has the signature $(-,-,+,+,+,+,+,+,+,+,+,+)$ relative to the basis { $\partial s_1,\partial s_2,\partial s_3,\partial s_4,\partial s_5,\partial s_6,\partial s_7,\partial s_8,\partial s_9,\partial s_{10},\partial s_{11},\partial s_{12}$}.

With the standard coordinate system $(s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9,s_{10},s_{11},s_{12})$ of $R_6^{12}$, the map $\overline{\mathfrak{J}}$ is defined as $(s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9,s_{10},s_{11},s_{12})=(-s_2,s_1,-s_4,s_3,-s_{7}cos\alpha -s_{6}sin\alpha ,-s_{8}cos\alpha +s_{5}sin\alpha ,s_{5}cos\alpha +s_{8}sin\alpha ,s_{6}cos\alpha -s_{7}sin\alpha ,-s_{10},s_9,-s_{12},s_{11})$, and it satisfies the property ${\overline{\mathfrak{J}}}^2=-I$.

The triplet $(\overline{\nabla }={\overline{\nabla }}^{\circ }+K,\tilde{\rho },\overline{\mathfrak{J}})$, where K fulfills Equation (12) (referred to in Definition 2), defines an indefinite Kaehler statistical structure on $\tilde{N}$. Next, consider a submanifold N of $R_6^{12}$ described as $E=(u_{1}cosh\alpha -u_2,u_1+u_{2}cosh\alpha ,u_{1}sinh\alpha ,u_{2}sinh\alpha ,sin{u}_{3}sinh{u}_4,sin{u}_{3}cosh{u}_4,-cos{u}_{3}cosh{u}_4,-cos{u}_{3}sinh{u}_4,u_5-u_6,u_5+u_6,0,0)$.

Then, N is an SGLS of an indefinite Kaehler statistical manifold $R_2^{12}$.

4. QSNM Connection

We study the notion of a QSNM connection in an indefinite Kaehler statistical manifold and analyze the associated theory of SGLSs, drawing inspiration from [7]. The characterization results for the integrability of distributions in reference to the basic structure of the SGLS submanifold were derived. The parallelism and geodesicity of distributions with respect to the lightlike second fundamental form, screen second fundamental form, and other components are discussed. Our analysis delves into the fundamental properties and structural aspects of these submanifolds, emphasizing their significance in preserving both the statistical and geometric characteristics of the underlying manifold equipped with QSNM.

Let us consider $\tilde{N}$ as an indefinite Kaehler statistical manifold with QSNM.

Let ${\overline{\nabla }}^{\circ }$ be a Levi–Civita connection on $\tilde{N}$, where ${\overline{\nabla }}^{\circ }={\displaystyle \frac{1}{2}}\{\overline{\nabla }+{\overline{\nabla }}^{*}\}$. We set

$${\tilde{\mathcal{D}}}_{E}F={\overline{\nabla }}_{E}F-K(E,F)+\sigma (F)\overline{\mathfrak{J}}E,$$

(20)

and

$${\tilde{\mathcal{D}}}_{E}F={\overline{\nabla }}_E^{*}F+K(E,F)+\sigma (F)\overline{\mathfrak{J}}E,$$

(21)

Since $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$ are torsion-free, from the relationship betweenhe dual connections, we obtain

$$({\tilde{\mathcal{D}}}_E\tilde{\rho })(F,G)=-\sigma (F)\tilde{\rho }(\overline{\mathfrak{J}}E,G)-\sigma (G)\tilde{\rho }(F,\overline{\mathfrak{J}}E),$$

(22)

and

$${\tilde{T}}^{\tilde{\mathcal{D}}}(E,F)=\sigma (F)\overline{\mathfrak{J}}E-\sigma (E)\overline{\mathfrak{J}}F,$$

(23)

for any $E,F,G\in \Gamma (T\tilde{N})$, where ${\tilde{T}}^{\tilde{\mathcal{D}}}$ represents the torsion tensor of the connection $\tilde{\mathcal{D}}$, and $\sigma$ is a 1-form corresponding to the vector field U on $\tilde{N}$ defined by $\sigma (E)=\tilde{\rho }(E,U)$. So, $\tilde{\mathcal{D}}$ is a QSNM connection. Furthermore, $\tilde{N}$ is equipped with a tensor field $\overline{\mathfrak{J}}$ of type (1, 1). The following relation holds for any $E,F\in \Gamma (T\tilde{N})$:

$${\tilde{\mathcal{D}}}_E\overline{\mathfrak{J}}F=\overline{\mathfrak{J}}{\tilde{\mathcal{D}}}_{E}F+\sigma (F)E+\sigma (\overline{\mathfrak{J}}F)\overline{\mathfrak{J}}E,$$

(24)

Let N be an SGLS of $\tilde{N}$ with a QSNM connection $\tilde{\mathcal{D}}$. Let $\mathcal{D}$ represent the linear connection on N induced by $\tilde{\mathcal{D}}$. Therefore, the corresponding Gauss formula is as follows:

$${\tilde{\mathcal{D}}}_{E}F={\mathcal{D}}_{E}F+{\tilde{\mathfrak{h}}}^l(E,F)+{\tilde{\mathfrak{h}}}^s(E,F),$$

(25)

for any $E,F\in \Gamma (TN)$, where ${\mathcal{D}}_{E}F\in \Gamma (TN)$ and ${\tilde{\mathfrak{h}}}^l$, ${\tilde{\mathfrak{h}}}^s$ represent the lightlike second fundamental form and the screen second fundamental form of N, respectively. Now, using (2) and (25) in (20), we obtain

$${\mathcal{D}}_{E}F={\nabla }_{E}F+\sigma (F)\varphi E-K(E,F),$$

(26)

$${\tilde{\mathfrak{h}}}^l(E,F)={\mathfrak{h}}^l(E,F),{\tilde{\mathfrak{h}}}^s(E,F)={\mathfrak{h}}^s(E,F)+\sigma (F)wE.$$

(27)

Further, from (22), (17), and (25), we obtain

$$({\mathcal{D}}_E\tilde{\rho })(F,G)=\tilde{\rho }({\tilde{\mathfrak{h}}}^l(E,F),G)+\tilde{\rho }(F,{\tilde{\mathfrak{h}}}^l(E,G))-\sigma (F)\tilde{\rho }(\varphi E,G)-\sigma (G)\tilde{\rho }(F,\varphi E),$$

(28)

and

$$T^{\mathcal{D}}(E,F)=\sigma (F)\varphi E-\sigma (E)\varphi F.$$

(29)

for any $E,F,G\in \Gamma (TN)$, where $T^{\mathcal{D}}$ is the torsion tensor of the induced connection $\mathcal{D}$ on N.

The following result holds for a screen generic lightlike submanifold (SGLS):

Theorem 2. 

Let N be an SGLS of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$. Then, the induced connection $\mathcal{D}$ on N is also a QSNM connection.

Suppose that ${\tilde{\mathfrak{h}}}^l$ is identically zero on N. Therefore,

$$({\mathcal{D}}_E\rho )(F,G)=-\sigma (F)\rho (\varphi E,G)-\sigma (G)\rho (F,\varphi E).$$

follows from (28).

Consequently, we arrive to the following outcome.

Theorem 3. 

Let N be an SGLS of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$. Then, the connection $\mathcal{D}$ on N is a quarter-symmetric metric connection if and only if ${\tilde{\mathfrak{h}}}^l$ is identically zero on N and the characteristic vector field $\zeta \in \Gamma (S(T{N}^{\perp }))$ such that $\sigma (E)=\rho (E,\zeta )$.

Corresponding to QSNM connection $\tilde{\mathcal{D}}$, the Weingarten formulae are

$${\tilde{\mathcal{D}}}_E{N}^{\prime }=-{\tilde{A}}_{N^{\prime }}E+{\tilde{\nabla }}_E^l{N}^{\prime }+{\tilde{D}}^s(E,N^{\prime }),$$

(30)

$${\tilde{\mathcal{D}}}_{E}W=-{\tilde{A}}_{W}E+{\tilde{\nabla }}_E^{s}W+{\tilde{D}}^l(E,W),$$

(31)

for any $E,F\in \Gamma (TN)$, $N^{\prime }\in \Gamma (ltr(TN))$, and $W\in \Gamma (S(T{N}^{\perp }))$. Using (4), (5), (30), (31), and (20) and then equating tangential and normal parts, we derive

$${\tilde{A}}_{N^{\prime }}E=A_{N^{\prime }}E-\sigma (N^{\prime })\varphi E+K(E,N^{\prime }),$$

(32)

$${\tilde{\nabla }}_E^l{N}^{\prime }={\nabla }_E^l{N}^{\prime },$$

(33)

$${\tilde{D}}^s(E,N^{\prime })=D^s(E,N^{\prime })+\sigma (N^{\prime })wE,$$

(34)

Consider P as the projection of $TN$ on $S(TN)$. Therefore, for any $E,F\in \Gamma (TN)$,

$${\mathcal{D}}_{E}PF={\mathcal{D}}_E^{\prime }PF+{\tilde{\mathfrak{h}}}^{\prime }(E,PF),{\mathcal{D}}_E\xi =-{\tilde{A}}_{\xi }^{\prime }E+{\tilde{\nabla }}_E^{\prime t}\xi ,$$

(35)

where $({\mathcal{D}}_E^{\prime }PF,{\tilde{A}}_{\xi }^{\prime }E)$ and $({\tilde{\mathfrak{h}}}^{\prime }(E,PF),{\tilde{\nabla }}_E^{\prime t}\xi )$ belong to $S(TN)$ and $Rad(TN)$, respectively. Thus, we obtain

$${\mathcal{D}}_E^{\prime }PF={\nabla }_E^{\prime }PF+\sigma (PF)\varphi E-K(E,PF),$$

(36)

$${\tilde{\mathfrak{h}}}^{\prime }(E,PF)={\mathfrak{h}}^{\prime }(E,PF)$$

(37)

and

$${\tilde{A}}_{\xi }^{\prime }E=A_{\xi }^{\prime }E-\sigma (\xi )\varphi E+K(E,\xi ),$$

(38)

$${\tilde{\nabla }}_E^{\prime t}\xi ={\nabla }_E^{\prime t}\xi ,$$

(39)

Theorem 4. 

Let N be an SGLS of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$. Then, the distribution $L_{\circ }$ is integrable if and only if

$$\rho ({\mathcal{D}}_E^{\prime }\overline{\mathfrak{J}}F-{\mathcal{D}}_F^{\prime }\overline{\mathfrak{J}}E,\varphi G)=\rho (B{\tilde{\mathfrak{h}}}^s(E,\overline{\mathfrak{J}}F)-B{\tilde{\mathfrak{h}}}^s(F,\overline{\mathfrak{J}}E),G),$$

$${\tilde{\mathfrak{h}}}^{\prime }(E,\overline{\mathfrak{J}}F)={\tilde{\mathfrak{h}}}^{\prime }(F,\overline{\mathfrak{J}}E)$$

for $E,F\in \Gamma (L_{\circ })$, $G\in \Gamma (L^{\prime })$ and $N^{\prime }\in \Gamma (ltr(TN))$.

Proof. 

The distribution $L_{\circ }$ is integrable if and only if

$$\rho ([E,F],G)=\tilde{\rho }([E,F],N^{\prime })=0$$

From Equations (23), (24), and (14), we have

$$\begin{array}{ccc}\hfill \rho ([E,F],G)& =& \tilde{\rho }({\tilde{\mathcal{D}}}_{E}F-{\tilde{\mathcal{D}}}_{F}E-\sigma (F)\overline{\mathfrak{J}}E+\sigma (E)\overline{\mathfrak{J}}F,G)\hfill \\ & =& \tilde{\rho }({\tilde{\mathcal{D}}}_E\overline{\mathfrak{J}}F-{\tilde{\mathcal{D}}}_F\overline{\mathfrak{J}}E,\overline{\mathfrak{J}}G)-\sigma (\overline{\mathfrak{J}}F)\tilde{\rho }(\overline{\mathfrak{J}}E,\overline{\mathfrak{J}}G)+\sigma (\overline{\mathfrak{J}}E)\tilde{\rho }(\overline{\mathfrak{J}}F,\overline{\mathfrak{J}}G)\hfill \end{array}$$

Using (25) and (35) along with Definition 3, we derive

$$\begin{array}{ccc}\hfill \rho ([E,F],G)& =& \tilde{\rho }({\mathcal{D}}_E\overline{\mathfrak{J}}F-{\mathcal{D}}_F\overline{\mathfrak{J}}E,\overline{\mathfrak{J}}G)-\tilde{\rho }(\overline{\mathfrak{J}}{\tilde{\mathfrak{h}}}^s(E,\overline{\mathfrak{J}}F)-\overline{\mathfrak{J}}{\tilde{\mathfrak{h}}}^s(F,\overline{\mathfrak{J}}E),G)\hfill \\ & =& \tilde{\rho }({\mathcal{D}}_E^{\prime }\overline{\mathfrak{J}}F-{\mathcal{D}}_F^{\prime }\overline{\mathfrak{J}}E,\varphi G)-\tilde{\rho }(B{\tilde{\mathfrak{h}}}^s(E,\overline{\mathfrak{J}}F)-B{\tilde{\mathfrak{h}}}^s(F,\overline{\mathfrak{J}}E),G)\hfill \end{array}$$

(40)

Similarly,

$$\begin{array}{ccc}\hfill \tilde{\rho }([E,F],N^{\prime })& =& \tilde{\rho }(\overline{\mathfrak{J}}[E,F],\overline{\mathfrak{J}}{N}^{\prime })\hfill \\ & =& \tilde{\rho }({\tilde{\mathcal{D}}}_E\overline{\mathfrak{J}}F-{\tilde{\mathcal{D}}}_F\overline{\mathfrak{J}}E-\sigma (\overline{\mathfrak{J}}F)\overline{\mathfrak{J}}E+\sigma (\overline{\mathfrak{J}}E)\overline{\mathfrak{J}}F,\overline{\mathfrak{J}}{N}^{\prime })\hfill \\ \hfill \tilde{\rho }([E,F],N^{\prime })& =& \tilde{\rho }({\tilde{\mathfrak{h}}}^{\prime }(E,\overline{\mathfrak{J}}F)-{\tilde{\mathfrak{h}}}^{\prime }(F,\overline{\mathfrak{J}}E),\overline{\mathfrak{J}}{N}^{\prime })\hfill \end{array}$$

(41)

From Equations (40) and (41), the proof is complete. □

Theorem 5. 

Let N be an SGLS of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$. Then, the distribution $L^{\prime }$ is integrable if and only if ${\mathcal{D}}_G\varphi V-{\mathcal{D}}_V\varphi G-{\tilde{A}}_{wV}G+{\tilde{A}}_{GZ}V$ has no component on $\Gamma (L_{\circ })$ and $\Gamma (Rad(TN))$ for $G,V\in \Gamma (L^{\prime })$ and $E\in \Gamma (L_{\circ })$.

Proof. 

The distribution $L^{\prime }$ is integrable if and only if

$$\rho ([G,V],E)=\tilde{\rho }([G,V],N^{\prime })=0$$

From Definition 2 and (21), (17), (25), and (31), we have

$$\begin{array}{ccc}\hfill \rho ([G,V],E)& =& \tilde{\rho }(\overline{\mathfrak{J}}{\tilde{\mathcal{D}}}_{G}V-\overline{\mathfrak{J}}{\tilde{\mathcal{D}}}_{V}G-\sigma (V){\overline{\mathfrak{J}}}^{2}G+\sigma (G){\overline{\mathfrak{J}}}^{2}V,\overline{\mathfrak{J}}E)\hfill \\ & =& \tilde{\rho }({\tilde{\mathcal{D}}}_G\overline{\mathfrak{J}}V-{\tilde{\mathcal{D}}}_V\overline{\mathfrak{J}}G-\sigma (\overline{\mathfrak{J}}V)\overline{\mathfrak{J}}G+\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}V,\overline{\mathfrak{J}}E)\hfill \\ & =& \tilde{\rho }({\mathcal{D}}_G\varphi V-{\mathcal{D}}_V\varphi G-{\tilde{A}}_{wV}G+{\tilde{A}}_{wG}V,\overline{\mathfrak{J}}E)\hfill \end{array}$$

This means that ${\mathcal{D}}_G\varphi V-{\mathcal{D}}_V\varphi G-{\tilde{A}}_{wV}G+{\tilde{A}}_{wG}V$ has no component on $\in \Gamma (L_{\circ })$.

Similarly,

$$\tilde{\rho }([G,V],N^{\prime })=\tilde{\rho }({\mathcal{D}}_G\varphi V-{\mathcal{D}}_V\varphi G-{\tilde{A}}_{wV}G+{\tilde{A}}_{wG}V,\overline{\mathfrak{J}}{N}^{\prime })$$

This means that ${\mathcal{D}}_G\varphi V-{\mathcal{D}}_V\varphi G-{\tilde{A}}_{wV}G+{\tilde{A}}_{wG}V$ has no component on $\Gamma (Rad(TN))$. □

Theorem 6. 

Let N be an SGLS of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$. Then, the distribution L is parallel if and only if

${\mathcal{D}}_E^{\prime }\varphi G-{\tilde{A}}_{wG}E+2K(E,\overline{\mathfrak{J}}G)+\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}E$ has no component in $\Gamma (L_{\circ })$, where $F\in L_{\circ }$,

${\tilde{\mathfrak{h}}}^{\prime }(E,\varphi G)+{\tilde{\mathfrak{h}}}^l(E,\varphi G)+{\tilde{D}}^l(E,wG)+2K(E,\overline{\mathfrak{J}}G)=0$, $F\in \Gamma (Rad(TN))$,

for $E,F\in \Gamma (L)$ and $G\in \Gamma (L^{\prime })$.

Proof. 

Employing Equations (20), (17), (25), and (31), we obtain

$$\begin{array}{ccc}\hfill \rho ({\mathcal{D}}_{E}F,G)& =& \tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,G)\hfill \\ & =& \tilde{\rho }(F,{\tilde{\mathcal{D}}}_{E}G-K(E,G)-\sigma (G)\overline{\mathfrak{J}}E)-\rho (K(E,F),G)\hfill \\ & =& \tilde{\rho }(\overline{\mathfrak{J}}F,\overline{\mathfrak{J}}{\tilde{\mathcal{D}}}_{E}G)-\tilde{\rho }(F,\sigma (G)\overline{\mathfrak{J}}E)-2\rho (K(E,F),G)\hfill \\ & =& \rho (\overline{\mathfrak{J}}F,{\mathcal{D}}_E^{\prime }\varphi G-{\tilde{A}}_{wG}E+2K(E,\overline{\mathfrak{J}}G)-\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}E)\hfill \end{array}$$

for $E,F\in \Gamma (L)$ and $G\in \Gamma (L^{\prime })$.

For $F\in \Gamma (L_{\circ })$, the parallelism of L yields the following:

$$0=\rho (\overline{\mathfrak{J}}F,{\mathcal{D}}_E^{\prime }\varphi G-{\tilde{A}}_{wG}E+2K(E,\overline{\mathfrak{J}}G)-\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}E)$$

This means that ${\mathcal{D}}_E^{\prime }\varphi G-{\tilde{A}}_{wG}E+2K(E,\overline{\mathfrak{J}}G)-\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}E$ has no component in $\Gamma (L_{\circ })$. Since $F\in \Gamma (Rad(TN))$,

$$0=\tilde{\rho }(\overline{\mathfrak{J}}F,{\tilde{\mathfrak{h}}}^{\prime }(E,\varphi G)+{\tilde{\mathfrak{h}}}^l(E,\varphi G)+{\tilde{D}}^l(E,wG)+2K(E,\overline{\mathfrak{J}}G))$$

This consequently yields the desired result. □

Theorem 7. 

Let N be an SGLS of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$. Then, distribution $L^{\prime }$ is parallel if and only if

${\mathcal{D}}_G\varphi V-{\tilde{A}}_{wG}V$ has no component in $\Gamma (L_{\circ })$ and $\Gamma (Rad(TN))$,

for $G,V\in \Gamma (L^{\prime })$, $E\in \Gamma (L_{\circ })$, and $N\in \Gamma (ltr(TN))$.

Proof. 

For $G,V\in \Gamma (L^{\prime })$ and $E\in \Gamma (L_{\circ })$, using Equations (17) and (20), we derive

$$\begin{array}{ccc}\hfill \rho ({\mathcal{D}}_{G}V,E)& =& \tilde{\rho }({\tilde{\mathcal{D}}}_{G}V,E)=\tilde{\rho }(\overline{\mathfrak{J}}{\tilde{\mathcal{D}}}_{G}V,\overline{\mathfrak{J}}E)\hfill \\ & =& \tilde{\rho }({\tilde{\mathcal{D}}}_G\overline{\mathfrak{J}}V-\sigma (V)G-\sigma (\overline{\mathfrak{J}}V)\overline{\mathfrak{J}}G,\overline{\mathfrak{J}})\hfill \\ & =& \rho ({\mathcal{D}}_G\varphi V-{\tilde{A}}_{wV}G,\overline{\mathfrak{J}}E)\hfill \end{array}$$

From the parellelism of the distribution $L^{\prime }$, we conclude that ${\mathcal{D}}_G\varphi V-{\tilde{A}}_{wG}V$ has no component in $\Gamma (L_{\circ })$. Similarly,

$$\tilde{\rho }({\mathcal{D}}_{G}V,N)=\tilde{\rho }({\mathcal{D}}_G\varphi V-{\tilde{A}}_{wV}G,\overline{\mathfrak{J}}N)$$

and ${\mathcal{D}}_G\varphi V-{\tilde{A}}_{wG}V$ have no components in $\Gamma (Rad(TN))$. □

We now present characterization theorems on the totally geodesic foliations and parallelism of the distributions for the SGLS of $\tilde{N}$ with a QSNM connection.

Definition 4. 

An SGLS of $\tilde{N}$ with QSNM connection is said to be L-geodesic if $\tilde{\mathfrak{h}}(E,F)=0$; $E,F\in \Gamma (L)$.

Therefore, N is called L-geodesic if ${\tilde{\mathfrak{h}}}^l(E,F)=0$ and ${\tilde{\mathfrak{h}}}^s(E,F)=0$ for any $E,F\in \Gamma (L).$

Also, N is called mixed geodesic if $\tilde{\mathfrak{h}}(E,F)=0$, for any $E\in \Gamma (L)$ and $F\in \Gamma (L^{\prime })$.

Theorem 8. 

For an SGLS N of $\tilde{N}$ with a QSNM connection $\tilde{\mathcal{D}}$, the distribution L defines a totally geodesic foliation in $\tilde{N}$ if and only if N is L-geodesic and L is parallel with respect to $\mathcal{D}$ on N.

Proof. 

We know that L defines a totally geodesic foliation in $\tilde{N}$ if and only if

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,\xi )=\tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,W)=\tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,G)=0$$

for $E,F\in \Gamma (L)$, $\xi \in \Gamma Rad(TN)$, $G\in \Gamma (L^{\prime })$ and $W\in \Gamma (S(T{N}^{\perp }))$.

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,\xi )=\tilde{\rho }({\tilde{\mathfrak{h}}}^l(E,F),\xi )$$

(42)

Also,

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,W)=\tilde{\rho }({\tilde{\mathfrak{h}}}^s(E,F),W)$$

(43)

and

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}F,G)=\tilde{\rho }({\mathcal{D}}_{E}F,G)$$

(44)

Hence, Equations (42)–(44) imply the desired result. □

Theorem 9. 

Let N be an SGLS of $\tilde{N}$ with a QSNM connection $\tilde{\mathcal{D}}$. Then, N is mixed geodesic if and only if the following hold for $E\in \Gamma (L)$,$G\in \Gamma (L^{\prime })$ and $W\in \Gamma (S(T{N}^{\perp }))$:

1. 

$\tilde{\rho }({\tilde{\mathfrak{h}}}^l(E,\varphi G)+{\tilde{D}}^l(E,wG),\xi )=0$;

2. 

$\tilde{\rho }({\tilde{A}}_{wG}E-{\mathcal{D}}_E\varphi G,BW)=\tilde{\rho }({\tilde{\mathfrak{h}}}^s(E,\varphi G)+{\tilde{\nabla }}_E^{s}wG,CW)$.

Proof. 

For $E\in \Gamma (L)$, $G\in \Gamma (L^{\prime })$, and $\xi \in \Gamma (Rad(TN))$,

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}G,\xi )=0$$

Employing Equation (25) along with the mixed geodesicity of N, we obtain

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}G,\xi )=\tilde{\rho }(\overline{\mathfrak{J}}{\tilde{\mathcal{D}}}_{E}G,\overline{\mathfrak{J}}\xi )$$

From Equations (18), (24), and (25), we have

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}G,\xi )=\tilde{\rho }({\tilde{\mathfrak{h}}}^l(E,\varphi G)+{\tilde{D}}^l(E,wG),\xi )-\tilde{\rho }(\sigma (G)E,\xi )-\tilde{\rho }(\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}E,\xi )$$

(45)

Also,

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}G,W)=\tilde{\rho }({\tilde{\mathcal{D}}}_E\overline{\mathfrak{J}}G-\sigma (G)E-\sigma (\overline{\mathfrak{J}}G)\overline{\mathfrak{J}}E,\overline{\mathfrak{J}}W)$$

$$\tilde{\rho }({\tilde{\mathcal{D}}}_{E}G,W)=\tilde{\rho }(-{\tilde{A}}_{wG}E+{\mathcal{D}}_E\varphi G+{\tilde{\mathfrak{h}}}^s(E,\varphi G)+{\tilde{\nabla }}_E^{s}wG,BW+CW)$$

(46)

The result follows from (45) and (46). □

Theorem 10. 

An SGLS N of $\tilde{N}$ with QSNM connection $\tilde{\mathcal{D}}$ is mixed geodesic if and only if the following hold for $E\in \Gamma (L)$, $G\in \Gamma (L^{\prime })$:

1. 

${\tilde{\mathfrak{h}}}^l(E,\varphi G)=-{\tilde{D}}^l(E,wG)$;

2. 

$w({\tilde{A}}_{wG}E-D_E\varphi G)=C({\tilde{\mathfrak{h}}}^s(E,\varphi G)+{\tilde{\nabla }}_E^{s}wG)$.

Proof. 

Employing Equations (18) and (14) in (24), we have

$$\overline{\mathfrak{J}}({\tilde{\mathcal{D}}}_E(\varphi G+wG))=-{\tilde{\mathcal{D}}}_{E}G+\sigma (G)\overline{\mathfrak{J}}E-\sigma (\overline{\mathfrak{J}}G)E$$

From Equations (25), (31), (18), and (19) and then taking the normal part of the resulting equation, we obtain

$$\begin{array}{ccc}\hfill -{\tilde{\mathfrak{h}}}^l(E,G)-{\tilde{\mathfrak{h}}}^s(E,G)& =& -w{\tilde{A}}_{wG}E+w{\mathcal{D}}_E\varphi G+C{\tilde{\mathfrak{h}}}^l(E,\varphi G)+C{\tilde{\mathfrak{h}}}^s(E,\varphi G)\hfill \\ & & +C{\tilde{\nabla }}_E^{s}wG+C{\tilde{D}}^l(E,wG)\hfill \end{array}$$

Therefore, the required result follows from the mixed geodesicity of N. □

Theorem 11. 

Let N be an SGLS of $\tilde{N}$ with a QSNM connection $\tilde{\mathcal{D}}$. Then, for $E\in \Gamma (L_{\circ })$, $G\in \Gamma (L^{\prime })$, we have

$${\mathcal{D}}_{E}G=-\varphi {\mathcal{D}}_E\varphi G+\varphi {\tilde{A}}_{wG}E-B{\tilde{\mathfrak{h}}}^s(E,\varphi G)-B{\tilde{\nabla }}_E^{s}wG+\sigma (G)\overline{\mathfrak{J}}E-\sigma (\overline{\mathfrak{J}}G)E$$

Proof. 

For $E\in \Gamma (L_{\circ })$ and $G\in \Gamma (L^{\prime })$,

$$\begin{array}{ccc}\hfill {\tilde{\mathcal{D}}}_{E}G& =& -\varphi {\mathcal{D}}_E\varphi G-w{\mathcal{D}}_E\varphi G-C{\tilde{\mathfrak{h}}}^l(E,\varphi G)-B{\tilde{\mathfrak{h}}}^s(E,\varphi G)-C{\tilde{\mathfrak{h}}}^s(E,\varphi G)\hfill \\ & & +\varphi {\tilde{A}}_{wG}E+w{\tilde{A}}_{wG}E-B{\tilde{\nabla }}_E^{s}wG-C{\tilde{\nabla }}_E^{s}wG-C{\tilde{D}}^l(E,wG)\hfill \\ & & +\sigma (G)\overline{\mathfrak{J}}E-\sigma (\overline{\mathfrak{J}}G)E\hfill \end{array}$$

follows from Equations (25), (31), (18), and (19). □

By taking the tangential part along with Equation (25), we obtain the required assertion.

The following lemma, presented below, will be useful for the next section:

Lemma 1. 

Let N be an SGLS of $\tilde{N}$ with a QSNM connection $\tilde{\mathcal{D}}$. Then,

$${\mathcal{D}}_E\varphi F-{\tilde{A}}_{wF}E=\varphi {\mathcal{D}}_{E}F+B{\tilde{\mathfrak{h}}}^s(E,F)+\sigma (F)E+\sigma (\overline{\mathfrak{J}}F)\varphi E$$

(47)

$${\tilde{\mathfrak{h}}}^l(E,\varphi F)+{\tilde{D}}^l(E,wF)=C{\tilde{\mathfrak{h}}}^l(E,F)$$

(48)

$${\tilde{\mathfrak{h}}}^s(E,\varphi F)+{\tilde{\nabla }}_E^{s}wF=w{\mathcal{D}}_{E}F+C{\tilde{\mathfrak{h}}}^s(E,F)+\sigma (\overline{\mathfrak{J}}F)wE$$

(49)

for $E,F\in \Gamma (TN)$.

Proof. 

Differentiating (17) with respect to $F\in \Gamma (TN)$ and using Equations (24), (25), and (31), we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}_E\varphi F& +& {\tilde{\mathfrak{h}}}^l(E,\varphi F)+{\tilde{\mathfrak{h}}}^s(E,\varphi F)-{\tilde{A}}_{wF}E+{\tilde{\nabla }}_E^{s}wF+{\tilde{D}}^l(E,wF)\hfill \\ & =& \varphi {\mathcal{D}}_{E}F+w{\mathcal{D}}_{E}F+B{\tilde{\mathfrak{h}}}^s(E,F)+C{\tilde{\mathfrak{h}}}^s(E,F)+\sigma (F)E+\sigma (\overline{\mathfrak{J}}F)\overline{\mathfrak{J}}E\hfill \end{array}$$

By comparing its parts, we obtain (47), (48), and (49). This concludes the proof. □

5. Totally Umbilical Screen Generic Lightlike Submanifold

This section analyzes the structure of a totally umbilical SGLS of $\tilde{N}$ with a QSNM connection. We examine the geometric conditions that characterize totally umbilical SGLSs, focusing on how the QSNM connection influences their extrinsic curvature properties. This exploration provides key insights into the behavior of such submanifolds, further enriching the study of lightlike geometry in the statistical frame.

A lightlike submanifold N of $\tilde{N}$ is known as totally umbilical with respect to $\overline{\nabla }$ (respectively, ${\overline{\nabla }}^{*}$) if $\mathfrak{h}(E,F)=H\tilde{\rho }(E,F)$ (respectively, ${\mathfrak{h}}^{*}(E,F)=H^{*}\tilde{\rho }(E,F)$) for all $E,F\in \Gamma (TN)$, where $H\in \Gamma (tr(TN))$ (respectively, $H^{*}\in \Gamma (tr(TN))$). Here, H (respectively, $H^{*}$) stands for the transversal curvature vector fields of N in $\tilde{N}$ with respect to $\overline{\nabla }$ (respectively, ${\overline{\nabla }}^{*})$.

Following this definition, an SGLS of $\tilde{N}$ with a QSNM connection is known as totally umbilical if there exist smooth vector fields ${\tilde{H}}^l\in \Gamma (ltr(TN))$ and ${\tilde{H}}^s\in \Gamma (S(T{N}^{\perp }))$ such that ${\tilde{\mathfrak{h}}}^l(E,F)={\tilde{H}}^l\tilde{\rho }(E,F)$ and ${\tilde{\mathfrak{h}}}^s(E,F)={\tilde{H}}^s\tilde{\rho }(E,F)$.

Theorem 12. 

Let N be a totally umbilical proper SGLS of $\tilde{N}$ with a QSNM connection. Then, $\tilde{\rho }(E,E)C{H}^s=0$ for $E,F\in \Gamma (L_{\circ })$.

Proof. 

Using Lemma 1, we have

$$\tilde{\rho }(E,\varphi F){\tilde{H}}^s+{\tilde{\nabla }}_E^{s}wF=w{\mathcal{D}}_{E}F+\tilde{\rho }(E,F)C{\tilde{H}}^s\forall E,F\in \Gamma (L_{\circ })$$

By taking $E=F$, we obtain the desired result. □

Lemma 2. 

Let N be a totally umbilical proper SGLS of $\tilde{N}$ with a QSNM connection. Then, $H^l=0$ for $E,F\in \Gamma (L_{\circ })$.

Proof. 

Equation (48) implies that

$$\tilde{\rho }(E,\overline{\mathfrak{J}}F){\tilde{H}}^l=\tilde{\rho }(E,F)C{\tilde{H}}^l$$

(50)

for $E,F\in \Gamma (L_{\circ })$. By replacing E and F in the above equation, we obtain

$$\tilde{\rho }(F,\overline{\mathfrak{J}}E){\tilde{H}}^l=\tilde{\rho }(F,E)C{\tilde{H}}^l$$

(51)

Subtracting (51) from (50), we obtain

$$2\tilde{\rho }(E,\overline{\mathfrak{J}}F){\tilde{H}}^l=0.$$

if we take $E=\overline{\mathfrak{J}}F$. As $L_{\circ }$ is non-degenerate, we obtain the result. □

Theorem 13. 

Let N be a totally umbilical proper SGLS of $\tilde{N}$ with a QSNM connection. If the distribution $L_{\circ }$ is integrable, then N is a totally geodesic SGLS of $\tilde{N}$ with respect to $\tilde{\mathcal{D}}$.

Proof. 

Considering Equation (49) for $E,F\in \Gamma (L_{\circ })$, we obtain

$$w{\mathcal{D}}_{E}F={\tilde{\mathfrak{h}}}^s(E,\varphi F)-C{\mathfrak{h}}^s(E,F)$$

Interchanging E and F and then subtracting, we obtain

$$w{\mathcal{D}}_{E}F-w{\mathcal{D}}_{F}E={\tilde{\mathfrak{h}}}^s(E,\varphi F)-{\tilde{\mathfrak{h}}}^s(F,\varphi E)$$

By utilizing the fact that N is a totally umbilical lightlike submanifold and applying Equation (29), we obtain

$$(\tilde{\rho }(E,\overline{\mathfrak{J}}F)-\tilde{\rho }(F,\overline{\mathfrak{J}}E)){\tilde{H}}^s=w[E,F]$$

Due to the non-degeneracy of $L_{\circ }$, we have ${\tilde{H}}^s=0$. The definition of totally umbilical leads to ${\tilde{\mathfrak{h}}}^s=0$.

Also, from Lemma 2, ${\tilde{H}}^l=0$. This leads to ${\tilde{\mathfrak{h}}}^l=0$. Thus, the result follows. □

6. Concluding Remarks

The study of statistical manifolds is a notable field of research due to its propitious requisitions in inference, neural networks, image analysis, clustering, and other diverse disciplines. The fusion of statistical structures with complex geometric frameworks has been a topic of increasing interest, yet progress in the domain of lightlike geometry within this context remains relatively limited. Given the interplay between statistical manifolds and indefinite Kaehler structures, exploring their intrinsic properties and potential applications presents a compelling research direction. We undertook a detailed investigation into the structure of screen generic lightlike submanifolds within an indefinite Kaehler statistical manifold, particularly focusing on their interaction with the quarter-symmetric non-metric connection. This study presents a geometric framework that envelops both SCR (Screen Cauchy Riemann) and generic lightlike submanifolds in the indefinite Kaehler statistical manifolds endowed with a QSNM, instrumental in preserving statistical properties.

This work lays a foundational framework for further research into complex and contact metric manifolds equipped with specialized connections. Such studies could reveal new geometric and statistical properties that are pivotal for advancements in applied mathematics, machine learning, and theoretical physics. By continuing this avenue of investigation, one could uncover deeper relationships between statistical structures, information geometry, and lightlike geometry, thereby fostering furtherance in both pure and applied mathematics. This study introduces the concept of screen generic lightlike submanifolds of an indefinite Kaehler statistical manifold with respect to a quarter-symmetric non-metric connection, supported by satisfactory expressions. The theory of this introductory study is also supported by examples. Claims for the integrability of the induced connection and distributions on the lightlike submanifold are proven. Characterization theorems on parallelism and geodesicity are presented. Results for a totally umbilic screen generic lightlike submanifold with a quarter-symmetric non-metric connection are also produced. This introductory paper can also be seen as a guide for researchers who want to study different types of lightlike manifolds and obtain new results.