Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model

1.   Introduction

Neural networks is useful for solving numerous complex optimization problems in electromagnetic theory. Applied physicist B. Bartlett presented unsupervised machine learning model for computing approximate electromagnetic field solutions ^[7]. In April 2019, the Event Horizon Telescope (EHT) collaboration released the first image of the shadow of a black hole with the help of deep learning algorithms. This image provides direct evidence for the existence of black holes and general theory of relativity and indirect evidence for the existence of lightlike geometry in the universe ^[19]. A statistical manifold is emerging branch of mathematics that generalizes the Riemannian manifold and is used to model the information; and also uses tools of differential geometry to study statistical inference, information loss and estimation ^[9]. Statistical manifolds are applicable to many areas such as neural networks, machine learning and artificial intelligence. On the other hand, the study of lightlike manifolds is one of the most important research areas in differential geometry with many applications in physics and mathematics, such as general relativity, electromagnetism and black hole theory (Please see ^{[7,8,10,11,19]}).

There exist qualified papers dealing with statistical manifolds and their submanifolds admitting various differentiable structures. In 1975, Efron ^[14] was the first researcher who emphasized the role of differential geometry in statistics and was further studied with the help of differential geometrical tools by Amari ^[1,2]. In 1989, Vos ^[30] obtained fundamental equations for submanifolds of statistical manifolds. Takano define a Sasaki-like statistical manifold and he study Sasaki-like statistical submersion with the property that the curvature tensor with respect to the affine connection of the total space satisfies the condition ^[28]. In ^[29], the authors define the concept of quaternionic Kahler-like statistical manifold and derive the main properties of quaternionic Kahler-like statistical submersions, extending in a new setting some previous results obtained by Takano concerning statistical manifolds endowed with almost complex and almost contact structures. The geometry of hypersurfaces of statistical manifolds was presented by Furuhata in ^[15,16]. Statistical manifolds admitting contact structures or complex structures and submanifolds of these kinds of manifolds were investigated in ^[18,24]. In degenerate case, lightlike hypersurfaces of statistical manifolds were introduced by the Bahadir and Tripathi ^[6]. Also, statistical lightlike hypersurfaces were studied by Jain, Singh and Kumar in ^[20]. In addition, many studies have been conducted in relation to these concepts ^{[3,4,5,21,22,25,26]}.

Motivated by these developments, we dedicate the present study to introduce the lightlike geometry of an indefinite Sasakian statistical manifold. In Section 2, we present basic definitions and results about statistical manifolds and lightlike hypersurfaces. In Section 3, we show that the induced connections on a lightlike hypersurface of a statistical manifold need not be dual and a lightlike hypersurface need not be a statistical manifold. Moreover, we show that the second fundamental forms are not degenerate. We conclude the section with an example. In Section $4$, we defined indefinite Sasakian statistical manifolds and obtain the characterization theorem for indefinite Sasakian statistical manifolds. This section is concluded with two examples. In Section $5$, we consider lightlike hypersurfaces of indefinite Sasakian statistical manifolds. We characterize the parallelness, totaly geodeticity and integrability of some distributions. In this section we also give two examples. In Section $6$, we prove that an invariant lightlike submanifold of indefinite Sasakian statistical manifold is an indefinite Sasakian statistical manifold.

2.   Preliminaries

Let $(\widetilde{M}, \widetilde{g})$ be an $(m+2)$-dimensional semi-Riemannian manifold with ${\rm index}(\widetilde{g}) = q\geq 1$ and $(M, g)$ be a hypersurface of $(\widetilde{M}, \widetilde{g})$ with the induced metric $g$ from $\widetilde{g}$. If the induced metric $g$ on $M$ is degenerate, then $M$ is called a lightlike (null or degenerate) hypersurface.

For a lightlike hypersurface $(M, g)$ of $(\widetilde{M}, \widetilde{g})$, there exists a non-zero vector field $\xi$ on $M$ such that

Here the vector field $\xi$ is called a null vector (^[11,12,13]). The radical or the null space ${\rm{Rad}}\; T_{x}M$ at each point $x\in M$ is defined as

The dimension of ${\rm{Rad}}\; T_{x}M$ is called the nullity degree of $g$. We recall that the nullity degree of $g$ for a lightlike hypersurface is equal to $1$. Since $g$ is degenerate and any null vector being orthogonal to itself, the normal space $T_{x}M^{\perp }$ is a null subspace. Also, we have

The complementary vector bundle ${\rm{S}}(TM)$ of ${\rm{Rad}}\; TM$ in $TM$ is called the screen bundle of $M$. We note that any screen bundle is non-degenerate. Therefore we can write the following decomposition:

Here $\perp$ denotes the orthogonal-direct sum. The complementary vector bundle ${\rm{S}}(TM)^{\perp }$ of ${\rm{S}}(TM)$ in $T\widetilde{M}$ is called the screen transversal bundle. Since ${\rm{Rad}}\; TM$ is a lightlike subbundle of ${\rm{S}}(TM)^{\perp }$, there exists a unique local section $N$ of ${\rm{S}}(TM)^{\perp }$ such that we have

Note that $N$ is transversal to $M$ and $\{\xi, N\}$ is a local frame field of ${\rm{S}}(TM)^{\perp }$ and there exists a line subbundle ${\rm{ltr}}(TM)$ of $T\widetilde{M}$. This set is called the lightlike transversal bundle, locally spanned by $N$. Hence we have the following decomposition:

where $\oplus$ is the direct sum but not orthogonal (^[11,12]).

For details of lightlike submanifolds, we may refer to ^[8,11,12,13].

Definition 1. ^[28] A statistical manifold is a triple ($\widetilde{M}$, $\widetilde{g}$, $\widetilde{D}$) formed of a semi-Riemannian manifold and a torsion free connection $\widetilde{D}$ subject to the following identity

$(\widetilde{D}_{X}\widetilde{g})(Y, Z) = (\widetilde{D}_{Y} \widetilde{g})(X, Z),$ for all $X, Y, Z\in \Gamma(T\widetilde{M})$.

Given statistical manifold ($\widetilde{M}$, $\widetilde{g}$, $\widetilde{D}$) the $\widetilde{g}-$ dual of $\widetilde{D}$, $\widetilde{D}^{\ast }$ is defined by the following identity

It is easy to check that $\widetilde{D}^{\ast }$ is torsion free and $(\widetilde{M}, \widetilde{g}, \widetilde{D}^{\ast })$ is a statistical manifold. Also, an equivalent of Definition 1 according to definition $D^{\ast }$ is as follows:

Definition 2. A statistical manifold is a quadruple $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$ formed of a semi-Riemannian manifold $(\widetilde{M}, \widetilde{g})$ and a pair of torsion free connection $(\widetilde{D}, \widetilde{D}^{\ast })$ subject the following identity

$Z\widetilde{g}(X, Y) = \widetilde{g}(X, \widetilde{D}_{Z}^{\ast }Y)+\widetilde{g}(\widetilde{D}_{Z}X, Y)$, for all $X, Y, Z\in \Gamma(T\widetilde{M})$.

A statistical manifold will be represented by $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$.

If $\widetilde{D}^{0}$ is Levi-Civita connection of $\widetilde{g}$, then we have

If we choose $\widetilde{D}^{\ast } = \widetilde{D}$ in the Eq (2.8), then Levi-Civita connection is obtained.

Lemma 2.1. For statistical manifold $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$, if we set

Then we have

for any $X, Y, Z\in\Gamma(TM)$.

Conversely, for a Riemannian metric $g$, if $\overline{\mathbb{K}}$ satisfies (2.10), the pair $(\widetilde{D} = \widetilde{\nabla}+\overline{\mathbb{K}}, \widetilde{g})$ is a statistical structure on $\widetilde{M}$ ^[18].

3.   Lightlike hypersurface of a statistical manifold

Let $(M, g)$ be a lightlike hypersurface of a statistical manifold $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$. Then, Gauss and Weingarten formulas with respect to dual connections are given as follows:

for all $X, Y\in \Gamma (TM), \; N\in \Gamma (ltrTM)$, where $D_{X}Y$, $\ D_{X}^{\ast }Y$, $A_{N}X$, $\ A_{N}^{\ast }X\in \Gamma (TM)$ and

Here, $D$, $D^{\ast }$, $B$, $B^{\ast }$, ${A}_{N}$ and $A_{N}^{\ast }$ are called the induced connections on $M$, the second fundamental forms and the Weingarten mappings with respect to $\widetilde{D}$ and $\widetilde{D}^{\ast }$, respectively (^[6,15]). Using Gauss formulas and the Eq (2.8), we obtain

From the Eq (3.5), we have the following result.

Theorem 3.1. ^[6] Let $(M, g)$ be a lightlike hypersurface of a statistical manifold $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$ . Then we have the following expressions:

${\bf (i)}$ Induced connections $D$ and $D^{\ast }$ need not be dual.

${\bf (ii)}$ A lightlike hypersurface of a statistical manifold need not be a statistical manifold with respect to the dual connections.

Using Gauss and Weingarten formulas in (3.5), we get

Proposition 3.2. ^[6] Let $(M, g)$ be a lightlike hypersurface of a statistical manifold $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$ . Then the following assertions are true:

${\bf (i)}$ Induced connections $D$ and $D^{\ast }$ are symmetric connection.

${\bf (ii)}$ The second fundamental forms $B$ and $B^{\ast }$ are symmetric.

Let $P$ denote the projection morphism of $\Gamma (TM)$ on $\Gamma (S(TM))$ with respect to the decomposition (2.4). For all $X, Y\in \Gamma (TM)$ and $\xi \in \Gamma (RadTM)$, we have

where $\nabla _{X}PY$ and $\overline{A}_{\xi }X$ belong to $\Gamma (S(TM))$. Also we can say that $\nabla$ and $\overline{\nabla }^{t}$ are linear connections on $\Gamma (S(TM))$ and $\Gamma (RadTM)$, respectively. Here, $\overline{h}$ and $\overline{A}$ are called screen second fundamental form and screen shape operator of $S(TM)$, respectively. If we define

One can show that

Therefore, we have

Here $C(X, PY)$ is called the local screen fundamental form of $S(TM)$.

Similarly, the relations of induced dual objects on $S(TM)$ are given by

Using (3.5), (3.11), (3.13) and Gauss-Weingarten formulas, the relationship between induced geometric objects are given by

Now, using the Eq (3.15), we can state the following result.

Proposition 3.3. ^[6] Let $(M, g)$ be a lightlike hypersurface of a statistical manifold $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$ . Then second fundamental forms $B$ and $B^{\ast }$ are not degenerate.

Additionally, due to $\widetilde{D}$ and $\widetilde{D}^{\ast }$ are dual connections, we obtain

Using (3.17) and (3.18) we get

Example 1. Let $(R_{2}^{4}, \widetilde{g})$ be a $4$-dimensional semi-Euclidean space with signature $(-, -, +, +)$ of the canonical basis $(\partial_{0}, \ldots, \partial_{3})$. Consider a hypersurface $M$ of $R_{2}^{4}$ given by

For simplicity, we set $f = \sqrt{x_{2}^{2}+x_{3}^{2}}$. It is easy to check that $M$ is a lightlike hypersurface whose radical distribution $RadTM$ is spanned by

Then the lightlike transversal vector bundle is given by

It follows that the corresponding screen distribution $S(TM)$ is spanned by

Then, by direct calculations we obtain

for any $X\in\Gamma(TM)$ (see ^[13], Example 2, pp. 48–49).

We define an affine connection $\widetilde{D}$ as follows:

Then we obtain

Then $\widetilde{D}$ and $\widetilde{D}^{*}$ are dual connections. Here, one can easily see that $T^{\widetilde{D}} = 0$ and $\widetilde{D}\widetilde{g} = 0$. Thus, we can easily see that $(R_{2}^{4}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$ is a statistical manifold.

4.   Indefinite Sasakian statistical manifolds

In order to call a differentiable semi-Riemannian manifold $(\widetilde{M}, \widetilde{g})$ of dimension $n = 2m+1$ as practically contact metric one, a $(1, 1)$ tensor field $\widetilde{\varphi}$, a contravariant vector field $\nu$, a $1-$ form $\eta$ and a Riemannian metric $\widetilde{g}$ should be admitted, which satisfy

for all the vector fields $X$, $Y$ on $\widetilde{M}$. When a practically contact metric manifold performs

$\widetilde{M}$ is regarded as an indefinite Sasakian manifold. In this study, we assume that the vector field $\nu$ is spacelike.

Definition 3. Let $(\widetilde{g}, \widetilde{\varphi}, \nu)$ be an indefinite Sasakian structure on $\widetilde{M}$. A quadruplet $(\widetilde{D} = \widetilde{\nabla} + \overline{\mathbb{K}}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is called a indefinite Sasakian statistical structure on $\widetilde{M}$ if $(\widetilde{D}, \widetilde{g})$ is a statistical structure on $\widetilde{M}$ and the formula

holds for any $X, Y \in \Gamma(T\widetilde{M})$. Then $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is said to an indefinite Sasakian statistical manifold.

An indefinite Sasakian statistical manifold will be represented by $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. We remark that if $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is an indefinite Sasakian statistical manifold, so is $(\widetilde{M}, \widetilde{D}^{\ast }, \widetilde{g}, \varphi, \nu)$ ^[17,18].

Theorem 4.1. Let $(\widetilde{M}, \widetilde{D}, \widetilde{g})$ be a statistical manifold and $(\widetilde{g}, \widetilde{\varphi}, \nu)$ an almost contact metric structure on $\widetilde{M}$. $(\widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is an indefinite Sasakian statistical struture if and only if the following conditions hold:

for all the vector fields $X$, $Y$ on $\widetilde{M}$.

Proof. Using (2.9) we get

for all the vector fields $X$, $Y$ on $\widetilde{M}$. If we consider Definition 3 and the Eq (4.4), we have the formula (4.7). If we write $\widetilde{D}^{*}$ instead of $\widetilde{D}$ in (4.7), we have

Substituting $\nu$ for $Y$ in (4.10), we have the Eq (4.8).

Conversely using (4.7), we obtain

Assume (4.2) and (4.8) as well, we get

From (4.3), we have the Eq (4.10).

Now, using (4.7) and (4.10), respectively, we have the following equations:

and

This last two equations verifies (4.4) and (4.6).

Example 2. Let $\widetilde{M} = (R_{2}^{5}, \widetilde{g})$ be a semi-Euclidean space, where $\widetilde{g}$ is of the signature $(-, +, -, +, +)$ with respect to canonical basis $\{\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \frac{\partial}{\partial y_{1}}, \frac{\partial}{\partial y_{2}}, \frac{\partial}{\partial z}\}$. Defining

where $i = 1, 2$. It can easily see that $(\widetilde{\varphi}, \nu, \eta, \widetilde{g})$ is an indefinite Sasakian structure on $R_{2}^{5}$. If we choose $\overline{\mathbb{K}}(X, Y) = \widetilde{g}(Y, \nu)\widetilde{g}(X, \nu)\nu$, then $(\widetilde{D} = \widetilde{\nabla} + \overline{\mathbb{K}}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is an indefinite Sasakian statistical structure on $\widetilde{M}$.

Example 3. In a $5-$ dimensional real number space $\widetilde{M} = R^{5}$, let $\{x_{i}, y_{i}, z\}_{1\leq i\leq 2}$ be cartesian coordinates on $\widetilde{M}$ and $\{\frac{\partial}{\partial x_{i}}, \frac{\partial}{\partial y_{i}}, \frac{\partial}{\partial z}\}_{1\leq i\leq 2}$ be the natural field of frames. If we define $1-$ form $\eta$, a vector field $\nu$ and a tensor field $\widetilde{\varphi}$ as follows:

It is easy to check (4.1) and (4.2). Then, $(\widetilde{\varphi}, \nu, \eta)$ is an almost contact structure on $R^{5}$. Now, we define metric $\widetilde{g}$ on $R^{5}$ by

with respect to the natural field of frames. Then we can easily see that $(\widetilde{\varphi}, \nu, \eta, \widetilde{g})$ is an indefinite Sasakian structure on $R^{5}$. We set the difference tensor field $\overline{\mathbb{K}}$ as

where $\lambda\in C^{\infty}(\widetilde{M})$. Then, $(\widetilde{D} = \widetilde{\nabla} + \overline{\mathbb{K}}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is an indefinite Sasakian statistical structure on $\widetilde{M}$.

5.   Lightlike hypersurfaces of indefinite Sasakian statistical manifolds

Definition 4. Let $(M, g, D, D^{\ast })$ be a hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. The quadruplet $(M, g, D, D^{\ast })$ is called lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ if the induced metric $g$ is degenerate.

Let $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ be a $(2m+1)-$ dimensional Sasakian statistical manifold and $(M, g)$ be a lightlike hypersurface of $\widetilde{M}$, such that the structure vector field $\nu$ is tangent to $M$. For any $\xi\in\Gamma(RadTM)$ and $N\in \Gamma (ltr TM)$, in view of (4.1)–(4.3), we have

Also using (3.1) and (4.8) we obtain

Proposition 5.1. Let $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ be a $(2m+1)-$ dimensional Sasakian statistical manifold and $(M, g, D, D^{\ast })$ be its lightlike hypersurface such that the structure vector field $\nu$ is tangent to $M$. Then we have

where $\xi$ is a local section of $RadTM$ and $N$ is a local section of $ltrTM$.

Proof. Using (4.8) and (3.1), we have

and

From (4.3) and (5.1), we have (5.7).

Proposition 5.1 makes it possible to make the following decompositions:

where $L_{0}$ is non-degenerate and $\widetilde{\varphi}-$ invariant distribution of rank $2m-4$ on $M$. If we denote the following distributions on $M$

then $L$ is invariant and $L^{'}$ is anti-invariant distributions under $\widetilde{\varphi}$. Also we have

Now, we consider two null vector field $U$ and $W$ and their $1-$ forms $u$ and $w$ as follows:

Then, for any $X\in\Gamma(T\widetilde{M})$, we have

where $S$ projection morphism of $T\widetilde{M}$ on the distribution $L$. Applying $\widetilde{\varphi}$ to last equation, we obtain

where $\varphi$ is a tensor field of type $(1, 1)$ defined on $M$ by $\varphi X = \widetilde{\varphi}SX$.

Again, we apply $\widetilde{\varphi}$ to (5.14) and using (4.1)–(4.3) we have

which means that

Now applying $\varphi$ to the Eq (5.15) and since $\varphi U = 0$, we have $\varphi^{3}+\varphi = 0$ which gives that $\varphi$ is an $f$–structure.

Definition 5. Let $(M, g, D, D^{\ast })$ be a hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. The quadruplet $(M, g, D, D^{\ast })$ is called screen semi-invariant lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$ if

We remark that a hypersurface of indefinite Sasakian statistical manifold is screen semi-invariant lightlike hypersurface.

Example 4. Let us recall the Example 2. Suppose that $M$ is a hypersurface of $R_{2}^{5}$ defined by

Then $RadTM$ and $ltr(TM)$ are spanned by $\xi = \frac{\partial}{\partial x_{1}}+\frac{\partial}{\partial y_{2}}$ and $N = \frac{1}{2}\{-\frac{\partial}{\partial x_{1}}+\frac{\partial}{\partial y_{2}}\}$, respectively. Applying $\widetilde{\varphi}$ to this vector fields, we have

Thus M is a screen semi-invariant lightlike hyperfurface of indefinite Sasakian statistical manifold $R_{2}^{5}$.

Example 5. Let M be a hypersurface of $(\widetilde{\varphi}, \nu, \eta, \widetilde{g})$ on $\widetilde{M} = R^{5}$ in Example 3. Suppose that $M$ is a hypersurface of $R_{2}^{5}$ defined by

Then the tangent space $TM$ is spanned by $\{U_{i}\}_{1\leq i\leq 4}$, where $U_{1} = \frac{\partial}{\partial x_{1}}$, $U_{2} = \frac{\partial}{\partial x_{2}}+\frac{\partial}{\partial y_{2}}$, $U_{3} = \frac{\partial}{\partial y_{1}}$, $U_{4} = \nu$. $RadTM$ and $ltr(TM)$ are spanned by $\xi = U_{2}$ and $N = \frac{1}{2}\{-\frac{\partial}{\partial x_{2}}+\frac{\partial}{\partial y_{2}}\}$, respectively. Applying $\widetilde{\varphi}$ to this vector fields, we have

Thus M is a screen semi-invariant lightlike hyperfurface of indefinite Sasakian statistical manifold $\widetilde{M}$.

In view of (5.11) and (5.12), we have

Thus $\langle U\rangle \oplus \langle W\rangle$ is non-degenerate vector budle of $S(TM)$ with rank $2$. If we consider (5.8) and (5.9), we get

and

Thus, for any $X\in\Gamma(TM)$, we can write

where P and Q are projections of $TM$ into $L$ and $L^{'}$. Thus, we can write $QX = u(X)U$. Using (4.1)–(4.3), (5.14) and (5.18), we have

where $\widetilde{\varphi}PX = \varphi X$. We can easily see that

for any $X, Y\in\Gamma(TM)$. Also we have the following identities:

Thus, we have the following proposition.

Proposition 5.2. Let $(M, g, D, D^{\ast })$ be a lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. Then $\varphi$ need not be an almost contact structure.

Lemma 5.3. Let $(M, g, D, D^{\ast })$ be a lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. For any $X, Y\in\Gamma(TM)$, we have the following identities:

Proof. Using Gauss and Weingarten formulas in (4.7) we obtain

If we take tangential and transversal parts of this last equation, we have (5.22) and (5.23).

Similarly, we have the following lemma.

Lemma 5.4. Let $(M, g, D, D^{\ast })$ be a lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. For any $X, Y\in\Gamma(TM)$, we have the following identities:

Lemma 5.3 and Lemma 5.4 are give us the following theorem.

Theorem 5.5. A lightlike hypersurface $M$ of an indefinite Sasakian statistical manifold $\widetilde{M}$ need not be a statistical manifold.

Proposition 5.6. Let $(M, g, D, D^{\ast })$ be a lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. For any $X, Y\in\Gamma(TM)$, we have the following expressions:

(i) If the vector field $U$ is parallel with respect to $\nabla^{*}$, then we have

(ii) If the vector field $U$ is parallel with respect to $\nabla$, then we have

Proof. Replacing $Y$ in (5.22) by $U$, we obtain

Applying $\varphi$ to this equation and using (5.15), we get

If $U$ is parallel with respect to $\nabla^{*}$ then $\varphi A_{N}X = 0$. From (5.14), we have $\widetilde{\varphi}(A_{N}X) = u(A_{N}X)N$. If $\widetilde{\varphi}$ is applied to the last equation and using (4.2), we obtain $A_{N}X = u(A_{N}X)U+\tau(A_{N}X)\nu$. Also, if we write $U$ instead of $Y$ in the Eq (5.23), we have $\tau(X) = 0$.

We can easily obtain the Eq (5.28) with a similar method.

Proposition 5.7. Let $(M, g, D, D^{\ast })$ be a lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. For any $X, Y\in\Gamma(TM)$, we have the following expressions:

(i) If the vector field $W$ is parallel with respect to $\nabla^{*}$, then we have

(ii) If the vector field $W$ is parallel with respect to $\nabla$, then we have

Proof. If we write $\xi$ instead of $Y$ in the Eq (5.22), we obtain

If $W$ is parallel with respect to $D$, using (3.14) and (5.12) in this equation, we obtain

Applying $\widetilde{\varphi}$ this and using (5.15) we have

If we take screen and radical parts of this last equation, we have (5.29).

Similarly, we can easily obtain the Eq (5.30).

Definition 6. (^[17,23]) Let $(M, g)$ be a hypersurface of a statistical manifold $(\widetilde{M}, \widetilde{g}, \widetilde{D}, \widetilde{D}^{\ast })$.

${\bf (i)}$ $M$ is called totally geodesic with respect to $\widetilde{D }$ if $B = 0$.

${\bf (ii)}$ $M$ is called totally geodesic with respect to $\widetilde{ D}^{\ast }$ if $B^{\ast } = 0$.

Theorem 5.8. Let $(M, g, D, D^{\ast })$ be a lightlike hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$.

(i) $M$ is totally geodesic with respect to $\widetilde{D }$ if and only if

(ii) $M$ is totally geodesic with respect to $\widetilde{D }^{*}$ if and only if

Proof. For any $Y\in\Gamma(L)$ we know that $u(Y) = 0$. Then the Eqs (5.22) and (5.25) are reduced to the equations, respectively

On the other hand, replacing $Y$ by $U$ in (5.22) and (5.25), respectively, we also have

If taking into account (5.35)–(5.38), we can easily obtain our assertion.

The following two theorems give a characterization of the integrability of distributions $L\bot\langle\nu\rangle$ and $L^{'}\bot\langle\nu\rangle$, respectively.

Theorem 5.9. Let $(M, g, D, D^{\ast })$ be a screen semi-invariant hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. The following assertions are equivalent:

(i) The distribution $L\bot\langle\nu\rangle$ is integrable.

(ii) $B^{*}(X, \varphi Y) = B^{*}(\varphi X, Y)$, for all $X, Y\in\Gamma(L\bot\langle\nu\rangle)$,

(iii) $B(X, \varphi Y) = B(\varphi X, Y)$, for all $X, Y\in\Gamma(L\bot\langle\nu\rangle)$.

Proof. We know that $X\in\Gamma(L\bot\langle\nu\rangle)$ if and only if $u(X) = \widetilde{g}(X, W) = 0$. For any $X, Y\in\Gamma(L\bot\langle\nu\rangle)$, using (3.1) and (5.14), we obtain

From (5.23), we have

This gives the equivalence between (i) and (ii). Similarly we can easily see that the relation (i) and (iii).

Theorem 5.10. Let $(M, g, D, D^{\ast })$ be a screen semi-invariant hypersurface of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. The following assertions are equivalent:

(i) The distribution $L^{'}\bot\langle\nu\rangle$ is integrable.

(ii) $A^{*}_{\widetilde{\varphi} X}Y-A^{*}_{\widetilde{\varphi} Y}X = g(X, \nu)Y-g(Y, \nu)X$, for all $X, Y\in\Gamma(L^{'}\bot\langle\nu\rangle)$.

(ii)$A_{\widetilde{\varphi} X}Y-A_{\widetilde{\varphi} Y}X = g(X, \nu)Y-g(Y, \nu)X$, for all $X, Y\in\Gamma(L^{'}\bot\langle\nu\rangle)$.

Proof. $X\in\Gamma(L^{'}\bot\langle\nu\rangle)$ if and only if $\varphi X = 0$. For any $X, Y\in\Gamma(L\bot\langle\nu\rangle)$, using (3.2), (3.3) and (5.14) in (4.7), we have

Therefore, we can get

This gives the equivalence between (i) and (ii). Similarly, the relationship between (i) and (iii) is easily seen.

6.   Invariant submanifolds

Let $(M, g, D, D^{\ast })$ be a lightlike submanifold of an indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. if $M$ is tangent to the structure vector field $\nu$, then $\nu$ belongs to $S(TM)$ (see ^[13]). Using this, we say that $M$ is an invariant lightlike submanifold of $\widetilde{M}$ if $M$ is tangent to the structure vector field $\nu$ and

Proposition 6.1. Let $(M, g, D, D^{\ast })$ be an invariant lightlike submanifold of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. For any $X, Y\in\Gamma(TM)$, we have the following identities:

where $h$ and $h^{*}$ are second fundemental forms for affine dual connections $\widetilde{D}$ and $\widetilde{D}^{*}$, respectively.

Proof. Using (5.14) and Gauss formula in (4.7), we obtain

If we take tangential and transversal parts of this last equation, our claim is proven.

Similarly to the above proposition, the following proposition is given for dual connection $D^{*}$.

Proposition 6.2. Let $(M, g, D, D^{\ast })$ be an invariant lightlike submanifold of indefinite Sasakian statistical manifold $(\widetilde{M}, \widetilde{D}, \widetilde{g}, \widetilde{\varphi}, \nu)$. For any $X, Y\in\Gamma(TM)$, we have the following identities:

where $h$ and $h^{*}$ are second fundemental forms for affine dual connections $\widetilde{D}$ and $\widetilde{D}^{*}$, respectively.

From the Eqs (6.3) and (6.5), we have

A lightlike submanifold may not be an indefinite Sasakian statistical manifold. The following theorem gives a situation where this can happen.

Theorem 6.3. An invariant lightlike submanifold of indefinite Sasakian statistical manifold is an indefinite Sasakian statistical manifold.

Proof. In a invariant lightlike submanifold, $u(X) = 0$, for any $X\in\Gamma(TM)$. Then from (5.14) we have

Since $\widetilde{\varphi}X = \varphi X$, using (4.1)–(4.3), we obtain

Then $(g, \varphi, \nu)$ is an almost contact metric structure.

Using (3.5), we get

This equation says that $D$ and $D^{*}$ are dual connections. Moreover torsion tensor of the connection $D$ is equal zero. Then, the Eqs (3.5) and (3.6) tell us that $(D, g)$ is a statistical structure.

If we consider Gauss formula and (4.8) we obtain

If we consider (6.2) and (6.10) in the theorem 4.1, our assertion are proven.

Example 6. Let $\widetilde{M} = \left(\mathbb{R}_{2}^{7}, \widetilde{g}, \widetilde{\phi}, \nu\right)$ be the manifold endowed with the usual Sasakian structure (see, for example, [^[13], p. 321$]$ for such a structure), in which $\widetilde{g}$ has signature $(-, +, +, -, +, +, +)$, with respect to the canonical basis $\left\{\partial x^{1}, \partial x^{2}, \partial x^{3}, \partial y^{1}, \partial y^{2}, \partial y^{3}, \partial z\right\}$. By choosing the difference tensor $\overline{\mathbb{K}}(X, Y) = \widetilde{g}(Y, \nu)\widetilde{g}(X, \nu)\nu$, we can easily see that $(\widetilde{D} = \widetilde{\nabla} + \overline{\mathbb{K}}, \widetilde{g}, \widetilde{\varphi}, \nu)$ is an indefinite Sasakian statistical structure on $\widetilde{M}$. Now, we recall the example in ^[27] as follows:

Suppose that $M$ is a submanifold of $\widetilde{M}$ given by

It is easy to see that the vector fields $\xi_{1}, \xi_{2}, \nu, Z_{1}, Z_{2}$, and given by

spans $T M$. Moreover, one can see that $\operatorname{Rad} T M = \operatorname{Span}\left\{\xi_{1}, \xi_{2}\right\}$ and $S(T M) =$ Span $\left\{Z_{1}, Z_{2}, \nu\right\}$. Furthermore, we note that $\widetilde{\phi} \xi_{2} = \xi_{1}$ and $\widetilde{\phi} Z_{2} = Z_{1}$. It follows that $\operatorname{Rad} T M$ and $S(T M)$ are invariant under $\widetilde{\phi}$. On the other hand, $l \operatorname{tr}(T M)$ is spanned by $N_{1}$ and $N_{2}$, where

Note that $\widetilde{\phi} N_{2} = N_{1}$; hence, $l \operatorname{lr}(T M)$ is invariant under $\widetilde{\phi}$. Therefore, $M$ is a five-dimensional invariant lightlike submanifold of indefinite Sasakian statistical manifold $\widetilde{M}$ and M is an indefinite Sasakian statistical manifold.

7.   Conclusions and future work

In this paper, we expanded the Sasakian statistical manifold concept to indefinite Sasakian statistical manifolds and introduced lightlike hypersurfaces of an indefinite Sasakian statistical manifold. Some relations among induced geometrical objects with respect to dual connections in a lightlike hypersurface of an indefinite statistical manifold are obtained. We also give some original examples in this context.. We hope that, this introductory study will bring a new perspective for researchers and researchers will further work on it focusing on new results not available so far on lightlike geometry

Conflict of interest

The author declares that there is no competing interest.