On Lightlike Geometry of Indefinite Sasakian Statistical Manifolds

In this study, we introduce indefinite sasakian statistical manifolds and 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 sasakian manifold are obtained. Some examples related to these concepts are also presented. Finally, we prove that an invariant lightlike submanifold of indefinite sasakian statistical manifold is an indefinite sasakian statistical manifold.


Introduction
Neural networks can be applied to solving numerous complex optimization problems in electromagnetic theory. Applied physicist B. Bartlett presented unsupervised machine learning model for computing approximate electromagnetic field solutions [2]. 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 is direct evidence of the existence of black holes and general theory of relativity [3]. This is also an indirect evidence of the existence of lightlike geometry in the universe.
A statistical manifold, a new branch of mathematics, is a generalization of 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 [4]. Statistical manifolds also have many application areas such as neural networks, machine learning and artificial intelligence.
Lightlike geometry is one of the most important research areas in differential geometry and has many applications in physics and mathematics, such as general relativity, electromagnetism and black hole theory.
In 1975, Efron [5] first emphasized the role of differential geometry in statistics. Differential geometrical tools were used by Amari to develop this idea [6], [7]. In 1989, Vos [8] obtained fundamental equations of geometry of submanifolds of statistical manifolds. In 2009, hypersurfaces of a statistical manifold are studied by Furuhata [9]. Many studies have been done on both statistical manifolds and lightlike geometry over the last few decades [10]- [25]. Hovewer, no study combining these two notions has been done in the literature so far.
Motivated by these circumstances, in this study, we 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. Finally, an example is given. In Section 4, we introduce indefinite sasakian statistical manifolds and we obtain the characterization theorem of 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.

Preliminaries
Let (M ,ḡ) be an (m+ 2)-dimensional semi-Riemannian manifold with index(ḡ) = q ≥ 1. Let (M, g) be a hypersurface of (M,ḡ) with g =ḡ| M . If the induced metric g on M is degenerate, then M is called a lightlike (null or degenerate) hypersurface ( [14], [15], [16]). In this case, there exists a null vector field ξ = 0 on M such that The radical or the null space of T The dimension of Rad T x M is called the nullity degree of g. We recall that the nullity degree of g for a lightlike hypersurface of (M ,ḡ) is 1. Since g is degenerate and any null vector being orthogonal to itself, T x M ⊥ is also null and where ⊕ is the direct sum but not orthogonal ( [14], [15]). In view of the splitting (2.6), we have the following Gauss and Weingarten formulas, respectively, for any X, Y ∈ Γ(T M), where ∇ X Y, A N X ∈ Γ(T M) and h(X, Y ), ∇ t X N ∈ Γ(ltr(T M)). If we set B(X, Y ) =ḡ(h(X, Y ), ξ) and τ (X) =ḡ(∇ t X N, ξ), then (2.7) and (2.8) become respectively. Here, B and A are called the second fundamental form and the shape operator of the lightlike hypersurface M, respectively [14]. Let P be the projection of S(T M) on M.
Then, for any X ∈ Γ(T M), we can write where η is a 1-form given by η(X) =ḡ(X, N). (2.12) From (2.9), we have for all X, Y, Z ∈ Γ(T M), where the induced connection ∇ is a non-metric connection on M. From (2.4), we have 14) for all X ∈ Γ(T M), W ∈ Γ(S(T M)), where ∇ * X W and A * ξ X belong to Γ(S(T M)). Here C, A * ξ and ∇ * are called the local second fundamental form, the local shape operator and the induced connection on S(T M), respectively. We also have Moreover, from the first and third equations of (2.16), we have Now we define some statistical basic concepts for all X, Y, Z ∈ Γ(T M).
A statistical manifold will be represented by ( M , g, D, D * ). If ∇ is Levi-Civita connection of g, then (2.20) for any X, Y, Z ∈ Γ(T M).
Conversely, for a Riemannian metric g, if K satisfies (2.21), the pair ( D = ∇ + K, g) is a statistical structure on M [19].
Let (M, g) be a submanifold of ( M, g). If (M, g, D, D * ) is a statistical manifold, then (M, g, D, D * ) is called a statistical submanifold of ( M , g, D, D * ), where D, D * are affine dual connections on M and D, D * are affine dual connections on M (see [7], [9], [8]).

Lightlike hypersurface of a statistical manifold
Let (M, g) be a lightlike hypersurface of a statistical manifold ( M, g, D, D * ). Then, Gauss and Weingarten formulas with respect to dual connections are given by [9] Here, D, D * , B, B * , A N and A * N are called the induced connections on M, the second fundamental forms and the Weingarten mappings with respect to D and D * , respectively. Using Gauss formulas and the equation (2.18), we obtain From the equation (3.5), we have the following result. (i) Induced connections D and D * need not be dual .
(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 Then the following assertions are true: (i) Induced connections D and D * are symmetric connection.
(ii) The second fundamental forms B and B * are symmetric.
Proof. We know that T D = 0. Moreover, Comparing the tangent and transversal components of (3.7), we obtain where T D is the torsion tensor field of D. Thus, second fundamental form B is symmetric and induced connection D is symmetric connection. Similarly, it can be shown that the second fundamental form B * is symmetric and the induced connection D * is a symmetric connection.
Let P denote the projection morphism of Γ(T M) on Γ(S(T M)) with respect to the decomposition (2.4). Then, we have for all X, Y ∈ Γ(T M) and ξ ∈ Γ(RadT M), where ∇ X P Y and A ξ X belong to Γ(S(T M)), ∇ and ∇ t are linear connections on Γ(S(T M)) and Γ(RadT M) respectively. Here, h and A are called screen second fundamental form and screen shape operator of S(T M), respectively.
Here C(X, P Y ) is called the local screen fundamental form of S (T M).
Similarly, the relations of induced dual objects on S(T M) are given by Using (3.5), (3.12), (3.14) and Gauss-Weingarten formulas, the relationship between induced geometric objects are given by Now, using the equation (3.16) we can state the following result. Additionally, due to D and D * are dual connections we obtain Example 3.4 Let (R 4 2 , g) be a 4-dimensional semi-Euclidean space with signature (−, −, +, +) of the canonical basis (∂ 0 , . . . , ∂ 3 ). Consider a hypersurface M of R 4 2 given by For simplicity, we set f = x 2 2 + x 2 3 . It is easy to check that M is a lightlike hypersurface whose radical distribution RadT M is spanned by .
Then the lightlike transversal vector bundle is given by It follows that the corresponding screen distribution S(T M) is spanned by Then, by direct calculations we obtain for any X ∈ Γ(T M) [16].
We define an affine connection D as follows Then using (2.19) we obtain Then D and D * are dual connections. Here, one can easily see that T D = 0 and D g = 0. From Definition 2.1, we say that (R 4 2 , g, D, D * ) is a statistical manifold.

Indefinite sasakian statistical manifolds
In order to call a differentiable semi-Riemannian manifold ( M , g) of dimension n = 2m + 1 as practically contact metric one, a (1, 1) tensor field ϕ, a contravariant vector field ν, a 1− form η and a Riemannian metric g should be admitted, which satisfy for all the vector fields X, Y on M . When a practically contact metric manifold performs M is regarded as an indefinite sasakian manifold. In this study, we assume that the vector field ν is spacelike. An indefinite sasakian statistical manifold will be represented by ( M , D, g, ϕ, ν). We remark that if ( M , D, g, ϕ, ν) is an indefinite sasakian statistical manifold, so is ( M , D * , g, ϕ, ν) [18], [19].
for all the vector fields X, Y on M . If we consider Definition 4.1 and the equation (4.4), we have the formula (4.7). If we write D * instead of D in (4.7), we have
Example 5.4 Let us recall the example 4.3, Suppose that M is a hypersurface of R 5 2 defined by Then RadT M and ltr(T M) are spanned by ξ = ∂ ∂x 1 + ∂ ∂y 2 and N = 1 2 {− ∂ ∂x 1 + ∂ ∂y 2 }, respectively. Applying ϕ to this vector fields, we have Thus M is a screen semi invariant lightlike hyperfurface of indefinite sasakian statistical manifold R 5 2 .
Example 5.5 Let M be a hypersurface of ( ϕ, ν, η, g) on M = R 5 in Example4.4, Suppose that M is a hypersurface of R 5 2 defined by Then the tangent space T M is spanned by RadT M and ltr(T M) are spanned by ξ = U 2 and N = 1 2 {− ∂ ∂x 2 + ∂ ∂y 2 }, respectively. Applying ϕ to this vector fields, we have Thus M is a screen semi invariant lightlike hyperfurface of indefinite sasakian statistical manifold M .
where ϕP X = ϕX. We can easily see that for any X, Y ∈ Γ(T M). Also we have the following identities: Proof. Using Gauss and Weingarten formulas in (4.7) we obtain Similarly, we have the following lemma Lemma 5.8 Let (M, g, D, D * ) be a lightlike hypersurface of indefinite Sasakian statistical manifold ( M , D, g, ϕ, ν). For any X, Y ∈ Γ(T M), we have the following identities: Lemme(5.7) and Lemma(5.8) are give us the following theorem. (i) If the vector field U is parallel with respect to ∇ * , then If the vector field U is parallel with respect to ∇, then Applying ϕ to this equation and using (5.15), we get If U is parallel with respect to ∇ * then ϕA N X = 0. From (5.14) we have ϕ(A N X) = u(A N X)N. Applying ϕ this and using (4.2) we obtain A N X = u(A N X)U + τ (A N X)ν. Also, if we write U instead of Y in the equation (5.23), we have τ (X) = 0.
If W is parallel with respect to D, using (3.15) and (5.12) in this equation, we obtain Applying ϕ 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 see the equation (5.30).
The following two theorems give a characterization of the integrability of distributions L⊥ ν and L ′ ⊥ ν , respectively.
Similar to the above proposition, the following proposition is given for dual connection D * .

Conclusion and future work
In the present paper, firstly we have studied lightlike geometry of statistical manifolds, Later, we have introduced lightlike geometry of an indefinite sasakian statistical manifold which is a new classification of statistical manifolds and we have given some results for its induced geometrical objects. Some examples related to these concepts are also presented. Finally, we prove that an invariant lightlike submanifold of indefinite sasakian statistical manifold is an indefinite sasakian statistical manifold. 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