Main Curvatures Identities on Lightlike Hypersurfaces of Statistical Manifolds and Their Characterizations

: In this study, some identities involving the Riemannian curvature invariants are presented on lightlike hypersurfaces of a statistical manifold in the Lorentzian settings. Several inequalities characterizing lightlike hypersurfaces are obtained. These inequalities are also investigated on lightlike hypersurfaces of Lorentzian statistical space forms.


Introduction
The concept of statistical manifolds is currently a new and attractive topic in differential geometry. It has many application areas such as neural networks, machine learning, artificial intelligence, and black holes [1][2][3][4]. A statistical structure on a Riemannian manifold was initially defined by S. Amari [5] in 1985, as follows: A Riemannian manifold ( M, g) with a Riemannian metric g and the Levi-Civita connection ∇ 0 is called a statistical manifold if there exists a pair of torsion-free affine connections ( ∇, ∇ * ), such that the following relation is satisfied for any tangent vector fields X, Y and Z on M There exist qualified papers dealing with statistical manifolds and their submanifolds admitting various differentiable structures. In this sense, the geometry of hypersurfaces of statistical manifolds was presented by H. Furuhata in Refs. [6,7]. Statistical manifolds admitting contact structures or complex structures and their submanifolds were investigated in Refs. [8,9]. Statistical manifolds involving Riemannian submersions were also studied in Refs. [10,11]. In degenerate cases, lightlike hypersurfaces of statistical manifolds were introduced by the first author and M. M. Tripathi [12]. In addition, statistical lightlike hypersurfaces were studied by V. Jain, A. P. Singh, and R. Kumar in Ref. [13].
Besides these facts, many sharp inequalities on submanifolds in various ambient spaces are obtained from isometric immersions. The main inequalities on submanifolds of real space forms were initially established by B.-Y. Chen [14][15][16][17]. Later, these inequalities have drawn the attention of many authors due to their interesting characterizations for the theory of submanifolds. Moreover, some inequalities involving curvature-like tensors were studied by M. M. Tripathi [18] and Chen-like inequalities and their characterizations were presented in submanifolds of statistical manifolds in Refs. [19][20][21][22].
In this paper, first of all, by examining the commonly known important submanifold types such as totally geodesic, totally umbilical, and minimal lightlike hypersurfaces with respect to the Levi-Civita connection, some relations are obtained. Then, various results are found by computing the curvature tensor fields such as the statistical sectional curvature, statistical screen Ricci curvature, and statistical screen scalar curvature. Finally, with the help of these curvature relations, various inequalities are established for hypersurfaces of statistical manifolds. The equality cases are also discussed.

Preliminaries
Let ( M, g) be an (m + 2)-dimensional Lorentzian manifold and (M, g) be a hypersurface of ( M, g) with the induced metric g from g. If g is degenerate, then M is called a lightlike (null or degenerate) hypersurface. For a lightlike hypersurface (M, g) of ( M, g), there exists a non-zero vector field ξ on M such that for all X ∈ Γ(TM). Here the vector field ξ is called a null vector [23][24][25]. The radical or the null space Rad T x M at each point x ∈ M is defined as 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 is equal to 1. Since g is degenerate and any null vector being orthogonal to itself, the normal space T x M ⊥ is a null subspace. In addition, we have Rad The complementary vector bundle S(TM) of 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 ⊥ denotes the orthogonal-direct sum. The complementary vector bundle S(TM) ⊥ of S(TM) in T M is called the screen transversal bundle. Since Rad TM is a lightlike subbundle of S(TM) ⊥ , there exists a unique local section N of S(TM) ⊥ such that we have Note that N is transversal to M and {ξ, N} is a local frame field of S(TM) ⊥ . Thus, there exists a line subbundle ltr(TM) of T M. This set is called the lightlike transversal bundle, locally spanned by N [23,24].
Let ∇ 0 be the Levi-Civita connection of M. The Gauss and Weingarten formulas are given for any X, Y ∈ Γ(TM). Here ∇ 0 is the induced linear connection on TM, B 0 is the second fundamental form on TM, A 0 N is the shape operator on TM, and τ 0 is a 1-form on TM. A lightlike hypersurface M is called totally geodesic if B 0 = 0. If there exists a λ ∈ R at every point of M such that for all X, Y ∈ Γ(TM) then M is called totally umbilical [23].

Lightlike Hypersurfaces of a Statistical Manifold
Let ( M, g) be a Lorentzian manifold. If there exists a torsion free connection ∇ satisfying the following: for all X, Y, Z ∈ Γ(T M) then ( g, ∇) is called a statistical structure. So, ( M, g, ∇) is a statistical manifold [6].
For a statistical manifold ( M, g, ∇), the dual of ∇, denoted by ∇ * , is defined by the following identity: It is easy to check that ∇ * is torsion free. If ∇ 0 is the Levi-Civita connection of g, then we can write Let (M, g) be a lightlike hypersurface of a statistical manifold ( M, g, ∇). From (6) and (7), the Gauss and Weingarten formulas with respect to dual connections can be expressed ∇ X N = −A * N X + τ * (X)N (15) and for all X, Y ∈ Γ(TM), N ∈ Γ(ltr TM), where ∇ X Y, ∇ * X Y, A N X, A * N X ∈ Γ(TM). Here, ∇, ∇ * are called the induced connections on M, B and B * are called the second fundamental forms, A N and A * N are called the Weingarten mappings with respect to ∇ and ∇ * , respectively. Using (7) in (14) and (15), we have and From the Gauss and Weingarten formulas, it is clear that both the induced connections ∇ and ∇ * are symmetric. In addition, both the second fundamental forms' B and B * are symmetric and bilinear, called the imbedding curvature tensors of the submanifold for ∇ and ∇, respectively. We note that a lightlike hypersurface of a statistical manifold does not need to be a statistical manifold with respect to ∇ and ∇ * [12].
Let P denotes the projection morphism of Γ(TM) on Γ(S(TM)) with respect to the decomposition (6). For any X, Y ∈ Γ(TM) and ξ ∈ Γ(Rad TM) we can write Here, ∇ X PY and A ξ X belong to Γ(S(TM)), ∇ and ∇ t are linear connections on Γ(S(TM)) and Γ(Rad TM), respectively. The tensor fields h and A are called the screen second fundamental form and the screen shape operator of S(TM) respectively. If we define then we can easily prove that Therefore, we can write from (19), (20) and (23) that for all X, Y ∈ Γ(TM). Here C(X, PY) is called the local screen fundamental form of S(TM). Similarly, the relations of induced dual objects on S(TM) with respect to ∇ are given by Using (25), (27) and Gauss-Weingarten formulas, the relationship between induced geometric objects are given by C(X, PY) = g(A N X, PY), C * (X, PY) = g(A * N X, PY).
As a result of (28), we obtain that the second fundamental forms B and B * are not degenerate. Additionally, due to ∇ and ∇ * are dual connections, we obtain Putting X = ξ in (30), (31) and using the fact that A ξ , A * ξ are S(TM)-valued tensor fields, we get Definition 1. A hypersurface M is a screen locally conformal lightlike hypersurface of a statistical manifold ( M, g, ∇) if there exist non-vanishing smooth functions ϕ and ϕ * on M such that In particular, M is called screen homothetic if ϕ and ϕ * are non-zero constants. totally geodesic with respect to ∇ * if B * = 0; 3.
totally tangentially umbilical with respect to ∇ if there exists a smooth function k such that B(X, Y) = kg(X, Y) for all X, Y ∈ Γ(TM); 4.
totally tangentially umbilical with respect to ∇ * if there exists a smooth function k * such that B * (X, Y) = k * g(X, Y), for any X, Y ∈ Γ(TM);

5.
totally normally umbilical with respect to ∇ if there exists a smooth function k such that A * N X = kX for any X, Y ∈ Γ(TM); 6.
totally normally umbilical with respect to ∇ * if there exists a smooth function k * such that A N X = k * X for all X, Y ∈ Γ(TM), where k * is smooth function.
From (6) and (7), we can consider a local quasi-orthonormal field {E 1 , . . . , E m , ξ, N} of frames of M along M where {E 1 , . . . , E m } is an orthonormal basis of Γ(S(TM)). Then, the mean curvature H with respect to ∇ is defined by The hypersurface M is called minimal with respect to ∇ if H = 0 at every point of M. Now, we will give some examples of lightlike hypersurfaces of statistical manifolds: Then it is easy to check that M is a lightlike hypersurface such that By direct calculations we obtain M as a totally geodesic lightlike hypersurface with respect to the Levi-Civita connection. Now let us define an affine connection ∇ as follows: Using (13) we obtain Using (4), (12) and the above calculations, one can choose for any X ∈ Γ(TM). Therefore ∇ and ∇ * are dual connections. Here one can easily see that these connections are torsion free and ∇ g = 0. Hence, from the definition statistical manifold, we see that (R 4 1 , g, ∇, ∇ * ) is a statistical manifold and M is totally geodesic with respect to ∇ and ∇ * .

Example 2. Let us consider a lightlike M in R
By straightforward computation we have Let us define an affine connection ∇ as follows: Then we obtain Therefore, ∇ and ∇ * are dual connections and (R 4 1 , g, ∇, ∇ * ) is a statistical Lorentzian manifold. It is clear that for any X, Y ∈ Γ(TM). Thus, we say that M is totally tangentially umbilical with respect to ∇ and k = − 2 t .
Further examples such as totally normally umbilical, minimal, and screen conformal lightlike hypersurfaces of a statistical manifold could be derived.
The following results are well known for lightlike hypersurfaces of statistical manifolds.

2.
A Theorem 1 ([12]). Let (M, g) be a lightlike hypersurface of ( M, g, ∇). Then, M is totally tangentially umbilical with respect to ∇ and ∇ * if and only if there exists a smooth function ρ such that the following equation satisfies for all X ∈ Γ(TM).

Proposition 3 ([12]
). Let (M, g) be a lightlike hypersurface of ( M, g, ∇). Then, M is screen locally conformal if and only if where ρ is non-vanishing smooth functions on M.  Proof. Suppose that M is totally normally umbilical. From Proposition 2 we have C = −C * . If we consider this fact and M is screen homothetic in Theorem 2, we see that M is totally geodesic with respect to the Levi-Civita connection ∇ 0 . The converse part of proof could be given similarly.
Theorem 4. Let (M, g) be a lightlike hypersurface of ( M, g, ∇). If M is totally umbilical with respect to ∇ 0 then the following relation satisfies: for any X ∈ Γ(S(TM)). (8) and (13), we can write B 0 = 1 2 (B + B * ). Using the fact that M is totally umbilical with respect to ∇ 0 , there exists a smooth function λ such that B 0 (X, Y) = λg(X, Y) for any X, Y ∈ Γ(TM). Now if we choose X and Y as orthonormal, then we have B(X, X) = 2λ − B * (X, X), and B(X, Y) = −B * (X, Y).

Proof. From
From (30), (31) and (38), we obtain and which imply (37). Corollary 1. Let (M, g) be a lightlike hypersurface of ( M, g, ∇). If M is totally umbilical with respect to ∇ 0 then M can not be totally tangentially umbilical with respect to ∇ and ∇ * .
Proof. Assume that M is totally tangentially umbilical with respect to ∇ and ∇ * . From Theorems 1 and 4, we get ρ = 0, which is a contradiction. Thus, M can not be totally tangentially umbilical with respect to ∇ and ∇ * .  . If M is totally umbilical with respect to ∇ 0 , then the following relation holds: Proposition 4. Let (M, g) be minimal with respect to the Levi-Civita connection ∇ 0 and the set {E 1 , . . . , E m } be an orthonormal basis of Γ(S(TM)). Then we have the following relation: Proof. If (M, g) is minimal with respect to ∇ 0 , then we have The proof is straightforward from (30), (31) and (41).

Main Curvature Relations
Let (M, g) be a lightlike hypersurface of a statistical manifold ( M, g, ∇). Denote the curvature tensors with respect to ∇ and ∇ * by R and R * , respectively. Using the Gauss and Weingarten formulas for ∇ and ∇ * , we obtain and where R and R * are the curvature tensor with respect to ∇ and ∇ * , respectively. The statistical manifold ( M, g) is called of constant curvature c if the following relation satisfies for any X, Y, Z ∈ Γ(TM) [13] R(X, Y)Z = c{g(Y, Z)X − g(X, Z)Y}.
Let Π = span{X, Y} be a two dimensional non-degenerate plane of T x M at x ∈ M. The statistical sectional curvature of Π with respect to ∇ and ∇ * is defined respectively by Ref. [13] and Suppose that ξ is a null vector of T x M. A plane Π of T x M is a null plane if it contains ξ and X such that g(ξ, X) = 0 and g(X, X) = 0. Then, the statistical null sectional curvatures are given respectively by and From the above equations and the Gauss-Weingarten formulas for M and S(TM), one can obtain the following proposition: Proposition 5. Let (M, g) be a lightlike hypersurface of ( M, g, ∇). Then we have the following equalities for any X, Y, Z, W ∈ Γ(TM): where and for any X, Y, Z ∈ Γ(TM) Proof. From (44) it follows that We obtain the claim of proposition by using the above relation in (51).
With similar arguments as in the proof of Lemma 1.8 in Ref. [28], we have the following lemma for the semi-Riemannian case: Lemma 1. For any statistical manifold M, the following identities hold for any tangent vector fields X, Y, Z on T M.
We note that if ( M, g, ∇) is of a constant curvature c with respect to ∇, then it is also of a constant curvature c with respect to ∇ * . Proposition 7. Let (M, g) be a lightlike hypersurface of M(c). Then we have for any unit vector X ∈ Γ(S(TM)) that Here κ * null denotes the null sectional curvature with respect to ∇ * .
Proof. Using (60) and the (ii) statement of Lemma 1, we get g( R * (X, Y)ξ, Z) = 0. Using this fact in (50), we obtain Putting Y = ξ and X = Z in (62), it follows that From (28) and (63) we write If we put X = Y = ξ in (30) and (31), we see that B(ξ, ξ) = 0. Hence we obtain the Equation (61) from (64). Proof. Since (M, g) be a screen homothetic lightlike hypersurface, we get from Proposition 7 that for any unit vector X ∈ Γ(S(TM)). This identity shows that the sign of κ * null and ϕ have the same signs.
Following the terminology used in Ref. [29], it is said to be any two vector field V and W are conjugate if B(V, W) = 0 [23]. Rad TM is a Killing distribution; 2.
Rad TM is a parallel distribution with respect to ∇; 3.
From Proposition 7 and Theorem 5, we get the following result: for any X, Y, Z, W ∈ Γ(S(TM)).
Proof. Using Theorem 2 and the statement ii) of Lemma 1 in (49) and (50), the proof of proposition is straightforward.
for any two linearly independent vector fields X, Y ∈ Γ(S(TM)).
Proof. From the definition of the Riemannian curvature tensor we write for any X, Y, W ∈ Γ(S(TM)). Hence, putting Z = Y and X = W in (66) and by choosing X and Y are unit vector field in Γ(S(TM)), we derive Since M is totally geodesic we have C * = −C and M is screen homothetic, we can write C = ϕB. Using these facts in (69), we get which is the claim of corollary. Proposition 9. Let (M, g) be a lightlike hypersurface of a statistical manifold ( M, g, ∇). If M is totally geodesic with respect to ∇ 0 then we have for any X, Y ∈ Γ(S(TM)).
Proof. Since M is a Lorentzian space form, we have from (44) that g( R(X, Y)ξ, N) = 0 for any X, Y ∈ Γ(S(TM)). Considering this fact in (55), we derive Since M is totally geodesic, we have B = B * and hence, from (28), we obtain A * N = −A N . Therefore, we write Considering the right hand sides of (56) and (71), we obtain (70).
Corollary 8. Let (M, g) be a screen conformal lightlike hypersurface of ( M, g, ∇). If M is totally geodesic with respect to ∇ 0 then we have for any X, Y ∈ Γ(S(TM)).
Proposition 10. Let (M, g) be a totally umbilical with respect to ∇ 0 . If M is screen conformal then for any orthonormal vector pair (X, Y) in Γ(S(TM)), we have and Proof. Using the fact that M is screen conformal in (49), we derive for any orthonormal vector pair (X, Y) in Γ(S(TM)). Since M is totally umbilical with respect to ∇ 0 , we derive from (38) that which implies (73). The proof of (74) could be shown in a similar way.
For k = m, π m = Span{E 1 , . . . , E m } = Γ(S(TM)). In this case, the statistical screen Ricci curvature is given by and the statistical screen scalar curvature is given by From (49), one can easily derive the following relation: where We can rewrite the last term of (75) as Ref. [30] m ∑ i,j=1 Then, (75) and (76) give us Hence, we obtain Thus, we have the following results: Theorem 6. Let (M, g) be a (m + 1)-dimensional lightlike hypersurface of ( M, g, ∇). Then The equality holds for all p ∈ M if and only if either M is a screen homothetic with ϕ * = −1 or M is a totally geodesic with respect to ∇ and ∇ * .
Proof. From (78), we clearly have the inequality (79). The equality case of this inequality holds for all p ∈ M if and only if we have which implies that B ij = −C * ji for all i, j ∈ {1, . . . , m} or B ij = C * ji = 0. These show that M is a screen homothetic with ϕ * = −1 or M is a totally geodesic with respect to ∇ and ∇ * . Theorem 7. Let (M, g) be a (m + 1)-dimensional lightlike hypersurface of ( M, g, ∇). Then The equality holds for all p ∈ M if and only if either M is a screen homothetic with ϕ * = 1 or M is a totally geodesic with respect to ∇ and ∇ * .
Proof. From (78), the proof of (80) is straightforward. The equality case of this inequality holds for all p ∈ M if and only if we have which implies that B ij = C * ji for all i, j ∈ {1, . . . , m} or B ij = C * ji = 0. These relations show that M is a screen homothetic with ϕ * = 1 or M is a totally geodesic with respect to ∇ and ∇ * .
Considering Lemma 2 in Theorem 6 and Theorem 7, we get the following results: Corollary 9. Let M be an (m + 1)-dimensional lightlike hypersurface of M(c). Then The equality holds for all p ∈ M if and only if either M is a screen homothetic with ϕ * = −1 or M is a totally geodesic with respect to ∇ and ∇ * .
The equality holds for all p ∈ M if and only if either M is a screen homothetic with ϕ * = 1 or M is a totally geodesic with respect to ∇ and ∇ * .
For the next inequalities, we rewrite the second term of (77) as Ref. [30] On combining (77) and (84), we derive the following results: Theorem 8. Let (M, g) be an (m + 1)-dimensional lightlike hypersurface of ( M, g, ∇). Then The equality holds for every point p ∈ M if and only if M is minimal.
Corollary 11. Let M be an (m + 1)-dimensional lightlike hypersurface of M(c). Then whereÃ is given by (85). The equality holds for every point p ∈ M if and only if M is minimal.
Finally, we give some inequalities on totally umbilical lightlike hypersurfaces with respect to the their Levi-Civita connections: Theorem 9. Let (M, g) be an (m + 1)-dimensional screen homothetic lightlike hypersurface of a statistical manifold ( M, g, ∇). Suppose that (M, g) is totally umbilical with respect to ∇ 0 . For any unit vector field X in Γ(S(TM)), we have Ric S(TM) (X) ≥ Ric S(TM) (X) + ϕ * mH(2λ − B(X, X)) − 2λϕ * B(X, X). (86) The equality case of (86) holds for all X ∈ Γ(S(TM)) if and only if M is totally umbilical with respect to ∇ * .
Proof. From Proposition 10, we have If we take trace in (87) then we get Therefore, we obtain Putting X = E 1 , the proof of (86) is straightforward from (34) and (88). The equality case of (89) holds for all unit vector fields X in Γ(S(TM)) if and only if B(X, E j ) = 0 for all j ∈ {1, . . . , m}. Thus, using the fact that B is bilinear we get B(X, Y) = 0 for any X, Y ∈ Γ(S(TM)). From the (ii) statement of Corollary 2 we get M as totally umbilical with respect to ∇ * .
With a similar arguments as Theorem 9, we obtain the following theorem: Theorem 10. Let (M, g) be an (m + 1)-dimensional screen homothetic lightlike hypersurface of ( M, g, ∇). Suppose that (M, g) is totally umbilical with respect to ∇ 0 . For any unit vector field X in Γ(S(TM)), we have The equality case of (89) holds for all X ∈ Γ(S(TM)) if and only if M is totally umbilical with respect to ∇ * . Theorem 11. Let (M, g) be an (m + 1)-dimensional screen homothetic lightlike hypersurface of ( M, g, ∇). Suppose that (M, g) is totally umbilical with respect to ∇ 0 . For all p ∈ M we have σ S(TM) (p) ≥ σ S(TM) (p) + ϕ * mH(2λm − 2λ − mH) with the equality holds if and only if M is totally umbilical with respect to ∇ * .
Proof. Taking trace in (88) we get which implies (90). From (91), the equality case of (90) satisfies if and only if B ij = 0 for all i, j ∈ {1, . . . , m}. From the (ii) statement of Corollary 2 we get M is totally umbilical with respect to ∇ * .

Future Works
In the last section, we obtained Chen-like inequalities of a lightlike hypersurface of statistical manifolds and Lorentzian statistical space forms including screen scalar and mean curvatures. We also considered equality cases. Many similar results can be seen in Refs. [31][32][33][34]. We can use these results for future projects, and give some characterizations of a lightlike hypersurface on a statistical manifold. The results stated here motivate further studies to obtain similar relationships for many kinds of invariants of similar nature for several statistical submersions. In particular, by introducing the curvature invariant δ(m 1 , . . . , m k ) on lightlike hypersurfaces of a statistical manifold, we can obtain similar relationships for lightlike hypersurfaces and the equality cases can be discussed.