Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

The purpose of this article is to establish some inequalities concerning the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequalities. Some examples are also provided.


Introduction
One of the basic problems in information geometry is to study the geometric properties of a statistical manifold, a concept introduced by Amari [1]. There is great interest in researching statistical manifolds with applications not only in information geometry but also in differential geometry, physics, statistics, machine learning, etc. The natural relationship between statistical manifolds and entropy has been investigated by many researchers. Mishra and Kumar studied in [2] the structure of statistical manifolds with respect to a relative α-entropy in a Bayesian framework to obtain a generalized Bayesian Cramér-Rao inequality. The entropic dynamics on statistical manifolds of Gibbs distributions are investigated in [3], which can provide new insight in fields such as economics and ecology. Very recently, the authors in [4] studied the information-geometric properties of the statistical manifold to reduce predictive uncertainly via data assimilation. On the other hand, statistical manifolds provide a setting for the theory of submanifolds, where a basic problem is to find out simple relationships between the main intrinsic and extrinsic curvature invariants of submanifolds [5]. In this regard, many geometers studied certain types of geometric inequalities in (statistical) submanifolds. Comprehensive surveys on such inequalities are provided by Chen in [5][6][7].
Recently, there has been growing interest in the study of optimal inequalities involving the extrinsic δ-Casorati curvatures defined by Decu, Haesen, and Verstraelen in [8,9]. A long time ago, Casorati introduced a new measure of curvature of a surface (now called the Casorati curvature) following a common idea of curvature, more accurate than the Gauss and mean curvature [10]. Only in modern times did mathematical models involving Casorati curvature begin to be studied, e.g., in computer vision [11]. Furthermore, Verstraelen described qualitatively geometrical models for human early vision [12]. In this respect, the corresponding (visual) perceptions can be defined as the surfaces M 2 given by the Casorati curvatures C = 1 2 (k 2 1 + k 2 2 ) in E 3 , where k 1 and k 2 are the principal curvatures of M 2 in E 3 [12]. A geometrical interpretation of this curvature for submanifolds in Riemannian spaces was proved in [13]. In economics, the isotropical Casorati curvature of production surfaces was investigated in [14]. Initially, the δ-Casorati curvatures were studied in optimal inequalities for submanifolds in real space forms in [8,9]. Later, this topic was extensively studied (e.g., see [5]). Recently, Suceavȃ and Vajiac established inequalities involving some Chen invariants, mean curvature, and Casorati curvature for strictly convex Euclidean hypersurfaces [15].
Para-Kähler geometry refers to the algebra of para-complex numbers (or hyperbolic numbers) and, especially, to the study of para-Kähler structures and their derived forms. Para-complex numbers were defined by Graves [16] in 1845 as a generalization of complex numbers using the expression x + yj, where x and y ∈ R, and j satisfies j 2 = 1 and j = 1. Para-Kähler manifolds were first investigated as stratified spaces by Rashevskij [17] (1948). These manifolds were explicitly defined independently in 1949 by Rozenfeld [18] and Ruse [19]. This is a challenging topic now, related to many applications in mathematics, physics, and mechanics [20]. Defever, Deszcz, and Verstraelen considered para-Kähler manifolds that satisfy curvature conditions of the pseudosymmetric type, with applications in the theory of general relativity [21]. Mihai and Rosca dealt with CR-submanifolds of para-Kählerian manifolds, which carry skew-symmetric vector fields [22]. Recently, Fei and Zhang introduced in [23] the notion of a Codazzi-para-Kähler structure and studied the interaction of Codazzi couplings with para-Kähler geometry; essentially, a Codazzi-para-Kähler structure is simultaneously a statistical structure and a para-Kähler structure [23]. Very recently, Vîlcu investigated statistical manifolds endowed with almost product structures and para-Kähler-like statistical submersions [24].
In this paper, we establish optimal inequalities between the intrinsic scalar curvature and the extrinsic δ-Casorati curvatures of totally real spacelike submanifolds of statistical manifolds of the type para-Kähler space form. Moreover, we investigate the equality cases in these inequalities. We present also some examples.

Preliminaries
Let (M,ḡ) be a semi-Riemannian manifold, letḡ be a semi-Riemannian metric onM, and let∇ be an affine connection onM.
If the torsion tensor field of∇ vanishes and∇ḡ is symmetric, then (∇,ḡ) is called a statistical structure onM. Thus, (M,ḡ,∇) is named a statistical manifold [35]. Clearly, if (M, g,∇) is a statistical manifold, then (M,ḡ,∇ * ) is as well. Moreover, one easily can see that ∇ = (∇ * ) * and∇ 0 =∇ +∇ * 2 , where∇ 0 is the Levi-Civita connection ofM [36]. Let N be an n-dimensional submanifold of a 2m-dimensional statistical manifold (M, g,∇), with g as the induced metric on N and ∇ as the induced connection on N. (N, g, ∇) is then also a statistical manifold. For any X 1 , X 2 ∈ Γ(TN), we have the following formulas of Gauss [35]:∇ where the (0, 2)-tensors h and h * (bilinear and symmetric) are said to be the imbedding curvature tensor of N inM with respect to∇ and∇ * , respectively. The mean curvature vector fields of N for∇ and∇ * are given by, respectively, where tr g is the trace with respect to g.
For the Levi-Civita connection∇ 0 , we denote by h 0 = h+h * 2 the second fundamental form and by the mean curvature vector field of N. Let {e 1 , ..., e n } and {e n+1 , ..., e 2m } be orthonormal bases of the tangent space T x N and T ⊥ x N, respectively, at a point x ∈ N. The squared mean curvatures of N for ∇ and ∇ * then satisfy .., 2m}. Denote by C and C * the Casorati curvatures of the submanifold N, defined by the squared norms of h and h * , respectively, over the dimension n of N, as follows: We denote by C 0 the following expression: Consider V as an k-dimensional subspace of T x N, k ≥ 2, and {e 1 , . . . , e k } as an orthonormal frame of V. Thus, the Casorati curvatures C(V) and C * (V) of V are revealed by Denote by δ C (r; n − 1) andδ C (r; n − 1) the generalized normalized δ-Casorati curvatures of N, defined in [9] as if 0 < r < n(n − 1), and if r > n(n − 1), for an a(r) set as where r ∈ R + and r = n(n − 1). Additionally, denote by δ * C (r; n − 1) andδ * C (r; n − 1) the dual generalized normalized δ * -Casorati curvatures of the submanifold N, defined as follows: Furthermore, the dual normalized δ * -Casorati curvatures δ * C (n − 1) and δ * C (n − 1) of the submanifold N inM can be written as A statistical submanifold (N, g, ∇) of (M,ḡ,∇) is said to be totally geodesic with respect to∇ if the second fundamental form of N vanishes identically [35].
Let R andR be the (0, 4)-curvature tensors for the connections ∇ and∇, respectively. For the vector fields X 1 , X 2 , X 3 , X 4 tangent to N, the equation of Gauss on the connection ∇ is then [36] g(R(X 1 , Similarly, let R * andR * be the (0, 4)-curvature tensors for the connections ∇ * and∇ * , respectively.
Next, for the vector fields X 1 , X 2 , X 3 , X 4 tangent to N, the equation of Gauss on the connection∇ * becomes [36] In general, g(R(X 1 , X 2 )X 3 , X 4 ) is not skew-symmetric for X 3 , X 4 [37], i.e., In order to define the sectional curvature of a statistical manifold, the property (5) of R is inconvenient. To overcome this, we define according to [35] the statistical curvature tensor field denoted by S for the statistical manifold (N, g, ∇): for X 1 , X 2 , X 3 ∈ Γ(TN). Naturally, S is skew-symmetric relative to X 3 , X 4 . Moreover, S satisfies the first Bianchi identity. It follows that S is a Riemann-curvature-like tensor [37]. For a non-degenerate 2-dimensional subspace π of the tangent space T x N, at a point x ∈ N, we can define immediately the sectional curvature of (N, ∇, g) [35] by where {X, Y} is a basis of π.
The scalar curvature τ of (N, ∇, g) at a point x ∈ N is defined by the following expression: where {e 1 , ..., e n } is an orthonormal frame at x. Moreover, the normalized scalar curvature ρ of N is defined as An almost product structure on a smooth manifoldM is a (1, 1)-tensor field P = ±I, such that where I is the identity tensor field of type (1, 1) onM. A manifoldM is called an almost para-Hermitian manifold [7] ifM is endowed with an almost product structure P and a semi-Riemannian metricḡ such that for all vector fields X 1 , X 2 onM. Hence, the dimension of an almost para-Hermitian manifold denoted by (M, P,ḡ) is even, i.e., dimM = 2m, and the metric is neutral. If∇P = 0, then (M, P,ḡ) is named a para-Kähler manifold [7], where∇ is the Levi-Civita connection ofM.
Consequently, the notion of a para-Kähler-like statistical manifold is a generalization of the notion of the para-Kähler manifold (when, in particular, we have P = P * , i.e., Formula (11) reduces to Formula (10), and∇ is the Levi-Civita connection).
A para-Kähler-like statistical manifold (M,∇, P,ḡ) is called a statistical manifold of a type para-Kähler space form if the following formula holds [24]: for any vector fields X 1 , X 2 , X 3 , whereR is the curvature tensor of the connection∇, and c is a real constant. A submanifold N in an almost para-Hermitian (like) manifold (M, P,ḡ) is called totally real if P maps each tangent space T x N into its corresponding normal space T ⊥ x N. In the particular case, when P interchanges each tangent space with its corresponding normal space, then N is said to be Lagrangian.
Next, we consider the constrained extremum problem where N is a submanifold of a (semi)-Riemannian manifold (M,ḡ), and f :M → R is a function of differentiability class C 2 .

Theorem 1 ([38]
). If N is complete and connected, (grad f )(x 0 ) ∈ T ⊥ x 0 N for a point x 0 ∈ N, and the bilinear form F : is positive definite in x 0 , then x 0 is the optimal solution of the problem (13), where h is the second fundamental form of N. (14) is positive semi-definite on the submanifold N, then the critical points of f |N are global optimal solutions of the problem (13).

Main Inequalities
It is well-known that one of the most fundamental problems in submanifold theory is the following (see, e.g., [39]).
Problem. Establish a simple relationship between the main extrinsic invariants and the main intrinsic invariants of a submanifold.
The following theorem provides an answer to this problem.

Proof. From Formulas
For x ∈ N, let {e 1 , ..., e n } be an orthonormal frame in T x N, and let {e n+1 , ..., e 2m } be an orthonormal frame in T ⊥ x N. Considering X 1 = X 3 = e i and X 2 = X 4 = e j with i, j ∈ {1, ..., n}, from Equation (18), we obtain Using notations H 0 and C 0 given by (1) and (2), then Equation (19) can be written as We choose Q, a quadratic polynomial expressed by where V is a hyperplane of T x N. Let V be a hyperplane spanned by the tangent vectors e 1 , ..., e n−1 , without losing the generality. We see that Moreover, (22) becomes r(n − 1) + a(r)n n(n − 1) We choose q α , a quadratic form defined by q α : R n → R for any α ∈ {n + 1, ..., 2m}, Our aim is to study the constrained extremum problem min q α with the condition We solve the following system of first order partial derivatives: for every i ∈ {1, ..., n − 1}, α ∈ {n + 1, ..., 2m}.
The above system solutions are for any i ∈ {1, ..., n − 1}, α ∈ {n + 1, ..., 2m}. For p ∈ W, let F be a 2-form, F : where h is the second fundamental form of W in R n+1 , and ·,· is the standard inner product on R n . The Hessian matrix of q α is as follows: is a real constant. The hyperplane W is totally geodesic in R n , so we have ∑ n i=1 U i = 0, for a vector field X 1 ∈ T p W. We obtain From Remark 1, the critical point (h 0α 11 , ..., h 0α nn ) of q α is the global minimum point of the problem. We then achieve Q ≥ 0.
We have thus proved the inequalities (15) and (16), considering infimum and supremum, respectively, over all tangent hyperplanes V of T x N.
Similarly, we investigated the equality cases of the inequalities (15) and (16) are the solutions of the following system of linear homogeneous equations: We find that the critical points satisfy h 0α ij = 0, with i, j ∈ {1, ..., n} and α ∈ {n + 1, ..., 2m}. However, because of the conditions Q ≥ 0 and Q(h c ) = 0, the critical point h c is a minimum point of Q. Thus, both inequalities (15) and (16) satisfy the equality cases if and only if h α ij = −h * α ij , for i, j ∈ {1, ..., n}, α ∈ {n + 1, ..., 2m}. (15) and (16) hold for all unit tangent vectors at x if and only if x is a totally geodesic point with respect to the Levi-Civita connection. It is argued that the submanifold N is totally geodesic with respect to the Levi-Civita connection∇ 0 in Equation (17).

Corollary 2.
Let N be a spacelike Lagrangian submanifold of the para-Kähler space formM 2n n . We then have where r ∈ R such that 0 < r < n(n − 1), and (ii)δ where r ∈ R such that r > n(n − 1). Furthermore, the equality sign of (25) and (26) holds identically at all points x ∈ N if and only if N is a totally geodesic submanifold.

Remark 3.
In a future article, we will investigate the optimality of the inequalities in the previous corollary. Due to additional properties that the second fundamental form h has in the Lagrangian case (see [33], Lemma 3.2), it is expected that the inequalities can be improved, and the case of equality can be achieved for another class of submanifolds.

Remark 4.
In the main inequalities demonstrated in this section, we established several elementary relationships between some fundamental extrinsic and intrinsic curvature invariants of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. The geometric significance of these inequalities is as follows. It is known that curvature invariants play basic roles in Riemannian geometry and in mathematical physics. It is worth mentioning that, due to the fact that Riemannian invariants model the behavior of a Riemannian space, these invariants are called Riemannian DNA [7]. The extrinsic curvature invariant that measures the shape of submanifolds in the most natural, best agreement with our intuitive, common-sense idea or feeling of curvature very likely is the Casorati curvature [40]. Obtaining elementary relationships between extrinsic and intrinsic curvature invariants is a fundamental problem in modern Riemannian geometry, since it is essential to be able to control the extrinsic quantities relative to intrinsic ones [41]. As inequalities provide such elementary relationships, the relevance of the results stated in this paper is clear. By carefully analyzing the nature of the terms in the main inequalities proved above, one can deduce that the simplest intrinsic curvature invariant, namely, the (normalized) scalar curvature, has an upper bound expressed in terms of some basic extrinsic curvature invariants.

Examples
Example 1. We notice that any para-Kähler manifold is a para-Kähler-like statistical manifold, while para-Kähler space forms provide particular examples of statistical manifolds of type para-Kähler space forms. The prototype of flat para-Kähler spaces is given by the para-Kähler n-plane E 2n n (see [42]). The complete classification of para-Kähler space forms of nonzero para-sectional curvature was realized by Gadea and Montesinos-Amilibia [43]. In particular, any totally geodesic totally real spacelike submanifold in such spaces provides examples of submanifolds satisfying the equality case in the inequalities stated above at all points. In the case of non-totally geodesic points, it is clear that the inequalities are strict. Both totally geodesic and non-totally geodesic examples of spacelike Lagrangian submanifolds in E 2n n can be found in [32,33,44]. Next, we will present two such examples of submanifolds in E 2n n .
Example 2. The spacelike n-dimensional plane provides a very natural example of a totally geodesic spacelike Lagrangian submanifold of E 2n n . This submanifold satisfies the equality cases of the inequalities stated above at all points. Example 3. If (a 1 (t), a n+1 (t)) is a non-degenerate spacelike curve in E 2 1 , then the submanifold of E 2n n defined by f (t, x 2 , . . . , x n ) = (a 1 (t), 0, . . . , 0, a n+1 (t), x 2 , . . . , x n ), is a spacelike Lagrangian submanifold without totally geodesic points. In this case, the inequalities stated above are strictly satisfied at all points, and the case of equality cannot be reached.

Example 4.
On R 2n with the coordinates (x 1 , . . . , x n , y 1 , . . . , y n ), we consider the flat affine connection ∇ and the pseudo-Riemannian metric g expressed by where α = 0 is a real constant. Consider on R 2n the almost product structure P given by P(∂ x i ) = ∂ y i , P(∂ y i ) = ∂ x i , i = 1, . . . , n.
It is then easy to check that (R 2n , ∇, P, g) is a statistical manifold of the type para-Kähler space form, being a flat para-Kähler-like statistical manifold. Note that the conjugate connection ∇ * is also flat, and P * is given by Let X be an open set of R n , and define an isometric immersion u : X → R 2n by u(y 1 , . . . , y n ) = (0, . . . , 0, y 1 , . . . , y n ).
u then defines a spacelike Lagrangian submanifold of (R 2n , ∇, P, g). Moreover, we have h = −h * and the equality case holds in the inequalities stated above at all points.

Conclusions
In this paper, we proved new inequalities between extrinsic and intrinsic invariants (δ-Casorati curvatures and a scalar curvature) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Furthermore, we investigated the equality cases and obtained some examples. This research may be a challenge for new developments focused on new relationships in terms of various invariants, for different types of statistical submanifolds in this ambient space.