Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

: In this paper, we prove some inequalities in terms of the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.


Introduction
The problem of discovering simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds is a basic problem in submanifold theory [1]. In this respect, beautiful results focus on certain types of geometric inequalities. Moreover, another basic problem in this field is to study the ideal submanifolds in a space form, namely to investigate the submanifolds which satisfy the equality case of such inequalities [2].
The method of looking for Chen invariants answers the problems posed above. First, Chen demonstrated in [3] an optimal inequality for a submanifold on a real space form between the intrinsically defined δ-curvature and the extrinsically defined squared mean curvature. This approach initiated a new line of research and was extended to various types of submanifolds in several types of ambient spaces, e.g., submanifolds in complex space forms of constant holomorphic sectional curvature (see [4][5][6][7]). The submanifolds attaining the equality of these inequalities (called Chen ideal submanifolds) were also investigated. Recently, Chen et al. classified δ(2, n − 2)-ideal Lagrangian submanifolds in complex space forms in [8].
Moreover, new solutions to the above problems are given by the inequalities involving δ-Casorati curvatures, initiated in [9,10]. In the search for a true measure of curvature, Casorati in 1890 proposed the curvature which nowadays bears his name because it better corresponds with our common intuition of curvature than Gauss and mean curvature [11]. However, this notion of curvature was soon forgotten and was rediscovered by Koenderink working in the field of computer vision [12]. Verstraelen developed some geometrical models for early vision, presenting perception via the Casorati curvature of sensation [13]. A geometrical interpretation of this type of curvature for submanifolds in Remark 1. If (M,g,∇) is a statistical manifold, then we remark that 1. (∇ * ) * =∇; 2. (M,g,∇ * ) is also a statistical manifold; 3.∇ always has a dual connection∇ * satisfying where∇ 0 is the Levi-Civita connection onM.
Let M be an m-dimensional submanifold of a 2n-dimensional statistical manifold (M,g) and g the induced metric on M. The Gauss formulas are given bỹ for any X, Y ∈ Γ(TM), where h and h * are symmetric and bilinear (0, 2)-tensors, called the imbedding curvature tensor of M inM for∇ and∇ * , respectively. Denote the curvature tensor fields of ∇ and∇ by R andR, respectively. Then, the Gauss equation concerning the connection∇ is ( [41]) for any X, Y, Z, W ∈ Γ(TM).
In addition, denote the curvature tensor fields of the connections ∇ * and∇ * by R * andR * , respectively. Then the Gauss equation concerning the connection∇ * is ( [41]) for any X, Y, Z, W ∈ Γ(TM).
If M is a submanifold of a statistical manifold (M,g,∇), then (M, g, ∇) is also a statistical manifold with the induced metric g and the induced connection ∇.
If π = span R {u 1 , u 2 } is a 2-dimensional subspace of T p M, for p ∈ M, then the sectional curvature of M is defined by [40]: Let {e 1 , ..., e m } be an orthonormal basis of the tangent space T p M, for p ∈ M, and let {e m+1 , ..., e 2n } be an orthonormal basis of the normal space T ⊥ p M. The scalar curvature τ at p is given by and the normalized scalar curvature ρ of M is defined as The mean curvature vector fields of M, denoted by H and H * , are given by From Equation (1), we get 2h 0 = h + h * and 2H 0 = H + H * , where h 0 and H 0 are the second fundamental form and the mean curvature field of M, respectively, with respect to the Levi-Civita connection ∇ 0 on M.
The squared mean curvatures of the submanifold M inM have the expressions where h α ij =g(h(e i , e j ), e α ) and h * α ij =g(h * (e i , e j ), e α ), for i, j ∈ {1, ..., m}, α ∈ {m + 1, ..., 2n}. Denote by C and C * the Casorati curvatures of the submanifold M, defined by the squared norms of h and h * , respectively, over the dimension m, as follows: Let L be an s-dimensional subspace of T p M, s ≥ 2 and let {e 1 , . . . , e s } be an orthonormal basis of L. Hence, the Casorati curvatures C(L) and C * (L) of L are given by The normalized δ-Casorati curvatures δ C (m − 1) andδ C (m − 1) of the submanifold M n are given by Moreover, the dual normalized δ * -Casorati curvatures δ * C (m − 1) and δ * C (m − 1) of the submanifold M inM are defined as Denote by δ C (r; m − 1) andδ C (r; m − 1), the generalized normalized δ-Casorati curvatures of M, defined in [10] as where r ∈ R + and r = m(m − 1). Furthermore, denote by δ * C (r; m − 1) andδ * C (r; m − 1) the dual generalized normalized δ * -Casorati curvatures of the submanifold M, defined as follows: , for a(r) set above. A statistical submanifold (M, g, ∇) of (M,g,∇) is called totally geodesic with respect to the connection∇ if the second fundamental form h of M for∇ vanishes identically [40].
For a holomorphic statistical manifold, the following formula holds: for any X, Y, Z, W ∈ Γ(TM). A holomorphic statistical manifold (M,∇,g, J) is said to be of constant holomorphic sectional curvature c ∈ R if the following formula holds [42]: for any X, Y, Z ∈ Γ(TM), whereS is the statistical curvature tensor field ofM.
Let M be an m-dimensional statistical submanifold of a holomorphic statistical manifold (M,∇,g, J). For any vector field X tangent to M we can decompose where PX and FX are the tangent component and the normal component, respectively, of JX. Given a local orthonormal frame {e 1 , e 2 , · · · , e m } of M, then the squared norm of P is expressed by Next, we consider the constrained extremum problem where M is a Riemannian submanifold of a Riemannian manifold (M,g), and f :M → R is a function of differentiability class C 2 .
Theorem 1 ([44]). If M is complete and connected, (grad f )(p) ∈ T ⊥ p M for a point p ∈ M, and the bilinear form A : is positive definite in p, then p is the optimal solution of the Problem (14). (15) is positive semi-definite on the submanifold M, then the critical points of f |M are global optimal solutions of the Problem (14).

Main Inequalities
Theorem 2. Let M be an m-dimensional statistical submanifold of a 2n-dimensional holomorphic statistical manifold (M,∇,g, J) of constant holomorphic sectional curvature c. Then we have for any real number r such that 0 < r < m(m − 1), where δ 0 C (r; m − 1) = δ C (r;m−1)+δ * C (r;m−1) 2 and C 0 = C+C * 2 ; and (ii) for any real number r such that r > m(m − 1), whereδ 0 Moreover, the equality cases of Inequalities (16) and (17) hold identically at all points p ∈ M if and only if the following condition is satisfied: where h and h * are the imbedding curvature tensors of the submanifold associated to the dual connections∇ and ∇ * , respectively.
For p ∈ M, we choose {e 1 , ..., e m } and {e m+1 , ..., e 2n } orthonormal bases of T p M and T ⊥ p M, respectively. For X = Z = e i and Y = W = e j with i, j ∈ {1, ..., m}, from the Equation (19), it follows that Denoting 2H 0 = H + H * and 2C 0 = C + C * , Equation (20) becomes Let P be the quadratic polynomial defined by where L is a hyperplane of T p M. We consider that the hyperplane L is spanned by the tangent vectors e 1 , ..., e m−1 , without loss of generality. Therefore, we get Then, Equation (23) yields Let f α be a quadratic form defined by f α : R m → R for any α ∈ {m + 1, ..., 2n}, We investigate the constrained extremum problem where k α is a real constant. We obtain the system of first-order partial derivatives: for every i ∈ {1, ..., m − 1}, α ∈ {m + 1, ..., 2n}. It follows that the constrained critical point is where h is the second fundamental form of Q in R m+1 and ·,· is the standard inner product on R m . The Hessian matrix of f α is given by is a real constant. The condition ∑ m i=1 X i = 0 is satisfied, for a vector field X ∈ T p Q, as the hyperplane Q is totally geodesic in R m . Then, we achieve Applying Remark 3, the critical point (h 0α 11 , ..., h 0α mm ) of f α is the global minimum point of the problem. Since f α (h 0α 11 , ..., h 0α mm ) = 0, we get P ≥ 0. We have then proved Inequalities (16) and (17), considering infimum and supremum, respectively, over all tangent hyperplanes L of T p M.
In addition, we study the equality cases of Inequalities (16) and (17). First, we find out the critical points of P h c = (h 0 m+1 as the solutions of following system of linear homogeneous equations: The critical points satisfy h 0α ij = 0, with i, j ∈ {1, ..., m} and α ∈ {m + 1, ..., 2n}. On the other hand, we know that P ≥ 0 and P (h c ) = 0, then the critical point h c is a minimum point of P. Consequently, the cases of equality hold in both Inequalities (16) and (17) if and only if h α ij = −h * α ij , for i, j ∈ {1, ..., m}, α ∈ {m + 1, ..., 2n}. (18), the submanifold M is totally geodesic with respect to the Levi-Civita connection∇ 0 . Then, the equality cases of Inequalities (16) and (17) hold for all unit tangent vectors at p if and only if p is a totally geodesic point with respect to the Levi-Civita connection.
where 2δ 0 C (m − 1) = δ C (m − 1) + δ * C (m − 1) and 2C 0 = C + C * , and (ii) Moreover, the equality cases of Inequalities (24) and (25) hold identically at all points if and only if h and h * satisfy the condition in Equation (18), which implies that M is a totally geodesic submanifold with respect to the Levi-Civita connection.
Let G be a g-natural metric on R 4 and J a complex structure defined by Oproiu ([45]) such that R 4 is Kählerian, as follows: Let the function u be defined as u(x 1 , x 2 , y 1 , y 2 ) = 1+ √ 1+4t 2 . Therefore, the function v becomes v(x 1 , x 2 , y 1 , y 2 ) = 1. Then, for the metric G and the complex structure J, there exists a tensor field K such that (R 4 ,∇ := ∇ G + K,g := G, J) is a special Kähler manifold [46]. Notice that a holomorphic statistical structure of holomorphic curvature 0 is nothing but a special Kähler manifold [43].

Conclusions
In this research study, we provided new solutions to the fundamental problem of finding simple relationships between various invariants (intrinsic and extrinsic) of the submanifolds. In this respect, we obtained inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. In addition, we characterized the equality cases. These results may stimulate new research aimed at obtaining similar relationships in terms of various invariants, for statistical submanifolds in other ambient spaces.

Conflicts of Interest:
The authors declare no conflict of interest.