Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and ∇ * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant φ -sectional curvature. In addition, we give examples to show that the total space is a sphere.


Introduction
A statistical model in information geometry has a Fisher metric as a Riemannian metric with an affine connection, whose connection is constructed from the average of the probability distribution.In the statistical models, a pair of a Fisher information metric and an affine connection gives the geometric structure, called the Chentsov-Amari connection [1], whose geometric structure is a generalization of a pair of a Riemannian metric and a Levi-Civita connection.By generalizing the geometric structure, a statistical structure has been studied in information geometry.Applying this idea to Sasakian manifolds, one arrived at the definition of a Sasakian statistical structure as a generalization of a Sasakian structure.In other words, it is a triple of an affine connection, a Riemannian metric, and a Sasakian structure on an odd dimensional manifold [2].The geometry of such a manifold is closely related to affine geometry and Hessian geometry.In such manifolds, there are the fundamental equations such as Gauss formula, Weingarten formula and the equations of Gauss, Codazzi and Ricci in submanifolds of a statistical manifold [3].
On the other hand, it is well-known that the Casorati curvature as a new extrinsic invariant is defined as the normalized square of the length of the second fundamental form, introduced by Casorati ( [4,5]).Geometric meanings of Casorati curavature were found in visual perception of shape and appearance ( [6][7][8]).Some optimal inequalities involving Casorati curvatures were proved in [9][10][11][12][13][14][15] for several submanifolds in real, complex and quaternionic space forms with various connections.Moreover, Lee et al. established that the normalized scalar curvature is bounded by Casorati curvatures of submanifolds in a statistical manifold of constant curvature [16].In Kenmotsu statistical manifolds, Decu et al. investigate curvature properties and establish optimizations in terms of a new extrinsic invariant (the normalized δ-Casorati curvature) and an intrinsic invariant (the scalar curvature) [17].
In our paper, we establish optimizations of the normalized scalar curvature (the intrinsic invariant) for a new extrinsic invariant (generalized normalized Casorati curvatures) on Legendrian and slant submanifolds in a Sasakian statistical space form.Moreover, we provide some examples for special Sasakian statistical sphere S 2m+1 of statistical sectional curvature 1.

Preliminaries
Let (M m , g) be a m-dimensional Riemannian manifold with an affine connection ∇.We denote by Γ(TM) the collection of all vector fields on M.
Definition 1 ([18]).A pair ∇, g is called a statistical structure on M if ∇ is a torsion free connection on M and the covariant derivative ∇g is symmetric.
Definition 2. A statistical manifold (M, g, ∇) is a Riemannian manifold, endowed with a pair of torsion-free affine connections ∇ and ∇ * satisfying for any vector fields X, Y and Z.The connections ∇ and ∇ * are called dual connections.

Remark 1.
(a) ∇ If {e 1 , ..., e n } is an orthonormal basis of the tangent space T p M and {e n+1 , ..., e m } is an orthonormal basis of the normal space T ⊥ p M, then the scalar curvature τ at p is defined as g S e i , e j e j , e i and the normalized scalar curvature ρ of M is defined as .
Then it is well-known that the squared mean curvatures of the submanifold M in M are defined by and the squared norms of h and h * over dimension n is denoted by C and C * are called the Casorati curvatures of the submanifold M, respectively.Therefore, we have Similarly, the dual normalized δ * -Casorati curvatures δ * C (n − 1) and δ * C (n − 1) of the submanifold M are defined as and The generalized normalized δ-Casorati curvatures δ C (t; n − 1) and δ C (t; n − 1) of the submanifold M are defined for any positive real number t = n(n − 1) as Moreover, the dual generalized normalized δ-Casorati curvatures δ * C (t; n − 1) and δ * C (t; n − 1) of the submanifold M are defined for any positive real number t = n(n − 1) as The following lemma plays a key role in the proof of our main theorem.

Lemma 1 ([20]).
Let Then, the constrained extremum problem min x∈Γ f (x) has a global solution as follows: Definition 4. A triple (g, ϕ, ξ) is called an almost contact metric structure on M if the following equations hold where ϕ is a section of TM ⊗ TM * and ξ is the structure vector field on M.
Definition 5. A quadraple ∇, g, ϕ, ξ is called a Sasakian statistical structure on M if ∇, g is a statistical structure.
A submanifold M n normal to ξ in a Sasakian statistical manifold M 2m+1 is said to be a C-totally real submanifold.In this case, ϕ For submanifolds tangent to ξ, there is a θ-slant submanifold of a Sasakian statistical manifold as follows [21]: A submanifold M n tangent to ξ in a Sasakian statistical manifold is called a θ-slant submanifold if for any vector X ∈ T p M, linearly independent on ξ p , the angle between ϕX and T p M is a constant θ ∈ [0, π 2 ], called the slant angle of M in M. In particular, if θ = 0 and θ = π 2 , M is invariant and anti-invariant, respectively.
Let p ∈ M and the set {e 1 , e 2 , where C 0 = 1 2 (C + C * ).Define a quadratic polynomial in the components of the second fundamental form h 0 by where L is a hyperplane of T p M. Without loss of generality, we can assume that L is spanned by e 1 , • • • , e n−1 .Then we derive and the constrained extremum problem min f α subject to where c α is a real constant.Comparing (10) with the quadratic function in Lemma 1, we see that Therefore, we have the critical point h 0α 11 , • • • , h 0α nn , given by which implies Therefore, we derive Therefore, we have the following theorem: Theorem 3. Let M be an n-dimensional C-totally real submanifold of a (2m + 1)-dimensional Sasakian statistical manifold M, ∇, g, ϕ, ξ .When 0 < t < n 2 − n, the generalized normalized δ-Casorati curvature . The equality case holds identically at any point p ∈ M if and only if h = −h * .
For a unit hypersphere S 2n+1 in R 2n+2 , the unit normal vector field N of S 2n+1 provides the structure vector field ξ = −JN with the standard almost complex structure J on R 2n+2 = C n+1 .In addition, ϕ = π • J is the natural projection of the tangent space of R 2n+2 onto the tangent space of S 2n+1 .Then we obtain the standard Sasakian structure (g, ϕ, ξ) on S 2n+1 .From [2], we can construct a Sasakian statistical structures on S 2n+1 of constant statistical sectional curvature 1.Therefore, we have the following optimal inequality: Example 1.Let M be an n-dimensional C-totally real submanifold of S 2m+1 .Then, the generalized normalized 2 in Theorem 3, we have an optimization for a normalized δ-Casoratic curvature as follows: Corollary 1.Let M be an n-dimensional C-totally real submanifold of a (2m + 1)-dimensional Sasakian statistical manifold M, ∇, g, ϕ, ξ .Then, the normalized δ-Casorati curvature δ 0 C (n − 1) on M satisfies Proof.Taking t = n(n−1) 2 in δ 0 C (t, n − 1), we have the following relation: p in any point p ∈ M. Therefore, we have an optimal inequality for the normalized δ-Casorati curvature δ 0 C (n − 1).
Theorem 4. Let M be an n-dimensional θ-slant submanifold of a (2m + 1)-dimensional Sasakian statistical manifold M, ∇, g, ϕ, ξ .When 0 < t < n 2 − n, the generalized normalized δ-Casorati curvature δ 0 Proof.Let p ∈ M and the set {e 1 , e 2 , g h * (e i , e j ), h(e i , e j ) By using a similar argument as in the proof of Theorem 3, we get Therefore, we have an ineqaulity as follows: If M is an invariant submanifold, then θ = 0. Then we obtain Corollary 2. Let M n be an n-dimensional invariant submanifold of a (2m +
(2) In any optimization throughout our paper, the equality cases hold if and only if a submanifold is totally geodesic from h = −h * .(3) In the case for t > n 2 − n, the methods of finding the above inequalities are analogous.