Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h∗ of the submanifold (associated with the dual connections) satisfy h=−h∗, i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.


Introduction
A fundamental problem in the general theory of Riemannian submanifolds is to establish simple relationships between the main intrinsic invariants and the main extrinsic invariants of the submanifolds [1]. Obviously, such simple relationships can be provided by certain types of inequalities. Furthermore, the investigation of ideal submanifolds in a space form, namely the study of submanifolds satisfying the equality case of such inequalities, is another basic problem of this field [2].
On one hand, it is well known that the theory of Chen invariants provides solutions to these problems. Chen proved initially in [1] some optimal inequalities between the intrinsic δ-curvatures of Chen and the extrinsic squared mean curvature of submanifolds in a real space form. Later, these sharp inequalities (called Chen inequalities) have been extended for different types of submanifolds in various ambient spaces, for example, arbitrary Riemannian manifolds, Kähler manifolds, and Kenmotsu space forms (see [3] and the references therein). The Chen ideal submanifolds have also been investigated, i.e., the submanifolds that do realize an optimal equality in Chen inequalities (see, e.g., the recent work [4]).
Let M be an (m + 1)-dimensional submanifold of a (2n + 1)-dimensional statistical manifold (M,ḡ) and g the induced metric on M. Then, the Gauss formulas are as follows: for any X, Y ∈ Γ(TM), where h and h * are symmetric (0, 2)-tensors, called the imbedding curvature tensor of M inM for∇, and the imbedding curvature tensor of M inM for∇ * , respectively. Denote by R andR the curvature tensors fields associated with ∇ and∇, respectively. Then the Gauss equation for the submanifold M ofM (with respect to the connection∇) is [33] g(R(X, Y)Z, for any X, Y, Z, W ∈ Γ(TM). Similarly, if R * andR * denote the curvature tensors fields associated with the connections ∇ * and ∇ * , respectively, then the Gauss equation with respect to the connection∇ * is [33] for any X, Y, Z, W ∈ Γ(TM).
If (M,ḡ,∇) is a statistical manifold and M is a submanifold ofM, then (M, g, ∇) is also a statistical manifold with the induced connection ∇ and the induced metric g.
For a statistical manifold (M, g, ∇), the tensor field S ∈ Γ(TM (1,3) ) called the statistical curvature tensor field of (M, g, ∇) is defined as [46]: for X, Y, Z ∈ Γ(TM). For a statistical structure (ḡ,∇) onM, we setK =∇ −∇ 0 according to [45], which implies: It follows that the tensor fieldK ∈ Γ(TM (1,2) ) satisfies for any X, Y, Z ∈ Γ(TM). We denoteK Let π = span R {v, w} be a two-dimensional subspace of T p M, for p ∈ M. The sectional curvature of M for π is defined by [46]: Let {e 1 , ..., e m+1 } be an orthonormal basis of the tangent space T p M, for p ∈ M, and let {e m+2 , ..., e 2n+1 } 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: We remark that, from Equation (1), we derive 2h 0 = h + h * and therefore 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 Casorati curvatures of the submanifold M inM are defined by the squared norms of h and h * over the dimension (m + 1), denoted by C and C * , respectively, 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) andδ C (m) of the submanifold M n are given by Similarly, we can define the dual normalized δ * -Casorati curvatures δ * C (m) and δ * C (m) of the submanifold M inM, just replacing C by C * in the last two relations: The generalized normalized δ-Casorati curvatures δ C (r; m) andδ C (r; m) of M inM are defined in [6] as: if 0 < r < m(m + 1), and if r > m(m + 1), whereby a(r) is set as for any positive real number r, different from m(m + 1). Moreover, the dual generalized normalized δ * -Casorati curvatures δ * C (r; m) andδ * C (r; m) of the submanifold M inM are given by: if r > m(m + 1), whereby a(r) is set above. Obviously, the (dual) generalized normalized δ-and δ * -Casorati curvatures are a natural generalization of the (dual) normalized δ-and δ * -Casorati curvatures, due the fact that δ C (m),δ C (m), δ * C (m) and δ * C (m) can be recovered from δ C (r; m),δ C (r; m), δ * C (r; m) andδ * C (r; m), respectively, for some particular values of r as follows: We recall that a statistical submanifold (M, g, ∇) in (M,ḡ,∇) is called totally geodesic with respect to the connection∇ if the second fundamental form h of M for∇ vanishes identically [46].
Consider that (M,ḡ) is a (2n + 1)-dimensional almost contact metric manifold with the structure tensors (ḡ, φ, ξ), whereḡ ∈ Γ(TM (0,2) ) is the Riemannian metric onM, φ ∈ Γ(TM (1,1) ), ξ ∈ Γ(TM). These structure tensors satisfy: The almost contact metric manifold (M,ḡ) is said to be a Kenmotsu manifold if the formulas: We outline that the Kenmotsu geometry turns out to be a valuable chapter of contact geometry with many applications in theoretical physics, providing an excellent setting to model space time near black holes or bodies with large gravitational fields [47]. One reason to study the Kenmotsu manifolds is that this class of manifolds is one of the three classes in Tanno's classification of connected almost contact metric manifolds whose automorphism group has a maximum dimension. Another reason is that these manifolds are in some sense complementary to Sasaki manifolds: while some properties of Kenmotsu manifolds can be obtained deforming slowly properties of Sasaki manifolds, others are very different [22].
Let M be an (m + 1)-dimensional statistical submanifold of a Kenmotsu statistical manifold (M,∇,ḡ, φ, ξ). Then, any vector field X tangent to M can be decomposed uniquely into its tangent and normal components PX and FX, respectively, thus: Given a local orthonormal frame {e 1 , e 2 , · · · , e m+1 } of M, then the squared norm of P is expressed by: Next, we consider the constrained extremum problem The Hessian of the function f on the manifoldM is defined by the (0, 2)-tensor: where∇ 0 is the Levi-Civita connection onM. We recall the following result.

Theorem 1.
If the submanifold M is complete and connected, in addition to the gradient of f being normal at a point p to M, and the bilinear form A : is positive definite in p, then p is an optimal solution of the problem (13) [48]. (14) is positive semi-definite on the submanifold M, then the critical points of f |M, which coincide with the points where the gradient of f is normal to M, are global optimal solutions of the problem (13) [48].
In addition, the equality cases of Equations (15) and (16) hold identically at all points p ∈ M if and only if the imbedding curvature tensors h and h * of the submanifold associated with the dual connections∇ and∇ * satisfy h = −h * .
Proof. From Equations (2)-(4), we get: where X, Y, Z, W ∈ Γ(TM). Let {e 1 , ..., e m+1 } and {e m+2 , ..., e 2n+1 } be orthonormal bases of T p M and T ⊥ p M, respectively, for p ∈ M. Setting X = Z = e i and Y = W = e j for i, j ∈ {1, ..., m + 1}, and summing over 1 ≤ i, j ≤ m + 1 in Equation (18), we obtain: Because 2H 0 = H + H * and 2C 0 = C + C * , the latter relation becomes: Let P be the quadratic polynomial in the components of the second fundamental form defined by where L is a hyperplane of T p M. Suppose that the hyperplane L is spanned by the tangent vectors e 1 , ..., e m , avoiding loss of generality. Then, from Equations (20) and (21), we derive that P has the expression Moreover, from Equation (22), we get: Let f α be a quadratic form defined for any α ∈ {m + 2, ..., 2n + 1} by f α : R m+1 → R, We study the constrained extremum problem where k α is a real constant. The first order partial derivatives system is: for every i ∈ {1, ..., m}, α ∈ {m + 2, ..., 2n + 1}. The system solution, satisfying the constraint Q, is the critical point with the expression: for any i ∈ {1, ..., m}, α ∈ {m + 2, ..., 2n + 1}. Let p be an arbitrary point of Q, p ∈ Q. We consider that the 2-form A : T p Q × T p Q → R given by: where h is the second fundamental form of Q in R m+1 and ·,· is the standard inner product on R m+1 .
The Hessian matrix of f α is as follows: As the hyperplane Q is totally geodesic in R m+1 , considering a vector field X ∈ T p Q, that is satisfying the condition ∑ m+1 i=1 X i = 0, we get However, according to the Remark 2, the critical point (h 0α 11 , ..., h 0α m+1 m+1 ) is the only optimal solution, i.e., the global minimum point of problem. In addition, f α (h 0α 11 , ..., h 0α m+1 m+1 ) = 0. Thus, we obtain P ≥ 0 and this implies for every tangent hyperplane L of T p M. Consequently, we get immediately both inequalities Equations (15) and (16) from the above relation, taking infimum and supremum respectively, over all tangent hyperplanes L of T p M. Next, we investigate the equality cases of the inequalities Equations (15) and (16). First of all, we determine the critical points of P as the solutions of following system of linear homogeneous equations: We achieve h 0α ij = 0, with i, j ∈ {1, ..., m + 1} and α ∈ {m + 2, ..., 2n + 1}. Because P ≥ 0 and P (h c ) = 0, then the critical point h c is a minimum point of P. Therefore, the equality cases hold in both inequalities Equations (15) and (16) if and only if h α ij = −h * α ij , for i, j ∈ {1, ..., m + 1}, α ∈ {m + 2, ..., 2n + 1}, and the conclusion is now clear. (17) signifies that the submanifold M is totally geodesic with respect to the connection ∇ 0 . Hence, the equality case at all points in both inequalities (7) and (8)

Remark 3. Equation
The normalized δ-Casorati curvatures δ C (m) and δ * C (m) satisfy where 2δ 0 C (m) = δ C (m) + δ * C (m) and 2C 0 = C + C * . (ii) The normalized δ-Casorati curvaturesδ C (m) andδ * C (m) satisfy Moreover, the equality cases of Equations (23) and (24) hold identically at all points if and only if the imbedding curvature tensors h and h * of the submanifold associated with the dual connections∇ and∇ * satisfy Equation (17), i.e., M is a totally geodesic submanifold with respect to the Levi-Civita connection.
Moreover, the equality cases of Equations (25) and (26) hold identically at all points if and only if the imbedding curvature tensors h and h * of the submanifold associated to the dual connections∇ and∇ * satisfy Equation (17), i.e., M is a totally geodesic submanifold with respect to the Levi-Civita connection.

Conclusions
It is well known that many applications of Amari's dual geometries involve one or more submanifolds imbedded in a manifold [33]. In particular, it follows that it is of great interest to find simple relationships between various invariants of the submanifolds and manifolds. In this work, using the fundamental equations for statistical submanifolds, we established such relationships between some basic extrinsic and intrinsic invariants of statistical submanifolds in Kenmotsu statistical manifolds of constant φ-sectional curvature. The results stated here motivate further studies to obtain similar relationships for many kinds of invariants of similar nature, for statistical submanifolds in several ambient spaces, like holomorphic statistical manifolds [46], Sasakian statistical manifolds [38] and cosymplectic statistical manifold [49].