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We establish Chen inequality for the invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Recently, in co-operation with Chen, we proved a Chen first inequality for such submanifolds. The present authors previously initiated the investigation of statistical submanifolds in Hessian manifolds of constant Hessian curvature; this paper represents a development in this topic.
In 1985, Amari  introduced the notion of a statistical manifold, closely related with information geometry. At the same time, the geometry of statistical manifolds is not far from affine differential geometry, because it also involves dual connections (also called conjugate connections). A Hessian structure is a particular case of a statistical structure.
Opozda, in , defined a sectional curvature on a statistical manifold, which cannot be defined in a standard way (as in Riemannian geometry), because the dual connections are not metric.
The main Riemannian invariants are the curvature invariants, with many important applications, for example in physics. Among them, the most studied and well known are the sectional curvature, scalar curvature, and Ricci curvatures.
In submanifold theory, beside the study of the geometric properties of submanifolds, establishing sharp relationships between intrinsic and extrinsic invariants is another topic of interest.
In 1993 , Chen defined a new type of curvature invariants, which he called -invariants (or Chen invariants) (see also [4,5]). In the same paper, he proved the Chen first inequality for submanifolds in Riemannian space forms. The Chen first invariant of an n-dimensional Riemannian manifold is defined by , where and K are the scalar and sectional curvatures of , respectively.
Moreover, , where and are mutually orthogonal plane sections at . Chen and his coworkers studied this invariant for Lagrangian submanifolds in complex space forms (see [6,7]).
On the other hand, in , it is shown that a Hessian manifold of constant Hessian curvature c is a statistical manifold of null constant curvature and a Riemannian space form of constant sectional curvature (with respect to the sectional curvature defined by the Levi-Civita connection). The present authors (see ) initiated the study of statistical submanifolds in such manifolds. In the same paper , we established a Euler inequality and also a Chen–Ricci inequality.
The curvature invariants of statistical submanifolds in different ambient spaces were recently studied by several authors, for example in Kenmotsu statistical manifolds of constant -sectional curvature (see ). Also, a generalized Wintgen inequality for statistical submanifolds was obtained in .
In 2019, Chen and the present authors  proved a Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature.
The main goal of this paper is to establish a Chen-like inequality for the invariant on such submanifolds.
2. Statistical Manifolds and Statistical Submanifolds
A statistical manifold is an m-dimensional Riemannian manifold endowed with a pairing of torsion-free affine connections and satisfying
for any One says that the connections and are dual connections (see [1,13,14]); one has The pairing is said to be a statistical structure.
The dual connection of a torsion-free affine connection always exists and is given by
where is the Levi-Civita connection on .
The curvature tensor fields with respect to the dual connections and are denoted by and respectively.
The curvature tensor associated with is known as the Riemannian curvature tensor.
A statistical structure is called of constant curvature  if
If the constant curvature is 0, then it is known as a Hessian structure.
The curvature tensor fields and of the dual connections are related by
It is clear that if is a statistical structure of constant curvature , then is also a statistical structure of constant curvature (obviously, if is Hessian, is also Hessian ).
On a Hessian manifold (, denote by . The tensor field of type (1,3) defined by
is said to be the Hessian curvature tensor for (see [2,8]). One has
Then, the Hessian sectional curvatures can be defined on a Hessian manifold using . More precisely, if one considers and a plane section in and an orthonormal basis of , then the Hessian sectional curvature is defined by
independent of the choice of an orthonormal basis.
A Hessian manifold is of constant Hessian sectional curvature c if and only if (see )
for all vector fields on .
In , it is proved that a Hessian manifold of constant Hessian sectional curvature c is a Riemannian space form of constant sectional curvature .
Let be a statistical manifold and a submanifold of of dimension n. The induced connections ∇ and and the induced metric g define a statistical structure on the submanifold . The set of the sections of the normal bundle to is denoted by .
The corresponding Gauss formulae for the conjugate connections (see ) are
for any ; h, are symmetric and bilinear and they are known as the imbedding curvature tensor of in for and the imbedding curvature tensor of in for respectively. One remarks that and are dual statistical structures on
Since h and are bilinear, there exist linear transformations and on defined by
for any and The connections and defined by Equations and are Riemannian dual connections with respect to the induced metric on
Let and be orthonormal tangent and normal frames on M, respectively. Then, the mean curvature vector fields are defined by
for and .
The Gauss, Codazzi, and Ricci equations for statistical submanifolds, with respect to the dual connections, were established by Vos .
 Let and be dual connections on a statistical manifold , and let ∇ be the induced connection by on a statistical submanifold Let and R be the Riemannian curvature tensors for and respectively. Then,
where is the Riemannian curvature tensor of on and
 Let and be dual connections on a statistical manifold , and let be the induced connection by on a statistical submanifold Let and be the Riemannian curvature tensors for and respectively. Then,
where is the Riemannian curvature tensor of on and
3. Chen Inequality for the Chen Invariant
In , the present authors proved a Euler inequality and also a Chen–Ricci inequality for submanifolds in a Hessian manifold of constant Hessian curvature. Recently Chen and the present authors  obtained a Chen first inequality for such submanifolds. Herein, we establish a Chen inequality for the Chen invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. We state the following algebraic lemma which is used in the proof of our main result.
Let be an integer and let be n real numbers. Then, one has
Moreover, the equality holds if and only if .
We shall prove this Lemma by mathematical induction.
For , the inequality becomes
or equivalently , with the equality holding if and only if .
Let us put
The equality sign of Equation (18) holds if and only if .
By using , we find
On the other hand, obviously one has
because it it equivalent to
Inequalities (18) and (20) imply , with the equality sign holding if and only if we have equalities in Equations (19) and (20), i.e.,
Thus, and the proof is complete. ☐
Let be a Hessian manifold of constant Hessian curvature c. Then, it is flat with respect to the dual connections and . Moreover, is a Riemannian space form of constant sectional curvature (with respect to the Levi-Civita connection ).
Let be an n-dimensional statistical submanifold of and , and and mutually orthogonal plane sections at p. Consider orthonormal bases of , of , and and orthonormal bases of and , respectively. We denote by the sectional curvature of the Levi-Civita connection on and by the second fundamental form of .
The sectional curvatures and of the plane sections and , respectively, are
Using Equations (11) and (14), we get
The equation of Gauss for the Levi-Civita connection implies
Analogously, we have
On the other hand, let be the scalar curvature of (with respect to the Hessian curvature tensor Q). Then, from Equations (11) and (14), we have
In addition, the equation of Gauss for the Levi-Civita connection implies
By subtracting Equations (21) and (22) from Equation (23), we obtain
Let H and denote the mean curvature vectors with respect to the dual connections ∇ and , respectively. Then, the above lemma implies
By summing the two above relations and substituting the result into Equation (24), we get
In summery, we may state our main result.
Let be a statistical submanifold in a Hessian manifold of constant Hessian curvature c. Then, for any and any plane sections and at p, we have
where and are the scalar curvature and the sectional curvature of with respect to the Riemann curvature tensor and τ and K with respect to the Hessian curvature tensor Q.
Moreover, the equality holds if and only if for any ,
An immediate consequence of Theorem 1 is the following.
Let be a statistical submanifold in a Hessian manifold of constant Hessian curvature c. If there exist a point and two mutually orthogonal plane sections and at p such that
then is nonminimal in , i.e., either or .
Theorem 1 represents a Chen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature.
Conceptualization, A.M. and I.M.; Methodology, A.M. and I.M.; Validation, A.M.; Investigation, A.M. and I.M.; Writing–original draft preparation, A.M. and I.M.; Writing–review and editing, A.M.; Visualization, A.M.; Supervision, I.M.; Project administration, I.M. Both authors contributed equally to this research. The research was carried out by both authors, and the manuscript was subsequently prepared together. All authors have read and agree to the published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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