Casorati Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature with Semi-Symmetric Metric Connection.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized δ-Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ-sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example.

1. Introduction

The study of simple relationships between the main intrinsic and extrinsic invariants of submanifolds is a fundamental problem in submanifold theory [1]. Recent research shows a growing trend in approaching this fascinating problem through an approach that proves some types of geometric inequalities (see, e.g., [2,3,4,5,6,7,8,9,10]).

The interest in such inequalities goes back in 1993, when B.-Y. Chen introduced the intrinsic δ-invariants, now called Chen invariants, satisfying optimal inequalities for submanifolds in real space forms [11]. Later, the notion of normalized δ-Casorati curvatures (extrinsic invariants) was defined in [12,13], giving rise to new inequalities. Unlike the Gauss and mean curvature, F. Casorati in 1890 proposed to measure the curvature of a surface at a point according to common intuition of curvature [14]. Currently, this measure is named the Casorati curvature, defined by , where and are the principal curvatures of the surface in . L. Verstraelen geometrically modeled the perception as the Casorati curvature of sensation in the context of early human vision [15]. The Casorati curvature is also assessed as a natural measure or a measure of the normal deviations from planarity in some models of computer vision [16,17]. In mechanics and modern computer science, the Casorati curvature has become known as bending energy [17].

The topic of -Casorati curvatures will appeal to more geometers focused on finding new solutions of the above problem. In this respect, some recent developments are devoted to inequalities on various submanifolds of a statistical manifold, notion defined by Amari [18] in 1985 in the realm of information geometry [3,4,5,6,7,8,9,10]. In this setting, the Fisher information metric is one of the most important metrics that can be considered on statistical models [19]. Actually, it is known that modulo rescaling is the only Riemannian metric invariant under sufficient statistics and it is seen as an infinitesimal form of the relative entropy [20]. In particular, Fisher information metrics play a key role in the multiple linear regressions by maximizing the likelihood [21]. Statistical manifolds are also applied in fields such as physics, machine learning, statistics, etc. There is a natural relationship between statistical manifolds and entropy. For example, P. Pessoa et al. studied the entropic dynamics on the statistical manifolds of Gibbs distributions in [22]. Since each point of the space is a probability distribution, a statistical manifold has a profound effect on the dynamics.

Initiated by K. Kenmotsu in 1972 [23] as a branch of contact geometry, Kenmotsu geometry has generated a wide range of applications in physics (thermodynamics, classical mechanics, geometrical optics, geometric quantization, classical mechanics) and control theory [24]. The Kenmotsu statistical manifold, defined by H. Furuhata in [25], is obtained locally as a warped product between a holomorphic statistical manifold and a real line. In [8], the authors established some Casorati inequalities for Kenmotsu statistical manifolds of constant -sectional curvature.

The concept of semi-symmetric metric connection on a Riemannian manifold was introduced by H.A. Hayden in [26]. Later, interesting properties of a Riemannian manifold with semi-symmetric metric connection were obtained by K. Yano in [27] and T. Imai in [28]. In addition, T. Imai investigated hypersurfaces of a Riemannian manifold with semi-symmetric metric connection [29]. Z. Nakao generalized Imai’s approach of hypersurfaces by studying submanifolds of a Riemannian manifold with semi-symmetric metric connection [30]. The geometric inequalities on submanifolds in various manifolds with semi-symmetric metric connection have been extensively proven (see, e.g., [31,32,33,34,35,36,37]). However, only a few results are dedicated to the ambient of statistical manifolds endowed with semi-symmetric metric connection. S. Kazan and A. Kazan obtained some geometric properties of Sasakian statistical manifolds with a semi-symmetric metric connection [38]. Furthermore, M.B.K. Balgeshir and S. Salahvarzi studied new curvature properties and equations of statistical manifolds with a semi-symmetric metric connection as well as their submanifolds [39].

In this article, we establish some basic inequalities between the normalized -Casorati curvatures (that are known to be extrinsic invariants) and the scalar curvature (a fundamental intrinsic invariant) of statistical submanifolds in Kenmotsu statistical manifolds having a constant -sectional curvature, which are endowed with semi-symmetric metric connection. Moreover, we investigated the equality cases of such inequalities. A nontrivial example is also constructed in the last part of the article.

2. Preliminaries

Let (, g) be a Riemannian manifold, with g a Riemannian metric on and an affine connection on . A triplet (, g, ) is called a statistical manifold if the torsion tensor field of vanishes and is symmetric [40]. With other words, the pair (, g) is a statistical structure on . Let be an affine connection of defined by for any X, Y, Z, where is the set of smooth tangent vector fields on .

Then is named the dual connection of with respect to g. Clearly, . Moreover, the Levi-Civita connection on is given by [41]. If , g, is a statistical manifold, then it is known that , g, is too.

Let M be a submanifold of a statistical manifold , g, with g the induced metric on M, and ∇ the induced connection on M. Then is a statistical manifold as well.

Denote by h and the imbedding curvature tensor of M in with respect to and , respectively. Then Gauss’s formulas [40] are expressed by: for any .

Furthermore, denote by R, , and the -curvature tensors for the connections ∇, , and , respectively. Thus, the Gauss equations for the connections and , respectively, hold as follows [41]: and for any .

We can define now the statistical curvature tensor field [40] on M and , denoted by S and , respectively: for any , and for any .

Set a tensor field by:

Furthermore, we have:

Then has the properties:

Next, we consider a -dimensional Kenmotsu manifold defined as an almost contact metric manifold which satisfies for any the relations: where , , is a 1-form on with .

A Kenmotsu manifold with a statistical structure is called a Kenmotsu statistical manifold [25] if the following formula holds for any : where is the tensor field defined in (5),

A Kenmotsu statistical manifold () is said to be of constant -sectional curvature c if and only if [25]: for any .

On the other hand, assume that is a linear connection on . Then is called a semi-symmetric connection if the torsion tensor of defined by satisfies for any the relation: where is a 1-form. Moreover, the connection is called a semi-symmetric metric connection on if we have (see [27]).

Next, we will denote by the -tensor field defined by

Let be a statistical manifold endowed with a semi-symmetric metric connection . Then satisfies for any [39]: where U is a vector field such that , is the difference tensor field defined in (5).

Let M be an -dimensional submanifold of a statistical manifold endowed with a semi-symmetric metric connection . Denote the induced connection and the second fundamental form on M with respect to . Then the Gauss formula with respect to is:

In addition, the Gauss equation with respect to is [39]: where and are the curvature tensor fields associated with the connections and , respectively.

We notice that coincides with the second fundamental form of the Levi-Civita connection (see, e.g., [39]). Thus, becomes:

According to Kazan et al. [38], the relations between the curvature tensor of and the curvature tensors and of the connections and are as follows: and for any .

On the other hand, since the induced connection of the semi-symmetric metric connection is also semi-symmetric metric connection [39], then the Gauss formula (10) becomes: where and R is the curvature tensor of the induced statistical connection ∇ on the submanifold M.

Similarly, we can obtain the Gauss formula involving the curvature tensor of the induced statistical connection on M as follows:

If and is a non-degenerate 2-plane, then the sectional curvature  is defined as [40]: where is a basis of .

The scalar curvature  of at a point is defined by: where is an orthonormal basis at x. On the other hand, the normalized scalar curvature of at a point is given by

The mean curvature vector fields of M are defined by, respectively:

It follows that we have and , where and are the second fundamental form and the mean curvature field of M, respectively, with respect to the Levi–Civita connection on M.

Then, the squared mean curvatures of the submanifold M in are given by: where and , for , .

The Casorati curvatures of the submanifold M in are defined by the squared norms of h and over the dimension , denoted by and , respectively, as follows: where and are defined above.

Let L be an s-dimensional subspace of , and let be an orthonormal basis of L. Then the Casorati curvatures and of L are given by:

The normalized δ-Casorati curvatures and of the submanifold M are given by: and

Furthermore, the dual normalized   and of the submanifold M in are defined as follows: and

The generalized normalized δ-Casorati curvatures  and of M in are defined in [13] by: if , and if , where is set as for any positive real number r, different from .

Moreover, the dual generalized normalized   and of the submanifold M in are given by: if , and if , where is expressed above.

Next, we consider the following constrained extremum problem where M is a submanifold of a Riemannian manifold , and is a function of differentiability class . In this setting, we recall the following result which we will use later.

([42]). If the Riemannian submanifold M is complete and connected, is positive definite in (20), where

If the bilinear form (21) is positive semi-definite on the submanifold M, then the critical points of (20). For more details see ([43], Remark 3.2).

Let (

Furthermore, the case of equality of any of the inequalities (22) and (23) holds at all points

for any

for any

From Equations (13) and (14), we obtain:

Moreover, using the definition (5), the formula (24) becomes:

Next, the relation (25) implies: for any , where is expressed by:

On the other hand, from the formula (6), we obtain: for any .

For , let and be orthonormal bases of and , respectively. Suppose and (, with ) in the relations (26) and (27), then we obtain:

On the other hand, from the Gauss formulas (15) and (16) we obtain: for any , where has the following expression: with . Now, we can easily see that we have for any .

For and , from (29) we have:

Next, from (28) and (30) it follows that:

We remind that any vector field admits a unique decomposition into its tangent and normal components and , respectively, as follows:

Next, by summation over , Equation (31) becomes: where is the squared norm of P expressed by

Let be a quadratic polynomial in the components of the second fundamental form given by:

We will prove that .

Consider, without loss of generality, that L is spanned by . Then, the expression of in (33) becomes:

Moreover, the above relation implies:

Furthermore, given by (34) can be written as:

The latter equation implies:

Now, suppose that is a quadratic form expressed by , for :

Our aim is to investigate the constrained extremum problem under the constraint where is a real constant. In this respect, we establish the following first order partial derivatives system: for all , .

By using the constraint Q defined by (35), the above system provides the critical point: for all , .

For Q, we define the 2-form by: where denotes the second fundamental form of Q in and stands for the standard inner product on .

We achieve also the Hessian matrix of with the expression: where is a real constant set as .

Assume that is a tangent vector field to the hyperplane Q at x such that . Then we have:

By using in (36), it follows that:

By virtue of the Remark 1, the critical point is the global minimum point of the problem. In particular, we have . As a result, we obtain the inequality , namely represented by the inequalities (22) and (23), related to the infimum and supremum, respectively, over all tangent hyperplanes L of .

Finally, we pursue the equality cases of the inequalities (22) and (23). For this purpose, we reveal the critical points of , i.e., the solutions of following equations system:

Since is a Kenmotsu statistical manifold, then we obtain the solution , for all and . Moreover, due to and , then has a minimum point indicated above. In conclusion, the case of equality of any of the inequalities (22) and (23) holds if and only if for , . □

3. Main Inequalities

As a consequence of Theorem 2, we can derive the following inequalities involving the normalized -Casorati curvatures and , the dual normalized -Casorati curvatures and , as well as the normalized scalar curvature of the submanifold.

Let (

Furthermore, the case of equality in any of the inequalities (38) and (39) holds at all points

where

where

The inequality (38) follows replacing in (22), by using (19) and remarking that we have the relation Similarly, we obtain inequality (39) replacing in (23), by taking account of (19) and □

As proved in Theorems 2 and 3, the equality case of any of the inequalities (22), (23), (38) and (39) is attained for those statistical submanifolds for which the imbedding curvature tensors h and (12), this condition implies the vanishing of the second fundamental form of the semi-symmetric metric connection. Hence, the equality case of any of the inequalities (22), (23), (38) and (39) holds at all points only for statistical submanifolds that are totally geodesic with respect to the semi-symmetric metric connection, or equivalently with respect to the Levi-Civita connection. This is a consequence of a result recently stated in [39] (see Corollary 4.4), where it was proved that for a statistical submanifold of a statistical manifold equipped with a semi-symmetric metric connection

4. Example

Let us consider the -dimensional Kenmotsu statistical manifold constructed in [25] (for details see Examples 3.3 and 3.10 in the above referenced article). For the sake of simplicity, we will limit to the case of dimension 5, but the example we are going to build can be extended to any odd dimension. We remind that and the structure tensors are defined by and

Denote by and the dual connections on such that . We obtain:

Moreover, we obtain:

For any , we assume that the -tensor field is given by: where and is the 1-form on dual to , that is .

Thus, it is known that is a Kenmotsu statistical manifold with constant -sectional curvature (see ([25]).

Next, we prove that admits a semi-symmetric metric connection. First, we assume that is an affine connection defined as follows:

Then the torsion tensor of satisfies the relations: for all .

It follows that is a semi-symmetric connection satisfying (7) with . Furthermore, the relation holds, which implies that is a semi-symmetric metric connection on the Kenmotsu statistical manifold of constant -sectional curvature .

Let M be a 3-dimensional submanifold of the Kenmotsu statistical manifold with coordinates given by:

Consider the following bases in the tangent bundle and normal bundle , respectively: and

Then we obtain: and it follows immediately that the submanifold M is totally geodesic with respect to the semi-symmetric metric connection . Moreover, we conclude that the inequalities (22), (23), (38) and (39) are all satisfied with equality sign.

5. Conclusions

The purpose of this paper is to establish new inequalities between intrinsic and extrinsic curvature invariants, related to the normalized -Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant -sectional curvature, which are endowed with semi-symmetric metric connection. In addition, we pursued the equality cases of these inequalities and provided a nontrivial example to illustrate the results. Therefore, we believe that the topic of this survey may be developed in new challenging approaches on various classes of submanifolds in some statistical manifolds endowed with semi-symmetric metric connection.