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.

1. 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]).

On the other hand, the study of δ-Casorati curvatures, initiated in [5,6], offers new solutions to the above problems. The Casorati curvature is a concept preferred by Casorati over the traditional Gauss curvature and the mean curvature because it corresponds better with the common intuition of curvature [7]. Koenderink [8] and Verstraelen [9] studied the meaning of the Casorati curvature in geometry and other fields, like human/computer vision. Notice that some results in terms of isotropical Casorati curvature of production surfaces were obtained in [10]. A geometrical interpretation of the Casorati curvature of submanifolds in Riemannian manifolds was given in [11]. Very recently, Brubaker et al. [12] gave a geometric interpretation of Cauchy–Schwarz inequality in terms of Casorati curvature. The first optimal inequalities involving the extrinsic δ-Casorati curvatures and the intrinsic scalar curvature of submanifolds in real space forms were proved in [5,6]. Later, this knowledge has been extended (e.g., see [13,14,15,16,17,18,19]). Submanifolds for which these equalities hold are called Casorati ideal submanifolds. Recently, Lee et al. [20] studied optimal inequalities in terms of -Casorati curvatures of submanifolds in Kenmotsu space forms. We recall that the Kenmotsu geometry is an area of contact geometry initiated by Kenmotsu in [21], with many applications, e.g., in physics (geometrical optics, classical mechanics, thermodynamics, geometric quantization) and control theory [22]. This geometry arose in a natural way in the paper [21], where the author proposed to investigate the geometric properties of the warped product of the complex space with the real line. It is indeed a natural problem since this product is one of the three classes in Tanno’s classification of connected almost contact Riemannian manifolds with an automorphism group of maximum dimension [22].

In 1985, Amari defined the notions of statistical manifold and conjugate connection in the basic study on information geometry [23]. It is well known that there is a deep relationship between statistical manifolds and entropy [24]. We notice that a characterization of the class of statistical manifolds having coordinate systems for which the relative entropy (Kullback–Leibler divergence) is a Bregman divergence was obtained by Nagaoka [25]. On the other hand, Dillen et al. studied in [26] the conjugate (dual) connections on a semi-Riemannian manifold. Moreover, the authors found in [26] a new formulation of Radon’s theorem in affine differential geometry, proving a necessary and sufficient condition for the existence of an affine immersion which realizes the induced affine connection and the induced affine second fundamental form. The geometry of statistical manifolds also provides interesting issues for differential geometry, statistics, machine learning, etc. (see, e.g., [27,28,29,30,31,32]). In particular, the differential geometry field is focused on topics such as submanifold theory of statistical manifolds [33], Hessian geometry [34], statistical submersions [35], complex manifold theory of statistical manifolds ([29,36,37]), contact theory on statistical manifolds [38], and quaternionic theory on statistical manifolds [39]. For the above problems, Aydin et al. obtained Chen–Ricci inequalities [40] and a generalized Wintgen inequality [41] for submanifolds in statistical manifolds of constant curvature. Moreover, Lee et al. established optimal inequalities involving the Casorati curvatures and the normalized scalar curvature on submanifolds of statistical manifolds of constant curvature [42]. These inequalities were extended by Aquib and Shahid [43] in the setting of statistical submanifolds in quaternion Kähler-like statistical space forms. On the other hand, Mihai et al. [44] proved an Euler inequality and a Chen–Ricci inequality for statistical submanifolds of Hessian manifolds of constant Hessian curvature.

Recently, Furuhata et al. [45] introduced the concept of Kenmotsu statistical manifold, which is locally obtained as the warped product of a holomorphic statistical manifold and a line. They proved that a Kenmotsu statistical manifold of constant -sectional curvature is constructed from a special Kähler manifold, which is an important example of holomorphic statistical manifold. In this respect, the authors considered the warped product of two statistical manifolds and investigated the statistical sectional curvature of this warped product. Then, they equipped a Kenmotsu manifold with a natural affine connection and studied how to construct a Kenmotsu statistical manifold of constant -sectional curvature as the warped product of a holomorphic statistical manifold and a line. It would thus be of interest to study inequalities concerning the extrinsic -Casorati curvatures for statistical submanifolds in Kenmotsu statistical manifolds. In this article, we establish inequalities in terms of the extrinsic normalized -Casorati curvatures and the intrinsic scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant -sectional curvature. The methodology is based on a constrained extremum problem. Furthermore, we study the equality cases. We derive that the equality at all points characterizes the submanifolds that are totally geodesic with respect to the Levi–Civita connection.

2. Preliminaries

Let , ) be a -dimensional Riemannian manifold with an affine connection . Denote by the set of sections of a vector bundle , e.g., means the set of all tensor fields on of type . Let be the torsion tensor field of . A pair is called a statistical structure [45] on if (1) the torsion tensor field of vanishes and (2) the Codazzi equation holds for any X, Y, Z . Any connection satisfying the condition for X, Y is called torsion-free.

A statistical manifold [33] is a Riemannian manifold , in which there exists a pair of torsion-free affine connections and that satisfy for any X, Y, Z . The statistical manifold is denoted by the triplet , , . The connections and are called dual connections.

If

Any torsion-free affine connection where

Let M be an -dimensional submanifold of a -dimensional statistical manifold and g the induced metric on M. Then, the Gauss formulas are as follows: for any , where h and are symmetric -tensors, called the imbedding curvature tensor of M in , and the imbedding curvature tensor of M in , respectively.

Denote by R and the curvature tensors fields associated with ∇ and , respectively. Then the Gauss equation for the submanifold M of (with respect to the connection ) is [33] for any .

Similarly, if and denote the curvature tensors fields associated with the connections and , respectively, then the Gauss equation with respect to the connection is [33] for any .

If (, , is a statistical manifold and M is a submanifold of , then is also a statistical manifold with the induced connection ∇ and the induced metric g.

For a statistical manifold , the tensor field called the statistical curvature tensor field of is defined as [46]: for .

For a statistical structure on , we set according to [45], which implies:

It follows that the tensor field satisfies for any . We denote

Let be a two-dimensional subspace of , for . The sectional curvature of M for is defined by [46]:

Let be an orthonormal basis of the tangent space , for , and let be an orthonormal basis of the normal space . The scalar curvature at p is given by and the normalized scalar curvature of M is defined as

The mean curvature vector fields of M, denoted by H and , are given by:

We remark that, from Equation (1), we derive and therefore , 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 calculated 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:

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

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

Similarly, we can define the dual normalized and of the submanifold M in , just replacing by in the last two relations:and

The generalized normalized δ-Casorati curvatures and of M in are defined in [6] as: if , and if , whereby 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 , whereby is set above.

Obviously, the (dual) generalized normalized - and -Casorati curvatures are a natural generalization of the (dual) normalized - and -Casorati curvatures, due the fact that , , and can be recovered from , , and respectively, for some particular values of r as follows:

We recall that a statistical submanifold in is called totally geodesic with respect to the connection if the second fundamental form h of M for vanishes identically [46].

Consider that is a -dimensional almost contact metric manifold with the structure tensors (), where is the Riemannian metric on , , . These structure tensors satisfy: where is a 1-form on such that , for any .

The almost contact metric manifold is said to be a Kenmotsu manifold if the formulas: hold for any .

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].

Denote () a Kenmotsu manifold and let () be a statistical structure on , where is given as in Equation (5). A quadruple () is called a Kenmotsu statistical structure [45] on if the relation: holds for any .

A manifold equipped with a Kenmotsu statistical structure is called a Kenmotsu statistical manifold. Notice that if () is a Kenmotsu statistical manifold, then () is too.

A Kenmotsu statistical manifold () is of constant -sectional curvature c if and only if [45]: for any .

Let M be an -dimensional statistical submanifold of a Kenmotsu statistical manifold . Then, any vector field X tangent to M can be decomposed uniquely into its tangent and normal components and , respectively, thus:

Given a local orthonormal frame 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 manifold is defined by the -tensor: where is the Levi–Civita connection on . We recall the following result.

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 is positive definite in p, then p is an optimal solution of the problem (13) [48].

If the bilinear form (14) is positive semi-definite on the submanifold M, then the critical points of (13) [48].

Let M be an

For any real number r such that where

For any real number r such that where

In addition, the equality cases of Equations (15) and (16) hold identically at all points

From Equations (2)–(4), we get: where .

Let and be orthonormal bases of and , respectively, for . Setting and for , and summing over in Equation (18), we obtain:

Because and , the latter relation becomes:

Let be the quadratic polynomial in the components of the second fundamental form defined by where L is a hyperplane of .

Suppose that the hyperplane L is spanned by the tangent vectors , avoiding loss of generality. Then, from Equations (20) and (21), we derive that has the expression

Moreover, from Equation (22), we get:

Let be a quadratic form defined for any by

We study the constrained extremum problem with the constraint where is a real constant.

The first order partial derivatives system is: for every , .

The system solution, satisfying the constraint Q, is the critical point with the expression: for any , .

Let p be an arbitrary point of Q, Q. We consider that the 2-form given by: where is the second fundamental form of Q in and is the standard inner product on .

The Hessian matrix of is as follows: where b is a real constant, namely

As the hyperplane Q is totally geodesic in , considering a vector field , that is satisfying the condition , we get

However, according to the Remark 2, the critical point is the only optimal solution, i.e., the global minimum point of problem. In addition, . Thus, we obtain and this implies for every tangent hyperplane L of . 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 .

Next, we investigate the equality cases of the inequalities Equations (15) and (16). First of all, we determine the critical points of as the solutions of following system of linear homogeneous equations:

We achieve , with and . Because and , then the critical point is a minimum point of . Therefore, the equality cases hold in both inequalities Equations (15) and (16) if and only if for , , and the conclusion is now clear.  ☐

Equation (17) signifies that the submanifold M is totally geodesic with respect to the connection

Let M be an

The normalized δ-Casorati curvatures where

The normalized δ-Casorati curvatures where

Moreover, the equality cases of Equations (23) and (24) hold identically at all points if and only if the imbedding curvature tensors h and (17), i.e., M is a totally geodesic submanifold with respect to the Levi–Civita connection.

The conclusion follows immediately from Theorem 2, taking account of Equations (8)–(10).  ☐

Corollary 1 is the statistical counterpart of the Theorem 1 from [20].

Any Kenmotsu manifold can be obtained locally as follows (see ([45], Proposition 3.2)). Let

The triple () is an almost contact metric structure on .

The pair is a Kähler structure on if and only if () is a Kenmotsu structure on .

Let ([45], Theorem 3.8). Then, it follows that [45], Proposition 3.9)).

Hence, from Theorem 2 and Corollary 1, we derive the following results.

Let

For any real number r such that where

For any real number r such that where

Moreover, the equality cases of Equations (25) and (26) hold identically at all points if and only if the imbedding curvature tensors h and (17), i.e., M is a totally geodesic submanifold with respect to the Levi–Civita connection.

Let

The normalized δ-Casorati curvatures where

The normalized δ-Casorati curvature where

Moreover, the equality cases of Equations (27) and (28) hold identically at all points if and only if the imbedding curvature tensors h and (17), i.e., M is a totally geodesic submanifold with respect to the Levi–Civita connection.

We consider the Kenmotsu manifold ([45], Example 3.3). We recall that and the structure tensors and

In the following, we setfor any [45], Example 3.10)).

Next, let M be any (25) and (26) are satisfied. In particular, the statistical submanifold (25) and (26) because M is a totally geodesic submanifold of

5. 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 -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].