Quaternion Statistical Submanifolds and Submersions

Abstract

This paper aims to develop a general theory of quaternion Kahlerian statistical manifolds and to study quaternion CR-statistical submanifolds in such ambient manifolds. It extends the existing theories of quaternion submanifolds and totally real submanifolds. Additionally, the work examines quaternion Kahlerian statistical submersions, including illustrative examples. The exploration also includes an analysis of the total space and fibers under certain conditions with example(s) in support. Moreover, Chen–Ricci inequality on the vertical distribution is derived for quaternion Kahlerian statistical submersions from quaternion Kahlerian statistical manifolds.

1. Introduction

A quaternion (or quaternionic) Kahlerian manifold is a Riemannian manifold whose holonomy group is contained within $Sp\left(n\right)Sp\left(1\right)$, the quaternionic unitary group. This condition ensures the existence of a quaternionic structure on the tangent bundle, characterized by a rank-3 subbundle of the endomorphism bundle that is locally spanned by almost complex structures satisfying the quaternionic relations. Such manifolds naturally arise in mathematical physics, particularly in the study of supersymmetry and string theory, as well as in representation theory and twistor theory.

Unlike their Kahlerian counterparts, quaternion Kahlerian manifolds generally do not admit a global complex structure. However, they possess a rich set of geometric properties, including a hypercomplex structure in special cases and a distinguished 4-form known as the quaternionic Kahler form. These features enable profound links between differential geometry, topology, and algebraic geometry.

Quaternion Kahlerian manifolds can be classified into two main categories: those with positive scalar curvature, which are compact and often related to symmetric spaces; and those with non-positive scalar curvature, which frequently appear in the study of moduli spaces and special holonomy. In addition, the subclass of hyperkähler manifolds, where the holonomy reduces further to $Sp\left(n\right)$, has garnered particular attention due to their rich integrable structures and applications in algebraic geometry.

The interplay between geometry and statistics has given rise to the field of information geometry, where statistical manifolds serve as a central object of study. A statistical manifold is a Riemannian manifold equipped with a pair of dual affine connections, enabling the geometric interpretation of probabilistic and information-theoretic concepts. Statistical manifolds represent a significant topic with extensive applications across various fields, including machine learning, image analysis, neural networks, physics, general relativity, and control systems, among others [1,2]. These manifolds arise from statistical models, where the points of the Riemannian manifold correspond to probability distributions. In recent years, the study of statistical manifolds has gained considerable attention, leading to numerous remarkable findings by researchers in the field [3,4,5,6]. When extended to a quaternionic setting, this leads to the theory of quaternion Kahlerian statistical manifolds, which integrate quaternionic geometry with statistical structures to explore deeper symmetries and interactions.

On the other hand, submersions play a fundamental role in differential geometry by establishing structured relationships between manifolds of differing dimensions. A submersion is a smooth map between manifolds that has surjective differential maps at every point. Among these, Riemannian submersions and statistical submersions stand out for their geometric and statistical significance, respectively. Building on the concept of Riemannian submersions, a statistical submersion is studied in [7]. Statistical submersions provide a unified framework for understanding the mappings between statistical manifolds, making them a cornerstone of modern research in differential geometry, information theory, and applied statistics. Their theoretical richness and practical applicability continue to inspire new avenues of study.

In this article, we begin by recalling some fundamental and essential notions. Consistent with the frameworks of holomorphic, Sasakian, and Kenmotsu statistical manifolds, we propose the statistical analog of a quaternion Kahlerian manifold, referred to as a quaternion Kahlerian statistical manifold, with an example. Naturally, this leads to the exploration of submanifold theory in the context of statistical manifolds. We examine key results related to statistical immersions and quaternion CR-statistical submanifolds in the same ambient manifold. Also, it is an interesting and important problem to study the quaternion Kahlerian statistical structure of constant Q-sectional curvature c. Furthermore, we derive several findings on quaternion Kahlerian statistical submersions, illustrated with example(s), and specifically investigate the relationship between the statistical curvature tensors of the base and target manifolds under such submersions.

2. Preliminaries

Definition 1.

Let $(M,g)$ be a Riemannian manifold of dimension r and D a torsion-free linear connection on M. Then, the pair $(D,g)$ is called a statistical structure on M if D is compatible with g, that is, $Dg$ is symmetric. A triple $(M,D,g)$ is known as a statistical manifold [3].

Now, let $D^{*}$ be an affine connection on $(M,D,g)$ such that for all ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}\in \Gamma \left(TM\right)$.

$$\begin{array}{c}\hfill {\mathbf{V}}_{\mathbf{1}}g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})=g(D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})+g({\mathbf{V}}_{\mathbf{2}},D_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{3}}).\end{array}$$

It follows that $D^{*}g$ is symmetric and $D^{*}$ is torsion-free. Note that $(M,D^{*},g)$ is known as the dual statistical manifold of $(M,D,g)$ and that $D^{*}$ is the dual connection of D.

The concept of dual connections, named because ${\left(D^{*}\right)}^{*}=D$, was first introduced by S. Amari in his pioneering work. It has since found applications in diverse fields, such as information theory, statistical physics, and neural networks. We also write the Levi–Civita connection $D^0$ on M as $D^0={\displaystyle \frac{1}{2}}(D+D^{*})$.

Let N be a submanifold of M and g be the induced metric on N. Then, $(N,D^N,g)$ is a statistical submanifold in a statistical manifold $(M,D,g)$ if $(D^N,g)$ is the induced statistical structure on N.

A pair $(D:=D^0+K,g)$ becomes a statistical structure if and only if the difference tensor $K\in \Gamma \left(T{M}^{(1,2)}\right)$ satisfies the following conditions:

• $K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}=K_{{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{1}}$;
• $g(K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})=g({\mathbf{V}}_{\mathbf{3}},K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}})$.

Furthermore, we have the relation $K=D^0-D^{*}={\displaystyle \frac{1}{2}}(D-D^{*})$.

Remark 1.

The curvature tensor field of D is indicated by $\overline{\mathcal{R}im}$ and is indicated by ${\overline{\mathcal{R}im}}^{*}$ for $D^{*}$, and ${\overline{\mathcal{R}im}}^0$ for $D^0$ [6,8].

1. 

The statistical curvature tensor field ${\overline{S}}^{dual}$ of $(D,g)$ is defined as

$$\begin{array}{c}\hfill {\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}={\displaystyle \frac{1}{2}}(\overline{\mathcal{R}im}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}+{\overline{\mathcal{R}im}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}),\end{array}$$

for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}\in \Gamma \left(TM\right)$.

2. 

Let an orthonormal basis of $T_{p}M$ be $\{v_1,v_2,\cdots ,v_r\}$. For a plane $v_i\wedge v_j$ of $T_{p}M$, the statistical sectional curvature ${\overline{\mathcal{K}}}^{{\overline{S}}^{dual}}(v_i\wedge v_j)$ of $(D,g)$ is defined by

$$\begin{array}{c}\hfill {\overline{\mathcal{K}}}^{{\overline{S}}^{dual}}(v_i\wedge v_j)=g({\overline{S}}^{dual}(v_i,v_j)v_j,v_i)\end{array}$$

for $1\le i<j\le r$. Note that the sectional curvature ${\overline{\mathcal{K}}}^0$ of g is obtained using ${\overline{\mathcal{R}im}}^0$ instead of ${\overline{S}}^{dual}$.

3. 

The statistical scalar curvature ${\overline{\tau }}^{{\overline{S}}^{dual}}$ of $(D,g)$ is defined as

$$\begin{array}{ccc}\hfill {\overline{\tau }}^{{\overline{S}}^{dual}}& =& \sum _{1\le i<j\le r}g({\overline{S}}^{dual}(v_i,v_j)v_j,v_i)\hfill \\ & =& 2\sum _{1\le i<j\le r}{\overline{\mathcal{K}}}^{{\overline{S}}^{dual}}(v_i\wedge v_j).\hfill \end{array}$$

The scalar curvature ${\overline{\tau }}^0$ of g is given by using ${\overline{\mathcal{R}im}}^0$ instead of ${\overline{S}}^{dual}$.

Furuhata et al. [9] proposed a new notion of sectional curvature for statistical manifolds, distinct from earlier definitions that use either the statistical curvature tensor field $S^{dual}$ (as defined above) or the K-curvature tensor field $[K,K]$.

Definition 2.

Let $(M,D,g)$ be a statistical manifold. Set ${\overline{\mathcal{U}}}^{dual}$ as

$$\begin{array}{ccc}\hfill {\overline{\mathcal{U}}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}& =& {\overline{\mathcal{R}im}}^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}-[K_{{\mathbf{V}}_{\mathbf{1}}},K_{{\mathbf{V}}_{\mathbf{2}}}]{\mathbf{V}}_{\mathbf{3}}\hfill \\ & =& 2{\overline{\mathcal{R}im}}^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}-{\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}},\hfill \end{array}$$

where ${\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}={\overline{\mathcal{R}im}}^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}+[K_{{\mathbf{V}}_{\mathbf{1}}},K_{{\mathbf{V}}_{\mathbf{2}}}]{\mathbf{V}}_{\mathbf{3}}$.

1. 

For a plane $v_i\wedge v_j$$T_{p}M$, the ${\overline{\mathcal{U}}}^{dual}$ sectional curvature ${\overline{\mathcal{K}}}^{{\overline{\mathcal{U}}}^{dual}}(v_i\wedge v_j)$ of $(D,g)$ is given by

$$\begin{array}{c}\hfill {\overline{\mathcal{K}}}^{{\overline{\mathcal{U}}}^{dual}}(v_i\wedge v_j)=g({\overline{\mathcal{U}}}^{dual}(v_i,v_j)v_j,v_i).\end{array}$$

(1)

2. 

The ${\overline{\mathcal{U}}}^{dual}$ scalar curvature ${\overline{\tau }}^{{\overline{\mathcal{U}}}^{dual}}$ is

$$\begin{array}{ccc}\hfill {\overline{\tau }}^{{\overline{\mathcal{U}}}^{dual}}& =& \sum _{1\le i<j\le r}g({\overline{\mathcal{U}}}^{dual}(v_i,v_j)v_j,v_i)\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& =& 2\sum _{1\le i<j\le r}{\overline{\mathcal{K}}}^{{\overline{\mathcal{U}}}^{dual}}(v_i\wedge v_j).\hfill \end{array}$$

(3)

3. Quaternion Kahlerian Statistical Manifolds

Let M be a smooth manifold with a rank-3 subbundle $\vartheta$ of type $(1,1)$ on $End\left(TM\right)$ such that the sections of $\vartheta$ admit a local basis $\{{\mathbf{J}}_1,{\mathbf{J}}_2,{\mathbf{J}}_3\}$ satisfying

$$\left.\begin{array}{c}\hfill {\mathbf{J}}_1^2=-I,{\mathbf{J}}_2^2=-I,{\mathbf{J}}_3^2=-I,\\ \hfill {\mathbf{J}}_1{\mathbf{J}}_2={\mathbf{J}}_3,{\mathbf{J}}_2{\mathbf{J}}_3={\mathbf{J}}_1,{\mathbf{J}}_3{\mathbf{J}}_1={\mathbf{J}}_2,\end{array}\right\}$$

(4)

Then, $\vartheta$ is called an almost quaternion structure on M. A couple $(M,\vartheta )$ is known as an almost quaternion manifold. An almost quaternion manifold M is of dimension $4r$ provided $r>1$.

On an almost quaternion manifold $(M,\vartheta )$, a Riemannian metric g exists such that $g(\varphi {\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})+g({\mathbf{V}}_{\mathbf{1}},\varphi {\mathbf{V}}_{\mathbf{2}})=0$, for any cross-section $\varphi$. Consequently, $(M,g,\vartheta )$ is referred to as an almost quaternion metric manifold.

Let $\{{\mathbf{J}}_1,{\mathbf{J}}_2,{\mathbf{J}}_3\}$ be a canonical local basis of $\vartheta$ of $(M,g,\vartheta )$. Since each of ${\mathbf{J}}_1$, ${\mathbf{J}}_2$, and ${\mathbf{J}}_3$ is almost Hermitian with respect to g, gives

$$\begin{array}{ccc}& & {\omega }_1({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=g({\mathbf{J}}_1{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),{\omega }_2({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=g({\mathbf{J}}_2{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),\hfill \\ & & {\omega }_3({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=g({\mathbf{J}}_3{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),\hfill \end{array}$$

where ${\omega }_1,{\omega }_2,{\omega }_3$ are local 2-forms.

Definition 3.

An almost quaternion metric manifold M satisfies

$$\begin{array}{ccc}\hfill D_{{\mathbf{V}}_{\mathbf{1}}}^0{\mathbf{J}}_1& =& {\Omega }_3\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_2-{\Omega }_2\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_3,\hfill \\ \hfill D_{{\mathbf{V}}_{\mathbf{1}}}^0{\mathbf{J}}_2& =& -{\Omega }_3\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_1+{\Omega }_1\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_3,\hfill \\ \hfill D_{{\mathbf{V}}_{\mathbf{1}}}^0{\mathbf{J}}_3& =& {\Omega }_2\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_1-{\Omega }_1\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_2,\hfill \end{array}$$

for any vector field ${\mathbf{V}}_{\mathbf{1}}$. Then, M is called a quaternion Kahlerian manifold.

Following the framework used for holomorphic, Sasakian, Kenmotsu, nearly Sasakian, and nearly Kähler statistical manifolds (see [10,11,12,13]), we now introduce the statistical counterpart of a quaternion Kahlerian manifold, referred to as the quaternion Kahlerian statistical manifold.

Definition 4.

A triple $(D=D^0+K,g,\vartheta )$ is a quaternion Kahlerian statistical structure on a smooth manifold M if the following is the case:

1. 

$(D,g)$ is a statistical structure on M;

2. 

$(g,\vartheta )$ is an almost quaternion metric structure on M;

3. 

The formulas $K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_1{\mathbf{V}}_{\mathbf{2}}=-{\mathbf{J}}_1{K}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}$, $K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_2{\mathbf{V}}_{\mathbf{2}}=-{\mathbf{J}}_2{K}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}$, $K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_3{\mathbf{V}}_{\mathbf{2}}=-{\mathbf{J}}_3{K}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}$ hold for any ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TM\right)$.

A quaternion Kahlerian statistical manifold $(M,D,g,\vartheta )$ is a manifold equipped with a quaternion Kahlerian statistical structure.

Remark 2.

It is worth noting that $(M,D,g,\vartheta )$ is a quaternion Kahlerian statistical manifold, so is $(M,D^{*},g,\vartheta )$. Also, note that our Definition 3 is equivalent to Definition 4.

Example 1.

Let $(g,\vartheta )$ be an almost quaternion metric structure on M. Consider $F\in \Gamma \left(TM\right)$ a chosen vector field, and we construct

$$\begin{array}{ccc}\hfill K({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})& =& \{g({\mathbf{J}}_{o}F,{\mathbf{V}}_{\mathbf{1}})g({\mathbf{J}}_{o}F,{\mathbf{V}}_{\mathbf{2}})-g(F,{\mathbf{V}}_{\mathbf{1}})g(F,{\mathbf{V}}_{\mathbf{2}})\}{\mathbf{J}}_{o}F\hfill \\ & & -\{g({\mathbf{J}}_{o}F,{\mathbf{V}}_{\mathbf{1}})g(F,{\mathbf{V}}_{\mathbf{2}})+g(F,{\mathbf{V}}_{\mathbf{1}})g({\mathbf{J}}_{o}F,{\mathbf{V}}_{\mathbf{2}})\}F,\hfill \end{array}$$

(5)

for any ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TM\right)$. Then, for $o\in \{1,2,3\}$, K satisfies the following:

1. 

$K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}=K_{{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{1}}$;

2. 

$g(K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})=g({\mathbf{V}}_{\mathbf{2}},K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}})$;

3. 

$K_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}+{\mathbf{J}}_o{K}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}=0$.

Thus, we have a quaternion Kahlerian statistical structure $(D=D^0+K,g,\vartheta )$ on M.

Example 2.

Consider $({\mathbb{R}}_2^4,g)$, a semi-Riemannian manifold with the local coordinate system $(x,y,z,w)$ and

$$\begin{array}{c}\hfill g=\left(\begin{array}{cccc}2& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right),\end{array}$$

which admits the following quaternion Kahlerian structure $\left({\mathbf{J}}_o\right)$, $o=1,2,3$ such that

$$\begin{array}{ccc}& & {\mathbf{J}}_1=\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right),{\mathbf{J}}_2=\left(\begin{array}{cccc}0& 0& -1/2& 0\\ 0& 0& 0& -1/2\\ 2& 0& 0& 0\\ 0& 2& 0& 0\end{array}\right),\hfill \\ & & {\mathbf{J}}_3=\left(\begin{array}{cccc}0& 0& 0& -1/2\\ 0& 0& 1/2& 0\\ 0& -2& 0& 0\\ 2& 0& 0& 0\end{array}\right).\hfill \end{array}$$

By Example 1, we can construct a quaternion Kahlerian statistical structure $(D=D^0+K,g,\vartheta )$ on ${\mathbb{R}}_2^4$.

We know that the sectional curvature for the statistical structure is defined using the tensor field $S^{dual}$ rather than R, as it does not possess the necessary symmetries characteristic of the curvature tensor field to a Levi–Civita connection. So, we have the following:

Let $(M,D,g,\vartheta )$ be a quaternion Kahlerian statistical manifold, and let $c\in \mathbb{R}$. Suppose E is a non-null vector field on M. Then, $Q\left(E\right)$ is a 4-dimensional subspace of $T_{x}M$ at each $x\in M$, spanned by $\{E,{\mathbf{J}}_{1}E,{\mathbf{J}}_{2}E,{\mathbf{J}}_{3}E\}$. The sectional curvature of a 2-plane in $Q\left(E\right)$ is referred to as the Q-sectional curvature. The quaternion Kahlerian statistical structure is defined as having a constant Q-sectional curvature c if the curvature tensor ${\overline{S}}^{dual}$ for D and $D^{*}$ satisfies

$$\begin{array}{ccc}\hfill {\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}& =& {\displaystyle \frac{c}{4}}\{g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}){\mathbf{V}}_{\mathbf{1}}-g({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}){\mathbf{V}}_{\mathbf{2}}\hfill \\ & & +\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}){\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}-g({\mathbf{V}}_{\mathbf{3}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}){\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}]\hfill \\ & & +\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})-g({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})]{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{3}}\}.\hfill \end{array}$$

(6)

Proposition 1.

On quaternion Kahlerian statistical manifold $(M,D,g,\vartheta )$, we have

1. 

$\left(D_{{\mathbf{V}}_{\mathbf{3}}}{\omega }_o\right)({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=g({\mathbf{V}}_{\mathbf{2}},D_{{\mathbf{V}}_{\mathbf{3}}}\left({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}\right))-g({\mathbf{V}}_{\mathbf{2}},{\mathbf{J}}_o{D}_{{\mathbf{V}}_{\mathbf{3}}}^{*}{\mathbf{V}}_{\mathbf{1}})$;

2. 

$\left(D_{{\mathbf{V}}_{\mathbf{3}}}{\omega }_o\right)({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})-\left(D_{{\mathbf{V}}_{\mathbf{3}}}^{*}{\omega }_o\right)({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=0$;

3. 

$\left(D_{{\mathbf{V}}_{\mathbf{1}}}^0{\mathbf{J}}_o\right){\mathbf{V}}_{\mathbf{2}}=D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}-{\mathbf{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}$;

for $o\in \{1,2,3\}$.

Theorem 1.

Let $(D,g)$ be a statistical structure and $(g,\vartheta )$ a quaternion Kahlerian structure on M; $(D,g,\vartheta )$ is a quaternion Kahlerian statistical structure if and only if the following formula holds:

$$\begin{array}{c}\hfill D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}-{\mathbf{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}={\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}{\mathbf{V}}_{\mathbf{2}}-{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}{\mathbf{V}}_{\mathbf{2}},\end{array}$$

for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TM\right)$, where the indices are taken from $\{1,2,3\}$ modulo 3.

4. Statistical Immersions

In this section, we present the foundational concepts of statistical submanifolds in a quaternion Kahelrian statistical manifold, excluding aspects of information geometry.

For any ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in T_{x}N$, and $\xi \in T_x^{\perp }N$, $x\in N$, the Gauss and Weingarten formulas are, respectively, defined by [14]

$$\begin{array}{ccc}\hfill D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}& =& D_{{\mathbf{V}}_{\mathbf{1}}}^N{\mathbf{V}}_{\mathbf{2}}+\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),\hfill \\ \hfill D_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}& =& D^{N\ast }{\mathbf{V}}_{\mathbf{2}}+{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),\hfill \\ \hfill D_{{\mathbf{V}}_{\mathbf{1}}}\xi & =& -A_{\xi }\left({\mathbf{V}}_{\mathbf{1}}\right)+D_{{\mathbf{V}}_{\mathbf{1}}}^{N\perp }\xi ,\hfill \\ \hfill D_{{\mathbf{V}}_{\mathbf{1}}}^{*}\xi & =& -A_{\xi }^{*}\left({\mathbf{V}}_{\mathbf{1}}\right)+D^{N\ast \perp }\xi ,\hfill \end{array}$$

Here, $\mathbf{B}$ (${\mathbf{B}}^{*}$) is a symmetric and bilinear tensor, referred to as the embedding curvature tensors N in M for D, while $D^N$ ($D^{N\ast }$) represents the induced dual connection on N ($N\ast$). Moreover, the dual connections $D^{N\perp }$ and $D^{N\ast \perp }$ of the vector bundles $T^{\perp }N$ and $T^{\perp }N\ast$ are called dual normal connections.

Also, we know

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),\xi )=g(A_{\xi }^{*}\left({\mathbf{V}}_{\mathbf{1}}\right),{\mathbf{V}}_{\mathbf{2}}),g({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}),\xi )=g(A_{\xi }\left({\mathbf{V}}_{\mathbf{1}}\right),{\mathbf{V}}_{\mathbf{2}}).\end{array}$$

(7)

Definition 5.

Let $(N,D^N,g)$ be a statistical submanifold of $(M,D,g)$.

1. 

If $\mathbf{H}={\mathbf{H}}^{*}=0$, then N is doubly minimal.

2. 

If $\mathbf{B}=g\otimes \mathbf{H}$ and ${\mathbf{B}}^{*}=g\otimes {\mathbf{H}}^{*}$, then N is doubly totally umbilical such that $\mathbf{H}$ (${\mathbf{H}}^{*}$) is the mean curvature vector of N for D ($D^{*}$) in M.

3. 

If $\mathbf{B}={\mathbf{B}}^{*}=0$, then N is doubly totally geodesic.

We recall from [4] the concept of statistical immersion $h:(N,D^N,g)\to (M,D,g)$, where $g=h^{*}g$ and $g(D_{{\mathbf{V}}_{\mathbf{1}}}^N{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})=g(D_{{\mathbf{V}}_{\mathbf{1}}}{h}_{*}{\mathbf{V}}_{\mathbf{2}},h_{*}{\mathbf{V}}_{\mathbf{3}})$ for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TN\right)$. Then, the Gauss formula is

$$\begin{array}{c}\hfill D_{{\mathbf{V}}_{\mathbf{1}}}{h}_{*}{\mathbf{V}}_{\mathbf{2}}=h_{*}{D}_{{\mathbf{V}}_{\mathbf{1}}}^N{\mathbf{V}}_{\mathbf{2}}+\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}).\end{array}$$

Therefore, in our settings, we put the statistical structure $(D^N,g)$ on N induced by h from $(D,g)$. Then, $(D^N,g,\tilde{\vartheta })$ is a quaternion Kahlerian statistical structure on N, where $\tilde{\vartheta }$ denotes a rank-3 subbundle of type $(1,1)$ of $End\left(TN\right)$ such that the sections of $\vartheta$ admit a local basis $\{{\mathbf{J}}_1,{\mathbf{J}}_2,{\mathbf{J}}_3\}$ satisfying (4). Thus, we have the following:

Theorem 2.

Let $(M,D,g,\vartheta )$ be a quaternion Kahelrian statistical manifold and $h:(N,D^N,g,\tilde{\vartheta })\to (M,D,g,\vartheta )$ a quaternion isometric immersion. Then, the following holds:

1. 

h is doubly minimal.

2. 

$\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\tilde{J}}_o{\mathbf{V}}_{\mathbf{2}})={\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})$ holds for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TN\right)$.

Proof. 

We use Theorem 1 for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TN\right)$,

$$\begin{array}{ccc}\hfill D_{{\mathbf{V}}_{\mathbf{1}}}\left({\mathbf{J}}_o{h}_{*}{\mathbf{V}}_{\mathbf{2}}\right)-{\mathbf{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{*}{h}_{*}{\mathbf{V}}_{\mathbf{2}}& =& {\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}{h}_{*}{\mathbf{V}}_{\mathbf{2}}\hfill \\ & & -{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}{h}_{*}{\mathbf{V}}_{\mathbf{2}}\hfill \\ \hfill D_{{\mathbf{V}}_{\mathbf{1}}}{h}_{*}{\tilde{J}}_o{\mathbf{V}}_{\mathbf{2}}-h_{*}{\tilde{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{N\ast }{\mathbf{V}}_{\mathbf{2}}+{\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})& =& {\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}{h}_{*}{\mathbf{V}}_{\mathbf{2}}\hfill \\ & & -{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}{h}_{*}{\mathbf{V}}_{\mathbf{2}}.\hfill \end{array}$$

But we know $h_{*}{\tilde{J}}_o{\mathbf{V}}_{\mathbf{1}}={\mathbf{J}}_o{h}_{*}{\mathbf{V}}_{\mathbf{1}}$ holds, so

$$\begin{array}{ccc}\hfill & & D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o{h}_{*}{\mathbf{V}}_{\mathbf{2}}-h_{*}{\tilde{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{N\ast }{\mathbf{V}}_{\mathbf{2}}-{\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})\hfill \\ & =& {\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}{h}_{*}{\mathbf{V}}_{\mathbf{2}}-{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}{h}_{*}{\mathbf{V}}_{\mathbf{2}}.\hfill \end{array}$$

(8)

On the other hand, we have

$$\begin{array}{c}\hfill D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o{h}_{*}{\mathbf{V}}_{\mathbf{2}}=h_{*}{D}_{{\mathbf{V}}_{\mathbf{1}}}^N\left({\tilde{J}}_o{\mathbf{V}}_{\mathbf{2}}\right)+\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\tilde{J}}_o{\mathbf{V}}_{\mathbf{2}}).\end{array}$$

(9)

Equations (8) and (9) imply that

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\tilde{J}}_o{\mathbf{V}}_{\mathbf{2}})-{\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=0.\end{array}$$

It is easy to see that $\mathbf{H}={\mathbf{H}}^{*}=0$. Therefore, h is doubly minimal. □

We, respectively, symbolize the Riemannian curvature tensors of $D^N$ ($D^{N\ast }$) by R ($R^{*}$). Then, the corresponding Gauss equations, for conjugate affine connection, are given by [14]

$$\begin{array}{ccc}& & \overline{\mathcal{R}im}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})-R({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})\hfill \\ & =& g(\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}),{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{4}}))-g({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{4}}),\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})),\hfill \end{array}$$

(10)

and

$$\begin{array}{ccc}& & {\overline{\mathcal{R}im}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})-R^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})\hfill \\ & =& g({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}),\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{4}}))-g(\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{4}}),{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})),\hfill \end{array}$$

(11)

where

$$\overline{\mathcal{R}im}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})=g(\overline{\mathcal{R}im}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})$$

and

$${\overline{\mathcal{R}im}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})=g({\overline{\mathcal{R}im}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}}).$$

Thus, we have the Gauss formula for both affine connections:

$$\begin{array}{ccc}\hfill & & 2{\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})-2S^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})\hfill \\ & =& g(\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}),{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{4}}))-g({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{4}}),\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}))\hfill \\ & & +g({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}),\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{4}}))-g(\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{4}}),{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})),\hfill \end{array}$$

(12)

where $2S^{dual}=R+R^{*}$.

Also, we have

$$\begin{array}{ccc}\hfill & & 4{\overline{\mathcal{R}im}}^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})-4R^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})\hfill \\ & =& -g(\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})+{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}),\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{4}})+{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{4}}))\hfill \\ & & +g(\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}})+{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}),\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{4}})+{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{4}})),\hfill \end{array}$$

(13)

where $R^0$ denotes the Riemannian curvature tensor for Levi–Civita connection on N.

Proposition 2.

Let $(M,D,g,\vartheta )$ be a quaternion Kahlerian statistical manifold and N be a quaternion submanifold of M. Then, the following hold:

1. 

$(N,D^N,g,\vartheta )$ is a quaternion Kahlerian statistical manifold.

2. 

$\mathbf{B}({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})={\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})$,

for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TN\right)$.

Proof. 

We use Theorem 1

$$\begin{array}{ccc}\hfill {\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}{\mathbf{V}}_{\mathbf{2}}-{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}{\mathbf{V}}_{\mathbf{2}}& =& D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}-{\mathbf{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}\hfill \\ & =& \mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})+D_{{\mathbf{V}}_{\mathbf{1}}}^N{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}\hfill \\ & & -{\mathbf{J}}_o({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})+D_{{\mathbf{V}}_{\mathbf{1}}}^{N\ast }{\mathbf{V}}_{\mathbf{2}})\hfill \\ & =& \mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})-{\mathbf{J}}_o({\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})\hfill \\ & & +{\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}{\mathbf{V}}_{\mathbf{2}}-{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}{\mathbf{V}}_{\mathbf{2}}.\hfill \end{array}$$

On comparing normal components, we obtain $\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})={\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})$. Similarly, we have $\mathbf{B}({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})={\mathbf{J}}_o{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})$. □

5. Quaternion CR-Statistical Submanifolds

In this section, we give the definition of quaternion CR-statistical submanifolds and generalize classical theorems to our setting.

Definition 6.

A submanifold N in a quaternion Kahelrian statistical manifold $(M,D,g,\vartheta )$ is called a quaternion CR-statistical submanifold if the following hold:

1. 

$(N,D^N,g)$ is a statistical submanifold in $(M,D,g,\vartheta )$;

2. 

There exists a differentiable distribution $\mathcal{D}:x\to {\mathcal{D}}_x\subset T_{x}N$, $x\in N$ satisfying the following:

(a) 

$\mathcal{D}$ is quaternion, that is, ${\mathbf{J}}_o\left({\mathcal{D}}_x\right)={\mathcal{D}}_x\subset T_{x}N$;

(b) 

the orthogonal complementary distribution ${\mathcal{D}}^{\perp }$ is totally real, that is, ${\mathbf{J}}_o{\mathcal{D}}_x^{\perp }\subset T_x^{\perp }N$.

Definition 7.

A statistical submanifold N in a quaternion Kahelrian statistical manifold $(M,D,g,\vartheta )$ is called a quaternion CR-statistical submanifold if the differentiable distribution $\mathcal{D}:x\to {\mathcal{D}}_x\subset T_{x}N$, $x\in N$ exists and satisfies the following:

1. 

$\mathcal{D}$ is quaternion, that is, ${\mathbf{J}}_o\left({\mathcal{D}}_x\right)={\mathcal{D}}_x\subset T_{x}N$;

2. 

The orthogonal complementary distribution ${\mathcal{D}}^{\perp }$ is totally real, that is, ${\mathbf{J}}_o{\mathcal{D}}_x^{\perp }\subset T_x^{\perp }N$.

Moreover, the quaternion CR-statistical submanifold N in $(M,D,g,\vartheta )$ has the following characteristics:

• It reduces to a quaternion submanifold (totally real submanifold) if $dim{\mathcal{D}}_x^{\perp }=0$$(dim{\mathcal{D}}_x=0)$.
• It reduces to a generic submanifold if ${\mathbf{J}}_o{\mathcal{D}}^{\perp }=T^{\perp }N$ and $\mathcal{D}\ne 0$.
• It reduces to a Lagrangian submanifold if ${\mathcal{D}}^{\perp }=TN$ and ${\mathbf{J}}_o{\mathcal{D}}^{\perp }=T^{\perp }N$.
• It is called proper if N is neither a quaternion submanifold (that is, $\mathcal{D}\ne 0$) nor a totally real submanifold (that is, ${\mathcal{D}}^{\perp }\ne 0$).

Let N be a submanifold of a quaternion Kahlerian statistical manifold $(M,D,g,\vartheta )$. Then, we have for any $\mathbf{X}\in \Gamma \left(TM\right)$ the following decomposition:

$$\begin{array}{c}\hfill {\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}=J_o^T{\mathbf{V}}_{\mathbf{1}}+J_o^{\perp }{\mathbf{V}}_{\mathbf{1}},\end{array}$$

(14)

$o=1,2,3$. Here, $J_o^T{\mathbf{V}}_{\mathbf{1}}$ ($J_o^{\perp }{\mathbf{V}}_{\mathbf{1}}$) denote the tangential component (normal component) of ${\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}$. Similarly, for any $\xi \in \Gamma \left(T^{\perp }N\right)$, we write

$$\begin{array}{c}\hfill {\mathbf{J}}_o\xi =t_o\xi +f_o\xi ,\end{array}$$

where $t_o\xi$ and $f_o\xi$ are the tangential and the normal parts of ${\mathbf{J}}_o\xi$, respectively.

Consequently, we have the following decomposition of the tangent bundle $TM$ as a direct sum of vector bundles:

$$\begin{array}{c}\hfill TM=TN\oplus T^{\perp }N=(\mathcal{D}\oplus {\mathcal{D}}^{\perp })\oplus ({\mathbf{J}}_o{\mathcal{D}}^{\perp }\oplus \mu ),\end{array}$$

where $\mu$ denotes a subbundle of $T^{\perp }N$ defined by ${\mu }_p=\{\xi \in T_p^{\perp }N|\xi \perp {\mathbf{J}}_o{\mathcal{D}}_p^{\perp }\}$, $p\in N$.

Concerning the integrability of the distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$ for a quaternion CR-statistical submanifold:

Theorem 3.

On a quaternion CR-statistical submanifold $(N,D^N,g)$ of a quaternion Kahlerian statistical manifold $(M,D,g,\vartheta )$, we have that $\mathcal{D}$ is integrable if and only if $\mathbf{B}({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})$, for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \mathcal{D}$.

Proof. 

It is easy to derive

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})-\mathbf{B}({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})=J_o^{\perp }[{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}].\end{array}$$

□

Since, ${\mathcal{D}}^{\perp }$ is completely integrable. Therefore, we can have the following result:

Lemma 1.

On a quaternion CR-statistical submanifold $(N,D^N,g)$ of a quaternion Kahlerian statistical manifold $(M,D,g,\vartheta )$, we have $A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}=A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{1}}$, (or $A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}=A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{\mathbf{V}}_{\mathbf{1}}$), for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in {\mathcal{D}}^{\perp }$.

Proof. 

It is easy to obtain

$$\begin{array}{c}\hfill A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{1}}-A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}=J_o^T[{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}].\end{array}$$

□

Lemma 2.

On a Lagrangian submanifold $(N,D^N,g)$ of a quaternion Kahlerian statistical manifold $(M,D,g,\vartheta )$, we have that $\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})={\mathbf{J}}_o{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}$, for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TN\right)$.

Proof. 

For ${\mathbf{V}}_{\mathbf{1}}\in \Gamma \left(TN\right)$ and $\xi \in \Gamma \left(T^{\perp }N\right)$, we have

$$\begin{array}{ccc}\hfill {\Omega }_{o+2}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+1}\xi -{\Omega }_{o+1}\left({\mathbf{V}}_{\mathbf{1}}\right){\mathbf{J}}_{o+2}\xi & =& D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{J}}_o\xi -{\mathbf{J}}_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{*}\xi \hfill \\ & =& D_{{\mathbf{V}}_{\mathbf{1}}}^N{t}_o\xi +\mathbf{B}({\mathbf{V}}_{\mathbf{1}},t_o\xi )+J_o^T{A}_{\xi }^{*}{\mathbf{V}}_{\mathbf{1}}\hfill \\ & & +J_o^{\perp }{A}_{\xi }^{*}{\mathbf{V}}_{\mathbf{1}}+t_o{D}_{{\mathbf{V}}_{\mathbf{1}}}^{N\ast \perp }\xi .\hfill \end{array}$$

Considering the normal parts of the above equation

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{V}}_{\mathbf{1}},t_o\xi )+J_o^{\perp }{A}_{\xi }^{*}{\mathbf{V}}_{\mathbf{1}}=0.\end{array}$$

We put $\xi ={\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}$ in the above relation and obtain

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})={\mathbf{J}}_o{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{\mathbf{V}}_{\mathbf{1}}.\end{array}$$

□

Theorem 4.

For a Lagrangian submanifold $(N,D^N,g)$ in a quaternion Kahlerian statistical manifold $\left(M\right(c),D,g,\vartheta )$ of constant Q-sectional curvature c, if we assume that $A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}=A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}$ for ${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}\in \Gamma \left(TN\right)$, then N is of constant sectional curvature ${\displaystyle \frac{c}{4}}$.

Proof. 

By using (12) and Lemmas 1 and 2, we arrive at the desired result. □

Theorem 5.

Let $(N,D^N,g)$ be a Lagrangian submanifold of a quaternion Kahlerian statistical manifold $\left(M\right(c),D,g,\vartheta )$ of constant Q-sectional curvature c. If we assume that $A^{*}=A$, then N is of constant $\mathcal{U}$-sectional curvature ${\displaystyle \frac{c}{4}}$.

Proof. 

Here, we use Equations (12) and (13) and find that

$$\begin{array}{ccc}\hfill \mathcal{U}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}& =& 2R^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}-S^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}\hfill \\ & =& 2{\overline{\mathcal{R}im}}^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}-{\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}\hfill \\ & & +A_{\mathbf{B}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})}^{*}{\mathbf{V}}_{\mathbf{1}}+A_{{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})}{\mathbf{V}}_{\mathbf{1}}\hfill \\ & & -A_{\mathbf{B}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}})}^{*}{\mathbf{V}}_{\mathbf{2}}-A_{{\mathbf{B}}^{*}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}})}{\mathbf{V}}_{\mathbf{2}}.\hfill \end{array}$$

But from Lemmas 1 and 2, our above equation reduces to

$$\begin{array}{ccc}\hfill \mathcal{U}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}& =& 2{\overline{\mathcal{R}im}}^0({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}}-{\overline{S}}^{dual}({\mathbf{V}}_{\mathbf{1}},\mathbf{Y}){\mathbf{V}}_{\mathbf{3}}\hfill \\ & & +A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{\mathbf{V}}_{\mathbf{3}}+A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{3}}\hfill \\ & & -A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{3}}-A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}}\hfill \\ & =& {\displaystyle \frac{c}{4}}\{g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}){\mathbf{V}}_{\mathbf{1}}-g({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}){\mathbf{V}}_{\mathbf{2}}\}+A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{\mathbf{V}}_{\mathbf{3}}\hfill \\ & & +A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{3}}-A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}^{*}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{3}}\hfill \\ & & -A_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}}}{A}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}}\hfill \\ & =& {\displaystyle \frac{c}{4}}\{g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}}){\mathbf{V}}_{\mathbf{1}}-g({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}}){\mathbf{V}}_{\mathbf{2}}\}.\hfill \end{array}$$

The $\mathcal{U}$-sectional curvature is given by

$$\begin{array}{ccc}\hfill {\mathcal{K}}^{\mathcal{U}}({\mathbf{V}}_{\mathbf{1}}\wedge {\mathbf{V}}_{\mathbf{2}})& =& g(\mathcal{U}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{1}})\hfill \\ & =& {\displaystyle \frac{c}{4}}\{g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{2}})g({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{1}})-g({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{1}})\}={\displaystyle \frac{c}{4}}.\hfill \end{array}$$

□

6. Quaternion Kahlerian Statistical Submersions

The concept of statistical submersion between statistical manifolds was first proposed by N. Abe and K. Hasegawa [7]. Their work expanded upon foundational results established by B. O’Neill [15] regarding Riemannian submersions and geodesics. Later, statistical submersions with different statistical structures were studied (see [16]), and obtained several geometric properties (see [17,18,19,20]).

Consider two statistical manifolds, $(M,D,g)$ and $(\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$, with $dim\left(M\right)=m$ and $dim\left(\mathcal{B}\right)=n$. Assume that $m>n$. Then, a statistical submersion $\iota :M\to \mathcal{B}$ is a smooth mapping of M onto $\mathcal{B}$ if these conditions hold [7,21]:

• $\iota$ has maximal rank;
• The differential ${\iota }_{*}$ preserves the length of horizontal vectors;
• ${\iota }_{*}{\left(D_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{2}}\right)}_x={\left(D_{\mathcal{B}U}V\right)}_{\iota \left(x\right)}$;

where ${\iota }_{*}$ is related to U and V on $\mathcal{B}$ while ${\mathbf{V}}_{\mathbf{1}}$ and ${\mathbf{V}}_{\mathbf{2}}$ are basic vector fields on M.

The $(m-n)$-dimensional statistical submanifold ${\iota }^{-1}\left(y\right)$, $y\in \mathcal{B}$ is known as a fiber $\overline{M}$, equipped with the induced metric G. We will denote $s=m-n$ throughout. The affine connections on $\overline{M}$ are represented by $\overline{D}$ and ${\overline{D}}^{*}$.

A vector field on M is classified as horizontal if it is always orthogonal to the fibers, and as vertical if it is always tangent to the fibers. For each $q\in M$, the vertical and horizontal subspaces in the tangent space $T_{q}M$ of the total space M are denoted by ${\mathcal{V}}_q\left(M\right)$ and ${\mathcal{H}}_q\left(M\right)$, respectively.

The tangent bundle $TM$ can be decomposed as

$$\begin{array}{c}\hfill TM=\mathcal{H}\left(M\right)\oplus \mathcal{V}\left(M\right),\end{array}$$

where $\mathcal{H}\left(M\right)$ and $\mathcal{V}\left(M\right)$ are the horizontal and vertical distributions, respectively. The projection mappings onto these distributions are denoted by $\mathcal{H}:TM\to \mathcal{H}\left(M\right)$ and $\mathcal{V}:TM\to \mathcal{V}\left(M\right)$.

The geometry of statistical submersions is distinguished by the tensors $\mathcal{T}$ and $\mathcal{A}$, O’Neil’s tensors, of type $(1,2)$ (see [15]), along with their dual counterparts, ${\mathcal{T}}^{*}$ and ${\mathcal{A}}^{*}$, which are derived by replacing D with $D^{*}$, as described in [17].

Remark 3.

The following points are important to note:

1. 

${\left({\mathcal{T}}^{*}\right)}^{*}=\mathcal{T}$ and ${\left({\mathcal{A}}^{*}\right)}^{*}=\mathcal{A}$.

2. 

$\mathcal{T}$ and ${\mathcal{T}}^{*}$ are symmetric for vertical vector fields.

3. 

For vertical vector fields $\mathcal{T}$ coincides with the second fundamental form of the immersion of the fiber submanifolds.

4. 

$\mathcal{T}\equiv 0$ (or $\mathcal{A}\equiv 0$) if and only if ${\mathcal{T}}^{*}\equiv 0$ (or ${\mathcal{A}}^{*}\equiv 0$).

5. 

For horizontal vectors $\mathcal{A}$ is symmetric if and only if $\mathcal{H}$ is integrable with respect to D.

Denote by $R^{\overline{M}}$ and $R^{\mathcal{B}}$ the curvature tensor of each fiber $\overline{M}$ with respect to $\overline{D}$ and that on $\mathcal{B}$ of $D_{\mathcal{B}}$. Then, from [17], we have the following equations:

$$\begin{array}{ccc}\hfill g(\overline{\mathcal{R}im}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})& =& g(R^{\overline{M}}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}},{\mathbf{V}}_{\mathbf{4}})+g({\mathcal{T}}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}},{\mathcal{T}}_{{\mathbf{V}}_{\mathbf{2}}}^{*}{\mathbf{V}}_{\mathbf{4}})\hfill \\ & & -g({\mathcal{T}}_{{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{3}},{\mathcal{T}}_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{4}}),\hfill \end{array}$$

(15)

for all vertical vector fields ${\mathbf{V}}_{\mathbf{1}}$, ${\mathbf{V}}_{\mathbf{2}}$, ${\mathbf{V}}_{\mathbf{3}}$, and ${\mathbf{V}}_{\mathbf{4}}$ on $\mathcal{V}$.

$$\begin{array}{ccc}\hfill g(\overline{\mathcal{R}im}({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })& =& g({\left(D_{{\mathbf{V}}_{\mathbf{1}}}\mathcal{T}\right)}_{{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })\hfill \\ & & -g({\left(D_{{\mathbf{V}}_{\mathbf{2}}}\mathcal{T}\right)}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime }),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill g(\overline{\mathcal{R}im}({{\mathbf{V}}_{\mathbf{1}}}^{\prime },{{\mathbf{V}}_{\mathbf{2}}}^{\prime }){{\mathbf{V}}_{\mathbf{3}}}^{\prime },{{\mathbf{V}}_{\mathbf{4}}}^{\prime })& =& g(R^{\mathcal{B}}({{\mathbf{V}}_{\mathbf{1}}}^{\prime },{{\mathbf{V}}_{\mathbf{2}}}^{\prime }){{\mathbf{V}}_{\mathbf{3}}}^{\prime },{{\mathbf{V}}_{\mathbf{4}}}^{\prime })\hfill \\ & & -g({\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{2}}}^{\prime }}{{\mathbf{V}}_{\mathbf{3}}}^{\prime },{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{4}}}^{\prime })\hfill \\ & & +g({\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{3}}}^{\prime },{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{2}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{4}}}^{\prime })\hfill \\ & & -g({\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }+{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{3}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{4}}}^{\prime }),\hfill \end{array}$$

(17)

for all horizontal vector fields ${{\mathbf{V}}_{\mathbf{1}}}^{\prime }$, ${{\mathbf{V}}_{\mathbf{2}}}^{\prime }$, ${{\mathbf{V}}_{\mathbf{3}}}^{\prime }$, and ${{\mathbf{V}}_{\mathbf{4}}}^{\prime }$ on $\mathcal{H}$.

In [20], A.D. Vilcu and G.E. Vilcu introduced the concept of a quaternionic Kähler-like statistical manifold by defining a dualistic pair of tensor fields, ${\mathbf{J}}_o,{\mathbf{J}}_o^{*}$, and subsequently explored the key properties of quaternionic Kähler-like statistical submersions within this framework. In contrast, we consider a different notion of quaternion Kahlerian statistical manifolds, as discussed in the preceding sections. In this section, we define and establish some fundamental properties of quaternion Kahlerian statistical submersions.

Definition 8.

A quaternion Kahlerian statistical submersion is the statistical submersion $\iota :(M,D,g,\vartheta )\to (\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$, where $(M,D,g,\vartheta )$ is a quaternion Kahlerian statistical manifold and $(\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$ is a statistical manifold.

Thus, we have the following:

Theorem 6.

For a quaternion Kahlerian statistical submersion $\iota :(M,D,g,\vartheta )\to (\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$. If $\overline{M}$ is a ${\mathbf{J}}_o$-invariant submanifold of M, then ι is with isometric fiber, that is, $\mathcal{T}={\mathcal{T}}^{*}=0$.

Theorem 7.

For a quaternion Kahlerian statistical submersion $\iota :(M,D,g,\vartheta )\to (\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$. If $\overline{M}$ is a ${\mathbf{J}}_o$-invariant submanifold of M, then the horizontal distribution is completely integrable, that is, $\mathcal{A}={\mathcal{A}}^{*}=0$.

Proof. 

From Theorem 1, we obtain the following:

$$\begin{array}{ccc}\hfill g(D_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{\mathbf{J}}_o{{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathbf{V}}_{\mathbf{1}})-g({\mathbf{J}}_o{D}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathbf{V}}_{\mathbf{1}})& =& {\Omega }_{o+2}\left({{\mathbf{V}}_{\mathbf{1}}}^{\prime }\right)g({\mathbf{J}}_{o+1}{{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathbf{V}}_{\mathbf{1}})\hfill \\ & & -{\Omega }_{o+1}\left({{\mathbf{V}}_{\mathbf{1}}}^{\prime }\right)g({\mathbf{J}}_{o+2}{{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathbf{V}}_{\mathbf{1}})\hfill \\ & =& -{\Omega }_{o+2}\left({{\mathbf{V}}_{\mathbf{1}}}^{\prime }\right)g({{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathbf{J}}_{o+1}{\mathbf{V}}_{\mathbf{1}})\hfill \\ & & +{\Omega }_{o+1}\left({{\mathbf{V}}_{\mathbf{1}}}^{\prime }\right)g({{\mathbf{V}}_{\mathbf{2}}}^{\prime },{\mathbf{J}}_{o+2}{\mathbf{V}}_{\mathbf{1}})\hfill \\ & =& 0,\hfill \end{array}$$

for the vertical vector field ${\mathbf{V}}_{\mathbf{1}}$ and the horizontal vector fields ${{\mathbf{V}}_{\mathbf{1}}}^{\prime }$ and ${{\mathbf{V}}_{\mathbf{2}}}^{\prime }$ on M. Therefore, we can compute that

$$\begin{array}{c}\hfill {\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}\left({\mathbf{J}}_o{{\mathbf{V}}_{\mathbf{2}}}^{\prime }\right)={\mathbf{J}}_o{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }.\end{array}$$

In particular, we have

$$\begin{array}{c}\hfill {\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }=-{\mathbf{J}}_o{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{\mathbf{J}}_o{{\mathbf{V}}_{\mathbf{2}}}^{\prime }.\end{array}$$

(18)

On the other hand, using the relation ${\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }=-{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{2}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{1}}}^{\prime }$, we derive

$$\begin{array}{c}\hfill {\mathcal{A}}_{{\mathbf{J}}_o{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{\mathbf{Y}}^{\prime }={\mathbf{J}}_o{\mathcal{A}}_{{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}},\end{array}$$

which can be simplified further as

$$\begin{array}{c}\hfill {\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }=-{\mathbf{J}}_o{\mathcal{A}}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}.\end{array}$$

(19)

By combining Equations (18) and (19), we obtain

$$\begin{array}{c}\hfill {\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{\mathbf{J}}_o{{\mathbf{V}}_{\mathbf{2}}}^{\prime }={\mathcal{A}}_{{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}}^{*}{\mathbf{V}}_{\mathbf{2}}.\end{array}$$

(20)

For $o=1$, from Equations (18)–(20), we deduce

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }& =& -{\mathbf{J}}_1{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{\mathbf{J}}_1{{\mathbf{V}}_{\mathbf{2}}}^{\prime }\hfill \\ & =& -{\mathbf{J}}_1{\mathcal{A}}_{{\mathbf{J}}_1{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}^{*}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }\hfill \\ & =& J_1^2{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }\hfill \\ & =& -{\mathcal{A}}_{{{\mathbf{V}}_{\mathbf{1}}}^{\prime }}{{\mathbf{V}}_{\mathbf{2}}}^{\prime }.\hfill \end{array}$$

Thus, $\mathcal{A}=0$, and similarly, we conclude that ${\mathcal{A}}^{*}=0$. □

Example 3.

Let $({\mathbb{R}}_2^2,D^{{\mathbb{R}}_2^2},g_{{\mathbb{R}}_2^2})$ be a statistical manifold with local coordinate system $(x_1,y_1)$, $g_{{\mathbb{R}}_2^2}=-{\left(d{x}_1\right)}^2-{\left(d{y}_1\right)}^2$ and flat connection. We consider the quaternion Kahlerian statistical manifold $({\mathbb{R}}_2^4,D=D^0+K,g,\vartheta )$ from Example 2 and define the quaternion Kahlerian statistical submersion $\iota :({\mathbb{R}}_2^4,D=D^0+K,g,\vartheta )\to ({\mathbb{R}}_2^2,D^{{\mathbb{R}}_2^2},g_{{\mathbb{R}}_2^2})$ as the projection mapping $\iota (x_1,y_1,x_2,y_2)=(x_1,y_1)$. Then, we find $\mathcal{A}=0$. And here each fiber is a totally geodesic semi-Riemannian submanifold of ${\mathbb{R}}_2^4$.

For a vertical vector field ${\mathbf{V}}_{\mathbf{1}}$ and horizontal vector field ${{\mathbf{V}}_{\mathbf{1}}}^{\prime }$, we put

$$\begin{array}{c}\hfill {\mathbf{J}}_o{{\mathbf{V}}_{\mathbf{1}}}^{\prime }=P_o{{\mathbf{V}}_{\mathbf{1}}}^{\prime }+F_o{{\mathbf{V}}_{\mathbf{1}}}^{\prime },\end{array}$$

(21)

and

$$\begin{array}{c}\hfill {\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}}=t_o{\mathbf{V}}_{\mathbf{1}}+f_o{\mathbf{V}}_{\mathbf{1}},\end{array}$$

(22)

where $P_o$ and $t_o$ are horizontal parts and $F_o$ and $f_o$ are vertical parts.

We consider the curvature with respect to D of the total space satisfies

$$\begin{array}{ccc}\hfill \overline{\mathcal{R}im}(U,V)\mathbf{W}& =& {\displaystyle \frac{c}{4}}\{g(V,\mathbf{W})U-g(U,\mathbf{W})V\hfill \\ & & +\sum _{o=1}^3[g(\mathbf{W},{\mathbf{J}}_{o}V){\mathbf{J}}_{o}U-g(\mathbf{W},{\mathbf{J}}_{o}U){\mathbf{J}}_{o}V]\hfill \\ & & +\sum _{o=1}^3[g(U,{\mathbf{J}}_{o}V)-g({\mathbf{J}}_{o}U,V)]{\mathbf{J}}_o\mathbf{W}\},\hfill \end{array}$$

(23)

for $U,V,\mathbf{W}\in \Gamma \left(TM\right)$, where $c\in \mathbb{R}$. Then, from (16), (22) and (23), we find

$$\begin{array}{ccc}\hfill g({\left(D_{{\mathbf{V}}_{\mathbf{1}}}\mathcal{T}\right)}_{{\mathbf{V}}_{\mathbf{2}}}{\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })-g({\left(D_{{\mathbf{V}}_{\mathbf{2}}}\mathcal{T}\right)}_{{\mathbf{V}}_{\mathbf{1}}}{\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })& =& {\displaystyle \frac{c}{4}}\{g({\mathbf{V}}_{\mathbf{2}},{\mathbf{V}}_{\mathbf{3}})g({\mathbf{V}}_{\mathbf{1}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })-g({\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{3}})g({\mathbf{V}}_{\mathbf{2}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })\hfill \\ & & +\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})g({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })-g({\mathbf{V}}_{\mathbf{3}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}})g({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })]\hfill \\ & & +\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},{\mathbf{J}}_o{\mathbf{V}}_{\mathbf{2}})-g({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})]g({\mathbf{J}}_o{\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })\}\hfill \\ & =& {\displaystyle \frac{c}{4}}\{\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{2}})g(t_o{\mathbf{V}}_{\mathbf{1}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })-g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{1}})g(t_o{\mathbf{V}}_{\mathbf{2}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })]\hfill \\ & & +\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{2}})-g(f_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})]g(t_o{\mathbf{V}}_{\mathbf{3}},{{\mathbf{V}}_{\mathbf{1}}}^{\prime })\}.\hfill \end{array}$$

(24)

From Equation (24) and $\mathcal{T}=0$, we find that $c=0$ or

$$\begin{array}{ccc}& & \sum _{o=1}^3[t_{o}g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{1}}-t_{o}g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{1}}){\mathbf{V}}_{\mathbf{2}}]\hfill \\ & & +\sum _{o=1}^3{t}_o[g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{2}})-g(f_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})]{\mathbf{V}}_{\mathbf{3}}=0.\hfill \end{array}$$

If we consider the equation

$$\begin{array}{ccc}& & \sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{2}}){\mathbf{V}}_{\mathbf{1}}-g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{1}}){\mathbf{V}}_{\mathbf{2}}]\hfill \\ & & +\sum _{o=1}^3[g({\mathbf{V}}_{\mathbf{3}},f_o{\mathbf{V}}_{\mathbf{2}})-g(f_o{\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}})]{\mathbf{V}}_{\mathbf{3}}=0,\hfill \end{array}$$

it implies that $f_o=0$ for $o\in \{1,2,3\}$. Consequently, it follows that $t_o=0$ if $f_o\ne 0$ for $o\in \{1,2,3\}$. Hence, we conclude the following:

Theorem 8.

Let $\iota :(M,D,g,\vartheta )\to (\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$ be a quaternion Kahlerian statistical submersion with isometric fiber. If the total space satisfies (23), then one of the following is the case:

1. 

$c=0$;

2. 

Each fiber is an invariant of M satisfying (23);

3. 

Each fiber is an anti-invariant of M, which is of constant curvature ${\displaystyle \frac{c}{4}}$.

7. Chen–Ricci Inequality

Several researchers have established the Chen–Ricci inequality for specific types of submanifolds in various ambient spaces. Also, particular cases of these inequalities have been demonstrated in statistical contexts (see [22,23,24]). This section focuses on deriving the Chen–Ricci inequality for the vertical distribution in quaternion Kahlerian statistical submersions from quaternion Kahlerian statistical manifolds.

Let $\iota :(M\left(c\right),D,g,\vartheta )\to (\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$ be a quaternion Kahlerian statistical submersion from a $4r$-dimensional quaternion Kahlerian statistical manifold $\left(M\right(c),D,g,\vartheta )$ of constant Q-sectional curvature c onto an n-dimensional statistical manifold $(\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$. Then, we suppose the orthonormal bases $\{v_1,v_2,\cdots ,v_s\}$ and $\{{v_{s+1}}^{\prime },{v_{s+2}}^{\prime },\cdots ,{v_{4r}}^{\prime }\}$ on the vertical space and the horizontal space, respectively. Then, the 4rean curvature vectors $\mathcal{N}$, ${\mathcal{N}}^{*}$, and ${\mathcal{N}}^0$ with respect to D, $D^{*}$, and $D^0$ are given, respectively, by

$$\begin{array}{ccc}\hfill \mathcal{N}& =& {\displaystyle \frac{1}{s}}\sum _{\beta =1}^s{\mathcal{T}}_{v_{\beta }}{v}_{\beta },\hfill \\ \hfill {\mathcal{N}}^{*}& =& {\displaystyle \frac{1}{s}}\sum _{\beta =1}^s{\mathcal{T}}_{v_{\beta }}^{*}{v}_{\beta },\hfill \\ \hfill {\mathcal{N}}^0& =& {\displaystyle \frac{1}{s}}\sum _{\beta =1}^s{\mathcal{T}}_{v_{\beta }}^0{v}_{\beta }.\hfill \end{array}$$

We use (15) to find the vertical scalar curvature ${\tau }_{\mathcal{V}}^{\overline{M}}$ as

$$\begin{array}{ccc}\hfill 2{\tau }_{\mathcal{V}}^{\overline{M}}& =& \sum _{i,j=1}^{s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & =& {\displaystyle \frac{c}{4}}s(s-1)+9rc-g(\mathcal{T},{\mathcal{T}}^{*})+s^{2}g(\mathcal{N},{\mathcal{N}}^{*}),\hfill \end{array}$$

which yields from $4\left|\right|{\mathcal{T}}^0{\left|\right|}^2={\left|\right|\mathcal{T}\left|\right|}^2+2g(\mathcal{T},{\mathcal{T}}^{*})+\left|\right|{\mathcal{T}}^{*}{\left|\right|}^2$ and $4\left|\right|{\mathcal{N}}^0{\left|\right|}^2={\left|\right|\mathcal{N}\left|\right|}^2+2g(\mathcal{N},{\mathcal{N}}^{*})+\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2$ that

$$\begin{array}{ccc}\hfill 4{\tau }_{\mathcal{V}}^{\overline{M}}& =& {\displaystyle \frac{c}{2}}s(s-1)+18rc+\sum _{z=1}^{4r}\sum _{i,j=1}^s[{\left({\mathcal{T}}_{ij}^z\right)}^2+{\left({\mathcal{T}}_{ij}^{*z}\right)}^2-4{\left({\mathcal{T}}_{ij}^{0z}\right)}^2]\hfill \\ & & +s^2\left(4\right||{\mathcal{N}}^0{\left|\right|}^2-{\left|\right|\mathcal{N}\left|\right|}^2-\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2)\hfill \\ & =& {\displaystyle \frac{c}{2}}s(s-1)+18rc+{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2+{\displaystyle \frac{1}{2}}({\mathcal{T}}_{11}^z-{\mathcal{T}}_{22}^z-\cdots -{\mathcal{T}}_{ss}^z)\hfill \\ & & +2\sum _{z=1}^{4r}\{\sum _{j=2}^s{\left({\mathcal{T}}_{1j}^z\right)}^2-\sum _{2\le i\ne j\le s}[{\mathcal{T}}_{ii}^z{\mathcal{T}}_{jj}^z-{\left({\mathcal{T}}_{ij}^z\right)}^2]\}\hfill \\ & & +{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2+{\displaystyle \frac{1}{2}}({\mathcal{T}}_{11}^{*z}-{\mathcal{T}}_{22}^{*z}-\cdots -{\mathcal{T}}_{ss}^{*z})\hfill \\ & & +2\sum _{z=1}^{4r}\{\sum _{j=2}^s{\left({\mathcal{T}}_{1j}^{*z}\right)}^2-\sum _{2\le i\ne j\le s}[{\mathcal{T}}_{ii}^{*z}{\mathcal{T}}_{*jj}^z-{\left({\mathcal{T}}_{ij}^{*z}\right)}^2]\}\hfill \\ & & -4\sum _{z=1}^{4r}\sum _{i,j=1}^s{\left({\mathcal{T}}_{ij}^{0z}\right)}^2+s^2\left(4\right||{\mathcal{N}}^0{\left|\right|}^2-{\left|\right|\mathcal{N}\left|\right|}^2-\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill 4{\tau }_{\mathcal{V}}^{\overline{M}}& \ge & {\displaystyle \frac{c}{2}}s(s-1)+18rc-{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2+4s^2\left|\right|{\mathcal{N}}^0{\left|\right|}^2\hfill \\ & & -2\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}({\mathcal{T}}_{ii}^z{\mathcal{T}}_{jj}^z-{\left({\mathcal{T}}_{ij}^z\right)}^2)-2\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}({\mathcal{T}}_{ii}^{*z}{\mathcal{T}}_{*jj}^z-{\left({\mathcal{T}}_{ij}^{*z}\right)}^2)\hfill \\ & & -4\sum _{z=1}^{4r}\sum _{i,j=1}^s{\left({\mathcal{T}}_{ij}^{0z}\right)}^2.\hfill \end{array}$$

Here, the equality holds if and only if ${\mathcal{T}}_{11}^z={\mathcal{T}}_{22}^z+\cdots +{\mathcal{T}}_{ss}^z$, ${\mathcal{T}}_{1j}^z=0$, $j=\{2,3,\cdots ,s\}$, ${\mathcal{T}}_{11}^{*z}={\mathcal{T}}_{22}^{*z}+\cdots +{\mathcal{T}}_{ss}^{*z}$, and ${\mathcal{T}}_{1j}^{*z}=0$, $j=\{2,3,\cdots ,s\}$.

Again, from (15), we derive

$$\begin{array}{ccc}\hfill \sum _{2\le i\ne j\le s}g(\overline{\mathcal{R}im}(v_i,v_j)v_j,v_i)& =& \sum _{2\le i\ne j\le s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & & +\sum _{2\le i\ne j\le s}g({\mathcal{T}}_{v_i}{v}_j,{\mathcal{T}}_{v_j}^{*}{v}_i)\hfill \\ & & -\sum _{2\le i\ne j\le s}g({\mathcal{T}}_{v_j}{v}_j,{\mathcal{T}}_{v_i}^{*}{v}_i)\hfill \\ & =& \sum _{2\le i\ne j\le s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & & +\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}{\displaystyle \frac{1}{2}}[4{\left({\mathcal{T}}_{ij}^{0z}\right)}^2-{\left({\mathcal{T}}_{ij}^z\right)}^2-{\left({\mathcal{T}}_{ij}^{*z}\right)}^2]\hfill \\ & & +\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}{\displaystyle \frac{1}{2}}[{\mathcal{T}}_{ii}^z{\mathcal{T}}_{jj}^z+{\mathcal{T}}_{ii}^{*z}{\mathcal{T}}_{jj}^{*z}-4{\mathcal{T}}_{ii}^{0z}{\mathcal{T}}_{jj}^{0z}],\hfill \end{array}$$

which can be rewritten as

$$\begin{array}{ccc}& & -\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}[{\mathcal{T}}_{ii}^z{\mathcal{T}}_{jj}^z+{\mathcal{T}}_{ii}^{*z}{\mathcal{T}}_{jj}^{*z}-{\left({\mathcal{T}}_{ij}^z\right)}^2-{\left({\mathcal{T}}_{ij}^{*z}\right)}^2]\hfill \\ & =& -2\sum _{2\le i\ne j\le s}g(\overline{\mathcal{R}im}(v_i,v_j)v_j,v_i)+2\sum _{2\le i\ne j\le s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & & +4\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}[{\left({\mathcal{T}}_{ij}^{0z}\right)}^2-{\mathcal{T}}_{ii}^{0z}{\mathcal{T}}_{jj}^{0z}].\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill 4{\tau }_{\mathcal{V}}^{\overline{M}}& \ge & {\displaystyle \frac{c}{2}}s(s-1)+18rc-{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2+4s^2\left|\right|{\mathcal{N}}^0{\left|\right|}^2\hfill \\ & & -4\sum _{2\le i\ne j\le s}g(\overline{\mathcal{R}im}(v_i,v_j)v_j,v_i)+4\sum _{2\le i\ne j\le s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & & +8\sum _{z=1}^{4r}\sum _{2\le i\ne j\le s}[{\left({\mathcal{T}}_{ij}^{0z}\right)}^2-{\mathcal{T}}_{ii}^{0z}{\mathcal{T}}_{jj}^{0z}]-4\sum _{z=1}^{4r}\sum _{i,j=1}^s{\left({\mathcal{T}}_{ij}^{0z}\right)}^2.\hfill \end{array}$$

By the Gauss equation for the Levi–Civita connection, we have

$$\begin{array}{ccc}\hfill 4{\tau }_{\mathcal{V}}^{\overline{M}}& \ge & {\displaystyle \frac{c}{2}}s(s-1)+18rc-{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2\hfill \\ & & -4\sum _{2\le i\ne j\le s}g(\overline{\mathcal{R}im}(v_i,v_j)v_j,v_i)+4\sum _{2\le i\ne j\le s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & & +8\sum _{2\le i\ne j\le s}[-g({\overline{\mathcal{R}im}}^0(v_i,v_j)v_j,v_i)+g(R^{0\overline{M}}(v_i,v_j)v_j,v_i)]\hfill \\ & & +4(2{\overline{\tau }}_{\mathcal{V}}^0-2{\tau }_{\mathcal{V}}^{0\overline{M}})\hfill \\ & \ge & {\displaystyle \frac{c}{2}}s(s-1)+18rc-{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2\hfill \\ & & -4\sum _{2\le i\ne j\le s}g(\overline{\mathcal{R}im}(v_i,v_j)v_j,v_i)+4\sum _{2\le i\ne j\le s}g(R^{\overline{M}}(v_i,v_j)v_j,v_i)\hfill \\ & & +8({\overline{\mathcal{R}ic}}_{\mathcal{V}}^0\left(v_1\right)-Ri{c}_{\mathcal{V}}^{0\overline{M}}\left(v_1\right)).\hfill \end{array}$$

Further, we rewrite it as

$$\begin{array}{ccc}\hfill 4Ri{c}_{\mathcal{V}}^{\overline{M}}\left(v_1\right)& \ge & {\displaystyle \frac{c}{2}}s(s-1)+18rc-{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2\hfill \\ & & -(s-1)(s-2)c-36(r-1)c+8({\overline{\mathcal{R}ic}}_{\mathcal{V}}^0\left(v_1\right)-Ri{c}_{\mathcal{V}}^{0\overline{M}}\left(v_1\right))\hfill \\ & =& {\displaystyle \frac{c}{2}}(s-1)(4-s)+18(2-r)c+8({\overline{\mathcal{R}ic}}_{\mathcal{V}}^0\left(v_1\right)-Ri{c}_{\mathcal{V}}^{0\overline{M}}\left(v_1\right))\hfill \\ & & -{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2.\hfill \end{array}$$

Theorem 9.

Let $\iota :(M\left(c\right),D,g,\vartheta )\to (\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$ be a quaternion Kahlerian statistical submersion from a $4r$-dimensional quaternion Kahlerian statistical manifold $\left(M\right(c),D,g,\vartheta )$ of constant Q-sectional curvature c onto statistical manifold $(\mathcal{B},D_{\mathcal{B}},g_{\mathcal{B}})$. Then,

$$\begin{array}{ccc}\hfill 4Ri{c}_{\mathcal{V}}^{\overline{M}}\left(v_1\right)& \ge & {\displaystyle \frac{c}{2}}(s-1)(4-s)+18(2-r)c+8({\overline{\mathcal{R}ic}}_{\mathcal{V}}^0\left(v_1\right)-Ri{c}_{\mathcal{V}}^{0\overline{M}}\left(v_1\right))\hfill \\ & & -{\displaystyle \frac{s^2}{2}}{\left|\right|\mathcal{N}\left|\right|}^2-{\displaystyle \frac{s^2}{2}}\left|\right|{\mathcal{N}}^{*}{\left|\right|}^2.\hfill \end{array}$$

(25)

The equality holds if and only if ${\mathcal{T}}_{11}^z={\mathcal{T}}_{22}^z+\cdots +{\mathcal{T}}_{ss}^z$, ${\mathcal{T}}_{11}^{*z}={\mathcal{T}}_{22}^{*z}+\cdots +{\mathcal{T}}_{ss}^{*z}$, ${\mathcal{T}}_{1j}^z=0$ and ${\mathcal{T}}_{1j}^{*z}=0$, $j=\{2,3,\cdots ,s\}$.

Remark 4.

In future research, one can expand this study’s scope by deriving the Chen–Ricci inequality for the horizontal distribution in quaternion Kahlerian statistical submersions from quaternion Kahlerian statistical manifolds. Furthermore, it would be interesting to establish such an inequality that intricately links the vertical and horizontal distributions, unveiling deeper geometric interrelations. These advancements are expected to provide significant insights and enrich the theoretical framework of quaternion Kahlerian statistical geometry, paving the way for further exploration and applications in this domain.