Statistical Submanifolds Equipped with $\mathcal{F}$-Statistical Connections

Abstract

This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with $\mathcal{F}$-statistical connections are examined. The equations of Gauss and Codazzi of $\mathcal{F}$-statistical connections are obtained. Such structures when the statistical submanifolds are conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to $\mathcal{F}$-statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared $\mathcal{F}$-mean curvature of a statistical submanifold of statistical manifolds.

1. Introduction

An attractive and noteworthy issue in differential geometry is the concept of statistical manifolds which was initially defined by Amari [1]. Statistical manifolds have many application areas such as machine learning, general relativity, neural networks, physics, image analysis, control systems and many more [2,3,4]. A statistical structure on a Riemannian manifold is inspired from statistical mold, where probability distributions correspond to the manifold points. In the recent years, statistical manifolds have been actively investigated by many mathematicians and interesting results have been obtained [5,6,7,8,9,10,11,12,13].

Suppose that $\zeta$ is an open subset of ${\mathbb{R}}^n$ and $\chi$ is a sample space with parameter $\vartheta =({\vartheta }^1,\cdots ,{\vartheta }^n)$. A statistical model S is the set of probability density functions defined by

$$S=\{p(v;\vartheta ):{\int }_{\chi }p(v;\vartheta )dv=1,p(v;\vartheta )>0,\vartheta \in \zeta \subseteq {\mathbb{R}}^n\}.$$

The Fisher information matrix $h\left(\vartheta \right)=\left[h_{ls}\left(\vartheta \right)\right]$ on S is given as ([14])

$$h_{ls}\left(\vartheta \right):={\int }_{\chi }{\partial }_l{\ell }_{\vartheta }{\partial }_s{\ell }_{\vartheta }p(v;\vartheta )\mathrm{d}v=E_p\left[{\partial }_l{\ell }_{\vartheta }{\partial }_s{\ell }_{\vartheta }\right],$$

(1)

where $E_p[\ell ]$ is the expectation of $\ell \left(v\right)$ with respect to $p(v;\vartheta )$, ${\ell }_{\vartheta }=\ell (v;\vartheta ):=\log p(v;\vartheta )$ and ${\partial }_l:={\displaystyle \frac{\partial }{\partial {\vartheta }^l}}$. The space S with together the information matrices is a statistical manifold [15]. It is seen that $(S,h)$ is a Riemannian manifold. An affine connection ∇ with respect to $p(v;\vartheta )$ is described by

$${\Gamma }_{ls,k}=h({\nabla }_{{\partial }_l}{\partial }_s,{\partial }_k):=E_p\left[\left({\partial }_l{\partial }_s{\ell }_{\vartheta }\right){\partial }_k{\ell }_{\vartheta }\right].$$

(2)

Statistical submanifolds were described by Vos in 1989 [16]. In 2015, Milijevic showed that a semi-parallel totally real statistical submanifold with some natural conditions is totally geodesic if it is of non-zero constant curvature [17]. In 2015, Aydin et al. investigated curvature properties of statistical submanifolds [18]. Moreover, in 2017, they generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature [19]. In 2018, Aytimur and özgür focused on submanifolds of statistical manifolds with quasi-constant curvature and obtained similar inequalities [20]. Statistical submanifolds of Hessian manifolds with constant Hessian curvature are studied in [21]. Then, Uohashi described conformally flat statistical submanifold of a flat statistical manifold using a Hessian domain [22]. Lee et al. derived extremities for normalized $\delta$-Casorati curvature for statistical submanifolds in statistical manifold with constant curvature [23]. Alkhaldi et al. showed that normalized scalar curvature is bounded above by Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds of constant $\varphi$-sectional curvature in [24]. In 2020, Jain et al. studied lightlike submanifolds of indefinite statistical manifolds and presented some conditions for the induced statistical Ricci tensor on a lightlike submanifold of indefinite statistical manifolds to be symmetric [25]. In [26], Aquib proved some of the curvature properties of submanifolds and provided a couple of inequalities for totally real statistical submanifolds of quaternionic Kaehler-like statistical space forms. In 2019, Chen et al. obtained a Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature [27]. Decu et al. studied inequalities for the Casorati curvature of statistical manifolds in holomorphic statistical manifolds of constant holomorphic curvature in [28]. Lone et al. defined Golden-like statistical manifolds and obtained certain interesting inequalities [29]. Recently, Meli et al. considered null submanifold in [30], where they presented statistical structures on a null hypersurface in the Lorentz-Minkowski space using the null second fundamental form. For some of the recent works, we refer to [31,32,33,34,35,36,37].

In [38] Balcerzak considered the affine combination of two affine connections ${\nabla }^{\left(0\right)}$ and ${\nabla }^{\left(1\right)}$ on a pseudo-Riemannian manifold and introduced a new connection as ${\nabla }^{\left(\mathcal{F}\right)}=(1-\mathcal{F}){\nabla }^{\left(0\right)}+\mathcal{F}{\nabla }^{\left(1\right)}$, where $\mathcal{F}$ is a smooth function on the pseudo-Riemannian manifold. Until now, this type of connections on (sub)statistical manifolds have not been considered. This paper fills the gap. This work is intended as an attempt to introduce a new family of connections which is statistical on statistical submanifolds. We refer to these connections as $\mathcal{F}$-statistical connections.

In this paper, we describe and study a family of statistical connections which called $\mathcal{F}$-statistical connections on statistical manifolds in Section 2. The Gauss and Weingarten formulas for $\mathcal{F}$-statistical connections are given in Section 3. Also, the geometric structures such as the $\mathcal{F}$-second fundamental forms, $\mathcal{F}$-mean curvatures, and the relations among them on a submanifold of a statistical manifold are examined. Then we show that if a statistical submanifold N of a statistical manifold $\tilde{N}$ is $\mathcal{F}$-doubly autoparallel, then the fundamental form and its dual are vice versa with respect to 0-statistical connections. Also, the $\mathcal{F}$-fundamental form and its dual are vice versa if N is geodesic. Moreover, if N is minimal with respect to $\mathcal{F}$-statistical connection and its dual, then 0-mean curvature vector field and its dual are vice versa. We describe the concept of divergence for $\mathcal{F}$-mean curvature vector fields and $\mathcal{F}$-fundamental forms and we find the properties of them. In Section 4, the equations of Gauss and Codazzi of $\mathcal{F}$-statistical connections are obtained. Such structures with condition conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to $\mathcal{F}$-statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared $\mathcal{F}$-mean curvature of a statistical submanifold of statistical manifolds.

2. Statistical Manifolds

For an n-dimensional manifold $\tilde{N}$, consider $(U,v^i)$, $i=1,\dots ,n,$ as a local chart of the point $v\in U$. Considering the coordinates $\left(v^i\right)$ on $\tilde{N}$, we have the local field ${\displaystyle \frac{\partial }{\partial v^i}}{|}_p$ as frames on $T_p\tilde{N}$. Assume that $\tilde{h}$ is a pseudo-Riemannian metric on $\tilde{N}$. An affine connection $\tilde{\nabla }$ is called Codazzi connection if the Codazzi equations satisfy:

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{{\mathcal{V}}_1}\tilde{h}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)=\left({\tilde{\nabla }}_{{\mathcal{V}}_2}\tilde{h}\right)({\mathcal{V}}_1,{\mathcal{V}}_3),(=\left({\tilde{\nabla }}_{{\mathcal{V}}_3}\tilde{h}\right)({\mathcal{V}}_1,{\mathcal{V}}_2)),\end{array}$$

(3)

for all ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(T\tilde{N}\right)$, where

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{{\mathcal{V}}_1}\tilde{h}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)={\mathcal{V}}_1\tilde{h}({\mathcal{V}}_2,{\mathcal{V}}_3)-\tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2,{\mathcal{V}}_3)-\tilde{h}({\mathcal{V}}_2,{\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_3).\end{array}$$

(4)

The triple $(\tilde{N},\tilde{\nabla },\tilde{h})$ is called a statistical manifold% We prefer italics. if $\tilde{\nabla }$ is a statistical connection, i.e., a torsion-free Codazzi connection. In particular, it is known that if the cubic tensor field is zero, a torsion-free Codazzi connection $\tilde{\nabla }$ reduces to the Levi-Civita connection ${\tilde{\nabla }}^{\tilde{h}}$. Moreover, the affine connection ${\tilde{\nabla }}^{*}$ of $\tilde{N}$ defined by

$$\begin{array}{c}\hfill {\mathcal{V}}_1\tilde{h}({\mathcal{V}}_2,{\mathcal{V}}_3)=\tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2,{\mathcal{V}}_3)+\tilde{h}({\mathcal{V}}_2,{\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3),\end{array}$$

(5)

is called the (dual) conjugate connection of $\tilde{\nabla }$ with respect to $\tilde{h}$. Immediately, one can see

$$\begin{array}{c}\hfill {\tilde{\nabla }}^{\tilde{h}}={\displaystyle \frac{1}{2}}\left(\tilde{\nabla }+{\tilde{\nabla }}^{*}\right),\end{array}$$

(6)

and $(\tilde{N},\tilde{\nabla },{\tilde{h}}^{*})$ forms a statistical manifold.

For a statistical structure $(\tilde{\nabla },\tilde{h})$ on $\tilde{N}$, if we consider a $(1,2)$-tensor field $\tilde{K}:\Gamma \left(T\tilde{N}\right)\times \Gamma \left(T\tilde{N}\right)\to \Gamma \left(T\tilde{N}\right)$ described by

$$\begin{array}{c}\hfill {\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2={\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2-{\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2,\end{array}$$

(7)

it follows that $\tilde{K}$ satisfies

$$\begin{array}{c}\hfill {\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2={\tilde{K}}_{{\mathcal{V}}_2}{\mathcal{V}}_1,\tilde{h}({\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2,{\mathcal{V}}_3)=\tilde{h}({\mathcal{V}}_2,{\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_3).\end{array}$$

(8)

For an affine connection $\tilde{\nabla }$, the curvature tensor ${\tilde{R}}^{\tilde{\nabla }}$ is defined as

$$\begin{array}{c}\hfill {\tilde{R}}^{\tilde{\nabla }}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3={\tilde{\nabla }}_{{\mathcal{V}}_1}{\tilde{\nabla }}_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\tilde{\nabla }}_{{\mathcal{V}}_2}{\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_3-{\tilde{\nabla }}_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}{\mathcal{V}}_3,\forall {\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(T\tilde{N}\right).\end{array}$$

(9)

On a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$, we denote ${\tilde{R}}^{\tilde{\nabla }}$ and ${\tilde{R}}^{{\tilde{\nabla }}^{*}}$ by $\tilde{R}$ and ${\tilde{R}}^{*}$, respectively for brief. It is known that the following hold

$$\begin{array}{cc}\hfill & \tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4)=-\tilde{R}({\mathcal{V}}_2,{\mathcal{V}}_1,{\mathcal{V}}_3,{\mathcal{V}}_4),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\tilde{R}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4)=-{\tilde{R}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_1,{\mathcal{V}}_3,{\mathcal{V}}_4),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4)=-{\tilde{R}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_4,{\mathcal{V}}_3),\hfill \end{array}$$

(12)

where $\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4)=\tilde{h}(\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3,{\mathcal{V}}_4)$. Moreover, if $\tilde{R}=0$, the statistical manifold $\tilde{N}$ is called a flat.

A statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ is called conjugate symmetric if the curvature tensors of the connections $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$, are equal, i.e., the following holds

$$\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3={\tilde{R}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3.$$

Let $\tilde{N}$ be a smooth manifold and $\mathcal{F}\in C^{\infty }\left(\tilde{N}\right)$. The affine combination of two affine connections ${\tilde{\nabla }}^{\left(0\right)}$ and ${\tilde{\nabla }}^{\left(1\right)}$ on $\tilde{N}$ is the connection ${\tilde{\nabla }}^{\left(\tilde{\mathcal{F}}\right)}$ given by

$${\tilde{\nabla }}^{\left(\mathcal{F}\right)}=(1-\mathcal{F}){\tilde{\nabla }}^{\left(0\right)}+\mathcal{F}{\tilde{\nabla }}^{\left(1\right)}.$$

Immediately, we see that

$$\begin{array}{c}\hfill {\tilde{T}}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}=(1-\mathcal{F})T^{{\tilde{\nabla }}^{\left(0\right)}}+\mathcal{F}{\tilde{T}}^{{\tilde{\nabla }}^{\left(1\right)}},{\tilde{\nabla }}^{\left(\mathcal{F}\right)}\tilde{h}=(1-\mathcal{F}){\tilde{\nabla }}^{\left(0\right)}\tilde{h}+\mathcal{F}{\tilde{\nabla }}^{\left(1\right)}\tilde{h},\end{array}$$

where ${\tilde{T}}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}$, ${\tilde{T}}^{{\tilde{\nabla }}^{\left(0\right)}}$ and ${\tilde{T}}^{{\tilde{\nabla }}^{\left(1\right)}}$ are the torsion tensors of ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$, ${\tilde{\nabla }}^{\left(0\right)}$ and ${\tilde{\nabla }}^{\left(1\right)}$, respectively [38].

Definition 1.

For a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$, the family of connections ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ given by affine combination of the conjugate connections ${\tilde{\nabla }}^{\left(0\right)}:=\tilde{\nabla }$ and ${\tilde{\nabla }}^{\left(1\right)}:={\tilde{\nabla }}^{*}$, i.e.,

$${\tilde{\nabla }}^{\left(\mathcal{F}\right)}=(1-\mathcal{F})\tilde{\nabla }+\mathcal{F}{\tilde{\nabla }}^{*},\mathcal{F}\in C^{\infty }\left(\tilde{N}\right),$$

is called $\mathcal{F}$-statistical connection of $\tilde{N}$.

Assuming $\mathcal{F}=0$, $\mathcal{F}=1$ and $\mathcal{F}={\displaystyle \frac{1}{2}}$ in the above definition, we obtain the statistical connections $\tilde{\nabla }$, ${\tilde{\nabla }}^{*}$ and the Levi-Civita connections ${\tilde{\nabla }}^{\tilde{h}}$ on $\tilde{N}$, respectively.

Corollary 1.

Let $(\tilde{h},\tilde{\nabla })$ be a statistical structure on $\tilde{N}$. Then

1. 

$(\tilde{N},\tilde{h},{\tilde{\nabla }}^{\left(\mathcal{F}\right)})$ is also a statistical manifold.

2. 

The connection ${\tilde{\nabla }}^{(1-\mathcal{F})}$ is dual of ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$, i.e.,

$$\begin{array}{c}\hfill {\mathcal{V}}_1\tilde{h}({\mathcal{V}}_2,{\mathcal{V}}_3)=\tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2,{\mathcal{V}}_3)+\tilde{h}({\mathcal{V}}_2,{\tilde{\nabla }}_{{\mathcal{V}}_1}^{(1-\mathcal{F})}{\mathcal{V}}_3),\forall {\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right).\end{array}$$

(13)

3. 

${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ and ${\tilde{\nabla }}^{(1-\mathcal{F})}$ satisfy the following

$$\begin{array}{c}\hfill {\tilde{\nabla }}^{\left(\mathcal{F}\right)}={\tilde{\nabla }}^{\tilde{h}}-{\displaystyle \frac{1-2\mathcal{F}}{2}}\tilde{K},{\tilde{\nabla }}^{(1-\mathcal{F})}-{\tilde{\nabla }}^{\left(\mathcal{F}\right)}=(1-2\mathcal{F})\tilde{K},{\tilde{\nabla }}^{(1-\mathcal{F})}+{\tilde{\nabla }}^{\left(\mathcal{F}\right)}=2{\tilde{\nabla }}^{\tilde{h}}.\end{array}$$

(14)

Proof. 

From Definition 1, it follows that the $\mathcal{F}$-statistical connection ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ is torsion-free, i.e.,

$$\begin{array}{cc}\hfill {\tilde{T}}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}({\mathcal{V}}_1,{\mathcal{V}}_2)=& {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2-{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\mathcal{V}}_1-[{\mathcal{V}}_1,{\mathcal{V}}_2]\hfill \\ \hfill =& (1-\mathcal{F}){\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{F}{\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2-(1-\mathcal{F}){\tilde{\nabla }}_{{\mathcal{V}}_2}{\mathcal{V}}_1+\mathcal{F}{\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_1-[{\mathcal{V}}_1,{\mathcal{V}}_2]\hfill \\ \hfill =& (1-\mathcal{F})\left({\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2-{\tilde{\nabla }}_{{\mathcal{V}}_2}{\mathcal{V}}_1\right)+\mathcal{F}\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2-{\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_1\right)-[{\mathcal{V}}_1,{\mathcal{V}}_2]\hfill \\ \hfill =& [{\mathcal{V}}_1,{\mathcal{V}}_2]-[{\mathcal{V}}_1,{\mathcal{V}}_2]=0.\hfill \end{array}$$

Moreover, it satisfies the following condition

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\tilde{h}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)=(1-2\mathcal{F})\left({\tilde{\nabla }}_{{\mathcal{V}}_1}h\right)({\mathcal{V}}_2,{\mathcal{V}}_3).\end{array}$$

In the same way, (2) and (3) follow. □

3. Statistical Submanifolds

Consider $\tilde{N}$ as an $(m+n)$-dimensional smooth manifold with a statistical structure $(\tilde{\nabla },\tilde{h})$ and N as an m-dimensional submanifold of $\tilde{N}$ with the induced metric h on it. Each tangent space of $\tilde{N}$ has the orthogonal decomposition

$$T_p\tilde{N}=T_{p}N\oplus T_p^{\perp }N,$$

where $T_p^{\perp }N:=\{{\zeta }_1\in T_p\tilde{N}|\tilde{h}({\zeta }_1,{\mathcal{V}}_1)=0,\forall {\mathcal{V}}_1\in T_{p}N\}$. The Gauss and Weingarten formulas for dual connections are described by ([39])

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2={\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2),{\tilde{\nabla }}_{{\mathcal{V}}_1}{\zeta }_1=-A_{{\zeta }_1}{\mathcal{V}}_1+D_{{\mathcal{V}}_1}{\zeta }_1,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2={\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2+{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2),{\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\zeta }_1=-A_{{\zeta }_1}^{*}{\mathcal{V}}_1+D_{{\mathcal{V}}_1}^{*}{\zeta }_1,\hfill \end{array}$$

(16)

for any ${\mathcal{V}}_1,{\mathcal{V}}_2\in \Gamma \left(TN\right)$ and ${\zeta }_1\in \Gamma \left(T^{\perp }N\right)$. It results that $(N,\nabla ,h)$ and $(N,\nabla ,h^{*})$ are statistical submanifolds, and ${\nabla }^{*}$ is the dual of ∇ with respect to h and the tensor fields $\mathcal{L}$, ${\mathcal{L}}^{*}$, A and $A^{*}$ satisfy

$$\begin{array}{cc}\hfill & \mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2)=\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_1),{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2)={\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_1),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & h(A_{{\zeta }_1}{\mathcal{V}}_1,{\mathcal{V}}_2)=\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2),{\zeta }_1),h(A_{{\zeta }_1}^{*}{\mathcal{V}}_1,{\mathcal{V}}_2)=\tilde{h}(\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2),{\zeta }_1).\hfill \end{array}$$

(18)

Furthermore, the Levi-Civita connections ${\tilde{\nabla }}^{\tilde{h}}$ and ${\nabla }^h$ on $\tilde{N}$ and N, respectively are related to the fundamental form $\widehat{\mathcal{L}}$ by

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\tilde{h}}{\mathcal{V}}_2={\nabla }_{{\mathcal{V}}_1}^h{\mathcal{V}}_2+\widehat{\mathcal{L}}({\mathcal{V}}_1,{\mathcal{V}}_2),{\tilde{\nabla }}_{{\mathcal{V}}_1}^{\tilde{h}}{\zeta }_1=-{\widehat{A}}_{{\zeta }_1}{\mathcal{V}}_1+{\widehat{D}}_{{\mathcal{V}}_1}{\zeta }_1.\end{array}$$

(19)

Lemma 1.

Let $\mathcal{F}\in C^{\infty }\left(N\right)$. The Gauss and Weingarten formulas for $\mathcal{F}$-statistical connections satisfy

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2=& (1-\mathcal{F})({\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2))+\mathcal{F}({\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2+{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2)),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\zeta }_1=& (1-\mathcal{F})(D_{{\mathcal{V}}_1}{\zeta }_1-A_{{\zeta }_1}{\mathcal{V}}_1)+\mathcal{F}(D_{{\mathcal{V}}_1}^{*}{\zeta }_1-A_{{\zeta }_1}^{*}{\mathcal{V}}_1),\hfill \end{array}$$

(21)

for any ${\mathcal{V}}_1,{\mathcal{V}}_2\in \Gamma \left(TN\right)$ and ${\zeta }_1\in \Gamma \left(T^{\perp }N\right)$.

Proof. 

Applying (15) in Definition 1, we see that

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2=& (1-\mathcal{F}){\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{F}{\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2\hfill \\ \hfill =& (1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2).\hfill \end{array}$$

(22)

Similarly, it follows (21). □

According to the above equations, we set

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2={\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2+{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2),{\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\zeta }_1=-A_{{\zeta }_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_1+D_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\zeta }_1,\end{array}$$

(23)

where

$$\begin{array}{cc}\hfill {\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2=& (1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2,{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2)=(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2),\hfill \\ \hfill D_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\zeta }_1=& (1-\mathcal{F})D_{{\mathcal{V}}_1}{\zeta }_1+\mathcal{F}{D}_{{\mathcal{V}}_1}^{*}{\zeta }_1,A_{{\zeta }_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_1=(1-\mathcal{F})A_{{\zeta }_1}{\mathcal{V}}_1+\mathcal{F}{A}_{{\zeta }_1}^{*}{\mathcal{V}}_1.\hfill \end{array}$$

In particular, setting $\mathcal{F}=0$, $\mathcal{F}=1$ and $\mathcal{F}={\displaystyle \frac{1}{2}}$ in (23) we have (15), (16) and (19), respectively.

Proposition 1.

For a statistical submanifold $(N,\nabla ,h)$ of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ with $\mathcal{F}\in C^{\infty }\left(N\right)$, we have

1. 

The triple $(N,h,{\nabla }^{\left(\mathcal{F}\right)})$ is a statistical submanifold in $(\tilde{N},{\tilde{\nabla }}^{\left(\mathcal{F}\right)},\tilde{h})$ induced by $(N,\nabla ,h)$.

2. 

The connection ${\nabla }^{(1-\mathcal{F})}$ is dual of ${\nabla }^{\left(\mathcal{F}\right)}$, i.e.,

$$\begin{array}{cc}\hfill {\mathcal{V}}_{1}h({\mathcal{V}}_2,{\mathcal{V}}_3)=& h({\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2,{\mathcal{V}}_3)+h({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}^{(1-\mathcal{F})}{\mathcal{V}}_3),\forall {\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right).\hfill \end{array}$$

3. 

The tensor fields ${\mathcal{L}}^{\left(\mathcal{F}\right)}$ and ${\mathcal{L}}^{(1-\mathcal{F})}$ are related by

$$\begin{array}{c}\hfill \widehat{\mathcal{L}}={\displaystyle \frac{{\mathcal{L}}^{\left(\mathcal{F}\right)}+{\mathcal{L}}^{(1-\mathcal{F})}}{2}}.\end{array}$$

4. 

For any ${\mathcal{V}}_1,{\mathcal{V}}_2\in \Gamma \left(TN\right)$ and ${\zeta }_1\in \Gamma \left(T^{\perp }N\right)$, we have

$$h(A_{{\zeta }_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_1,{\mathcal{V}}_2)=\tilde{h}({\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2),{\zeta }_1).$$

Proof. 

We prove only (1). The parts (2)–(4) are obtained in a similar way. Since $(\tilde{h},\tilde{\nabla })$ is a statistical structure on $\tilde{N}$ and $(\tilde{h},{\tilde{\nabla }}^{\left(\mathcal{F}\right)})$ is also a statistical structure, we can write

$$\begin{array}{cc}\hfill 0=& {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2-{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\mathcal{V}}_1-[{\mathcal{V}}_1,{\mathcal{V}}_2]\hfill \\ \hfill =& (1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2)\hfill \\ \hfill =& -(1-\mathcal{F}){\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_1-\mathcal{F}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_1-(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_1)-\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_1)-[{\mathcal{V}}_1,{\mathcal{V}}_2],\hfill \end{array}$$

which gives

$${\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2-{\nabla }_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\mathcal{V}}_1=[{\mathcal{V}}_1,{\mathcal{V}}_2],{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2)={\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_1).$$

On the other hand, for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$ we have

$$\begin{array}{cc}\hfill \left({\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}h\right)({\mathcal{V}}_2,{\mathcal{V}}_3)& ={\mathcal{V}}_{1}h({\mathcal{V}}_2,{\mathcal{V}}_3)-h({\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2,{\mathcal{V}}_3)-h({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3)\hfill \\ \hfill & ={\mathcal{V}}_1\tilde{h}({\mathcal{V}}_2,{\mathcal{V}}_3)-\tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2,{\mathcal{V}}_3)-\tilde{h}({\mathcal{V}}_2,{\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3)=\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\tilde{h}\right)({\mathcal{V}}_2,{\mathcal{V}}_3).\hfill \end{array}$$

Thus $\left({\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}h\right)({\mathcal{V}}_2,{\mathcal{V}}_3)=\left({\nabla }_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}h\right)({\mathcal{V}}_1,{\mathcal{V}}_3)$. Hence $(N,h,{\nabla }^{\left(\mathcal{F}\right)})$ forms a statistical submanifold of $\tilde{N}$. □

For a statistical submanifold $(N,\nabla ,h)$ of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$, for any tangent vector fields ${\mathcal{V}}_1,{\mathcal{V}}_2\in \Gamma \left(TN\right)$, we consider the difference tensor K on N as

$$\begin{array}{c}\hfill K_{{\mathcal{V}}_1}{\mathcal{V}}_2={\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2-{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2.\end{array}$$

(24)

From (7), (15) and the above equation, it follows that

$${\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2=K_{{\mathcal{V}}_1}{\mathcal{V}}_2+{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2)-\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2).$$

More precisely,

$${\left({\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2\right)}^{\top }=K_{{\mathcal{V}}_1}{\mathcal{V}}_2,{\left({\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2\right)}^{\perp }={\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2)-\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2).$$

Similarly, for ${\mathcal{V}}_1\in \Gamma \left(TN\right)$ and ${\zeta }_1\in \Gamma \left(T^{\perp }N\right)$ we have

$${\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1={\left({\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1\right)}^{\top }+{\left({\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1\right)}^{\perp },$$

where

$${\left({\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1\right)}^{\top }=A_{{\zeta }_1}{\mathcal{V}}_1-A_{{\zeta }_1}^{*}{\mathcal{V}}_1,{\left({\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1\right)}^{\perp }=D_{{\mathcal{V}}_1}^{*}{\zeta }_1-D_{{\mathcal{V}}_1}{\zeta }_1.$$

Moreover, (8) implies

$$\begin{array}{cc}\hfill & h({\left({\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1\right)}^{\top },{\mathcal{V}}_2)=\tilde{h}({\left({\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_2\right)}^{\perp },{\zeta }_1),\hfill \end{array}$$

which gives us

$$\begin{array}{cc}\hfill & h(A_{{\zeta }_1}{\mathcal{V}}_1-A_{{\zeta }_1}^{*}{\mathcal{V}}_1,{\mathcal{V}}_2)=h(A_{{\zeta }_1}{\mathcal{V}}_2-A_{{\zeta }_1}^{*}{\mathcal{V}}_2,{\mathcal{V}}_1)=\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2)-\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_2),{\zeta }_1).\hfill \end{array}$$

Regarding $\mathcal{F}\in C^{\infty }\left(N\right)$, (24) and the definitions of ${\nabla }^{\left(\mathcal{F}\right)}$ and ${\nabla }^{(1-\mathcal{F})}$ imply

$$\begin{array}{cc}\hfill {\nabla }^{\left(\mathcal{F}\right)}=& {\nabla }^h-{\displaystyle \frac{1-2\mathcal{F}}{2}}K,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\tilde{\nabla }}^{\left(\mathcal{F}\right)}=& {\nabla }^{\left(\mathcal{F}\right)}+\widehat{\mathcal{L}}-{\displaystyle \frac{1-2\mathcal{F}}{2}}({\mathcal{L}}^{*}-\mathcal{L}).\hfill \end{array}$$

(26)

Now, consider $\{e_1,\cdots ,e_m\}$ and $\{e_{m+1},\cdots ,e_{m+n}\}$ as orthonormal tangent and normal frames, respectively, on N. The $\mathcal{F}$-mean curvature vector field $H^{\left(\mathcal{F}\right)}$ with respect to ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ is defined by

$$\begin{array}{c}\hfill H^{\left(\mathcal{F}\right)}={\displaystyle \frac{1}{m}}\sum _{s=1}^m{\mathcal{L}}^{\left(\mathcal{F}\right)}(e_s,e_s)={\displaystyle \frac{1}{m}}\sum _{\gamma =1}^n\sum _{s=1}^m{\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{e}_{m+\gamma },\end{array}$$

(27)

where ${\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }=\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}(e_s,e_l),e_{m+\gamma })$. Moreover, the following identity holds

$$\begin{array}{c}\hfill H^{\left(\mathcal{F}\right)}=(1-\mathcal{F})H+\mathcal{F}{H}^{*},H^{(1-\mathcal{F})}=\mathcal{F}H+(1-\mathcal{F})H^{*}.\end{array}$$

(28)

For $\mathcal{F}=0,1$ and $\mathcal{F}={\displaystyle \frac{1}{2}}$, we use the notations $H,H^{*}$ and $\widehat{H}$, respectively. The above equation immediately leads to

$$\begin{array}{c}\hfill \widehat{H}={\displaystyle \frac{H^{\left(\mathcal{F}\right)}+H^{(1-\mathcal{F})}}{2}}.\end{array}$$

Moreover, setting

$$\left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2=\sum _{s,l=1}^m\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}(e_s,e_l),{\mathcal{L}}^{\left(\mathcal{F}\right)}(e_s,e_l)),\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2=\tilde{h}(H^{\left(\mathcal{F}\right)},H^{\left(\mathcal{F}\right)}),$$

and making use of the definitions ${\mathcal{L}}^{\left(\mathcal{F}\right)}$ and $H^{\left(\mathcal{F}\right)}$, we deduce

$$\begin{array}{c}\hfill \tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)},{\mathcal{L}}^{(1-\mathcal{F})})=2\left|\right|\widehat{\mathcal{L}}{\left|\right|}^2-{\displaystyle \frac{1}{2}}\left(\right||{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2),\end{array}$$

(29)

and

$$\begin{array}{c}\hfill \tilde{h}(H^{\left(\mathcal{F}\right)},H^{(1-\mathcal{F})})=2\left|\right|\widehat{H}{\left|\right|}^2-{\displaystyle \frac{1}{2}}\left(\right||H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2).\end{array}$$

(30)

In particular, for $\mathcal{F}=0$, we have

$$\begin{array}{c}\hfill \tilde{h}(\mathcal{L},{\mathcal{L}}^{*})=2\left|\right|\widehat{\mathcal{L}}{\left|\right|}^2-{\displaystyle \frac{1}{2}}{\left(\right|\left|\mathcal{L}\right||}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2),\end{array}$$

(31)

and

$$\begin{array}{c}\hfill \tilde{h}(H,H^{*})=2\left|\right|\widehat{H}{\left|\right|}^2-{\displaystyle \frac{1}{2}}{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2).\end{array}$$

(32)

Lemma 2.

The following relations hold

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2=& {(1-2\mathcal{F})}^2{\left(\right|\left|\mathcal{L}\right||}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2)+8\mathcal{F}(1-\mathcal{F})||\widehat{\mathcal{L}}{\left|\right|}^2,\hfill \end{array}$$

(33)

and

$$\begin{array}{cc}\hfill \left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2=& {(1-2\mathcal{F})}^2{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2)+8\mathcal{F}(1-\mathcal{F})||\widehat{H}{\left|\right|}^2.\hfill \end{array}$$

(34)

Proof. 

Using ${\mathcal{L}}^{\left(\mathcal{F}\right)}=(1-\mathcal{F})\mathcal{L}+\mathcal{F}{\mathcal{L}}^{*}$, we get

$$\left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2=\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)},{\mathcal{L}}^{\left(\mathcal{F}\right)})={(1-\mathcal{F})}^2{\left|\right|\mathcal{L}\left|\right|}^2+{\mathcal{F}}^2\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2+2\mathcal{F}(1-\mathcal{F})\tilde{h}(\mathcal{L},{\mathcal{L}}^{*}).$$

Similarly, it follows

$$\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2={\mathcal{F}}^2{\left|\right|\mathcal{L}\left|\right|}^2+{(1-\mathcal{F})}^2\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2+2\mathcal{F}(1-\mathcal{F})\tilde{h}(\mathcal{L},{\mathcal{L}}^{*}).$$

The above two equations imply

$$\left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2=(1+2{\mathcal{F}}^2-2\mathcal{F}){\left(\right|\left|\mathcal{L}\right||}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2)+4\mathcal{F}(1-\mathcal{F})\tilde{h}(\mathcal{L},{\mathcal{L}}^{*}).$$

Using (31) in the last equation, we obtain (33). In a similar way, (34) follows. □

Definition 2.

Considering $(N,\nabla ,h)$ as a statistical submanifold of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ with $\mathcal{F}\in C^{\infty }\left(N\right)$, M is called f-autoparallel with respect to ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ if ${\mathcal{L}}^{\left(\mathcal{F}\right)}=0$. In addition, if ${\mathcal{L}}^{\left(\mathcal{F}\right)}=0$ and ${\mathcal{L}}^{(1-\mathcal{F})}=0$, N is f-doubly autoparallel. The statistical submanifold N is called geodesic if it is ${\tilde{\nabla }}^h$-autoparallel. If $H^{\left(\mathcal{F}\right)}=0$ ($H^{(1-\mathcal{F})}=0$), N is called ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$-minimal (respectively, ${\tilde{\nabla }}^{(1-\mathcal{F})}$-minimal).

Proposition 2.

On a statistical submanifold $(N,\nabla ,h)$ of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$, we have

1. 

N is $\mathcal{F}$-doubly autoparallel if it is 0-doubly autoparallel.

2. 

If N is $\mathcal{F}$-doubly autoparallel, then $\mathcal{L}=-{\mathcal{L}}^{*}$.

3. 

${\mathcal{L}}^{\left(\mathcal{F}\right)}=-{\mathcal{L}}^{(1-\mathcal{F})}$ if N is geodesic.

4. 

If N is ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ and ${\tilde{\nabla }}^{(1-\mathcal{F})}$-minimal, then $H=-H^{*}$.

Proof. 

If N is 0-doubly autoparallel, we have $\mathcal{L}={\mathcal{L}}^{*}=0$ which gives ${\mathcal{L}}^{\left(\mathcal{F}\right)}={\mathcal{L}}^{(1-\mathcal{F})}=0$. Hence (1) follows. To prove (2), using $0={\mathcal{L}}^{\left(\mathcal{F}\right)}=(1-\mathcal{F})\mathcal{L}+\mathcal{F}{\mathcal{L}}^{*}$ and $0={\mathcal{L}}^{(1-\mathcal{F})}=\mathcal{F}\mathcal{L}+(1-\mathcal{F}){\mathcal{L}}^{*}$, we get $\mathcal{L}=-{\mathcal{L}}^{*}$. Considering the part (3) of Proposition 1 and $\widehat{\mathcal{L}}=0$, we get (3). Similar to proof (3) and using (28), (4) is obtained. □

Example 1.

The normal distribution manifold is described as

$$\tilde{N}=\{p(v,\mu ,\sigma )|p(v,\mu ,\sigma )={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}exp\{-{\displaystyle \frac{{(v-\mu )}^2}{2{\sigma }^2}}\},\mu \in \mathbb{R},\sigma >0\}.$$

Thus $\tilde{N}$ can be considered as a 2-dimensional manifold with a coordinate system $({\vartheta }^1,{\vartheta }^2)=(\mu ,\sigma )$. According to (1) and (2), the Fisher metric $\tilde{h}$ and the connection $\tilde{\nabla }$ are obtained by

$$\begin{array}{c}\hfill \tilde{h}({\displaystyle \frac{\partial }{\partial \mu }},{\displaystyle \frac{\partial }{\partial \mu }})={\displaystyle \frac{1}{{\sigma }^2}},\tilde{h}({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\displaystyle \frac{2}{{\sigma }^2}},\tilde{h}({\displaystyle \frac{\partial }{\partial \mu }},{\displaystyle \frac{\partial }{\partial \sigma }})=\tilde{h}({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \mu }})=0,\end{array}$$

and

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}{\displaystyle \frac{\partial }{\partial \mu }}=0,{\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}{\displaystyle \frac{\partial }{\partial \sigma }}={\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}{\displaystyle \frac{\partial }{\partial \mu }}=-{\displaystyle \frac{2}{\sigma }}{\displaystyle \frac{\partial }{\partial \mu }},{\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}{\displaystyle \frac{\partial }{\partial \sigma }}=-{\displaystyle \frac{3}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }}.\end{array}$$

From (5), it follows that

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}^{*}{\displaystyle \frac{\partial }{\partial \mu }}={\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{*}{\displaystyle \frac{\partial }{\partial \sigma }}={\displaystyle \frac{1}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }},{\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}^{*}{\displaystyle \frac{\partial }{\partial \sigma }}={\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{*}{\displaystyle \frac{\partial }{\partial \mu }}=0.\end{array}$$

Hence (7) yields

$$\tilde{K}({\displaystyle \frac{\partial }{\partial \mu }},{\displaystyle \frac{\partial }{\partial \mu }})={\displaystyle \frac{1}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }},\tilde{K}({\displaystyle \frac{\partial }{\partial \mu }},{\displaystyle \frac{\partial }{\partial \sigma }})=\tilde{K}({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \mu }})={\displaystyle \frac{2}{\sigma }}{\displaystyle \frac{\partial }{\partial \mu }},\tilde{K}({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\displaystyle \frac{4}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }}.$$

It follows that

$$\begin{array}{cc}\hfill & ({\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}\tilde{h})({\displaystyle \frac{\partial }{\partial \mu }},{\displaystyle \frac{\partial }{\partial \mu }})=0,({\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}\tilde{h})({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \mu }})=({\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}\tilde{h})({\displaystyle \frac{\partial }{\partial \mu }},{\displaystyle \frac{\partial }{\partial \mu }})={\displaystyle \frac{2}{{\sigma }^3}},\hfill \\ \hfill & ({\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}\tilde{h})({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\displaystyle \frac{8}{{\sigma }^3}},\hfill \end{array}$$

which give $(\tilde{N},\tilde{\nabla },\tilde{h})$ is a statistical manifold. For the submanifold

$$N=\left\{p(v,\mu ,\sigma )\right|p(v,\mu ,\sigma )\in \tilde{N},\mu =0,\sigma >0\},$$

we get the Fisher metric of N as

$$h({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\displaystyle \frac{2}{{\sigma }^2}}.$$

From (2), we have

$${\nabla }_{{\displaystyle \frac{\partial }{\partial \sigma }}}{\displaystyle \frac{\partial }{\partial \sigma }}=-{\displaystyle \frac{3}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }}.$$

Thus $(N,\nabla ,h)$ is a statistical submanifold because

$$({\nabla }_{{\displaystyle \frac{\partial }{\partial \sigma }}}h)({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\displaystyle \frac{8}{{\sigma }^3}}.$$

Suppose that $\mathcal{F}=\mathcal{F}\left(\sigma \right)$ is a function on N. We get ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ and ${\nabla }^{\left(\mathcal{F}\right)}$ on $\tilde{N}$ and N, respectively as

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}^{\left(\mathcal{F}\right)}{\displaystyle \frac{\partial }{\partial \sigma }}={\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{\left(\mathcal{F}\right)}{\displaystyle \frac{\partial }{\partial \mu }}=-{\displaystyle \frac{2}{\sigma }}(1-\mathcal{F}\left(\sigma \right)){\displaystyle \frac{\partial }{\partial \mu }},\hfill \\ & {\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \mu }}}^{\left(\mathcal{F}\right)}{\displaystyle \frac{\partial }{\partial \mu }}={\displaystyle \frac{\mathcal{F}\left(\sigma \right)}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }},{\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{\left(\mathcal{F}\right)}{\displaystyle \frac{\partial }{\partial \sigma }}={\displaystyle \frac{-3+4\mathcal{F}\left(\sigma \right)}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }},\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\nabla }_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{\left(\mathcal{F}\right)}{\displaystyle \frac{\partial }{\partial \sigma }}={\displaystyle \frac{-3+4\mathcal{F}\left(\sigma \right)}{\sigma }}{\displaystyle \frac{\partial }{\partial \sigma }}.\end{array}$$

The above equations imply

$$({\nabla }_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{(\mathcal{F})}h)({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})=-{\displaystyle \frac{8(2\mathcal{F}(\sigma )-1)}{\sigma 3}},$$

hence $(N,{\nabla }^{\left(\mathcal{F}\right)},h)$ is a statistical submanifold. It is evident from the above results that

$$\begin{array}{cc}\hfill & \mathcal{L}({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}{\displaystyle \frac{\partial }{\partial \sigma }}-{\nabla }_{{\displaystyle \frac{\partial }{\partial \sigma }}}{\displaystyle \frac{\partial }{\partial \sigma }}=0,\hfill \\ \hfill & {\mathcal{L}}^{*}({\displaystyle \frac{\partial }{\partial \sigma }},{\displaystyle \frac{\partial }{\partial \sigma }})={\tilde{\nabla }}_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{*}{\displaystyle \frac{\partial }{\partial \sigma }}-{\nabla }_{{\displaystyle \frac{\partial }{\partial \sigma }}}^{*}{\displaystyle \frac{\partial }{\partial \sigma }}=0,\hfill \end{array}$$

which give N is 0-doubly autoparallel and hence $\mathcal{F}$-doubly autoparallel.

Lemma 3.

The following relations hold:

$$\begin{array}{c}\hfill h({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)=-\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_4)),\end{array}$$

(35)

and

$$\begin{array}{c}\hfill h({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)=h({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_4),{\mathcal{V}}_3),\end{array}$$

(36)

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4\in \Gamma \left(TN\right)$.

Proof. 

From (1), we have

$$\begin{array}{c}\hfill \tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)={\mathcal{V}}_1\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)-\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\tilde{\nabla }}_{{\mathcal{V}}_1}^{(1-\mathcal{F})}{\mathcal{V}}_4).\end{array}$$

Since $\tilde{g}({\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)=0$ and using (23) in the above equation, we can write

$$\begin{array}{cc}\hfill \tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)& =-\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\nabla }_{{\mathcal{V}}_1}^{(1-\mathcal{F})}{\mathcal{V}}_4+{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_4))\hfill \\ \hfill & =-\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_4)).\hfill \end{array}$$

Thus (35) holds. To prove (36), considering $1-\mathcal{F}$ instead of $\mathcal{F}$ in (35), it follows that

$$\begin{array}{cc}\hfill \tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_4),{\mathcal{V}}_3)& =-\tilde{h}({\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_4),{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3))\hfill \\ \hfill & =\tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4).\hfill \end{array}$$

□

Consider $\tilde{\nabla }$ as an affine connection of a pseudo-Riemannian manifold $(\tilde{N},\tilde{h})$, the divergence of ${\mathcal{V}}_1\in \Gamma \left(T\tilde{N}\right)$ is introduced as the trace of the covariant derivative $\tilde{\nabla }{\mathcal{V}}_1$, i.e.,

$$di{v}^{\tilde{\nabla }}{\mathcal{V}}_1=tr\{{\mathcal{V}}_2\to {\tilde{\nabla }}_{{\mathcal{V}}_2}{\mathcal{V}}_1\}.$$

In general for a tensor field A of type $(1,n)$ on $\tilde{N}$, $di{v}^{\tilde{\nabla }}A$ is obtained by

$$di{v}^{\tilde{\nabla }}A=tr\{V_0\to \left({\tilde{\nabla }}_{V_0}A\right)({\mathcal{V}}_1,\dots ,V_n)\},\forall V_0,{\mathcal{V}}_1,\dots ,V_n\in \Gamma \left(T\tilde{N}\right).$$

Now, suppose that $(\tilde{N},\tilde{\nabla },\tilde{h})$ is a statistical manifold and $\mathcal{F}\in C^{\infty }\left(\tilde{N}\right)$. (7), (14) and the above equation provide the explicit formula for $di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}$ of ${\mathcal{V}}_1={{\mathcal{V}}_1}^i{\partial }_i\in \Gamma \left(T\tilde{N}\right)$:

$$\begin{array}{c}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{V}}_1=di{v}^{{\tilde{\nabla }}^{\tilde{h}}}{\mathcal{V}}_1-{\displaystyle \frac{1-2\mathcal{F}}{2}}(di{v}^{{\tilde{\nabla }}^{*}}{\mathcal{V}}_1-di{v}^{\tilde{\nabla }}{\mathcal{V}}_1).\end{array}$$

(37)

Proposition 3.

On an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$, where $(N,\nabla ,h)$ is an m-dimensional statistical submanifold of $\tilde{N}$ and $\mathcal{F}\in C^{\infty }\left(N\right)$, we have

$$\begin{array}{cc}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{L}}^{\left(\mathcal{F}\right)}=& (1-\mathcal{F})di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}\mathcal{L}+\mathcal{F}di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{L}}^{*},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}\mathcal{L}=& (1-\mathcal{F})di{v}^{\tilde{\nabla }}\mathcal{L}+\mathcal{F}di{v}^{{\tilde{\nabla }}^{*}}\mathcal{L},\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{L}}^{\left(\mathcal{F}\right)}-di{v}^{{\tilde{\nabla }}^{(1-\mathcal{F})}}& {\mathcal{L}}^{(1-\mathcal{F})}=\left(1-2\mathcal{F}\right)\left(di{v}^{\tilde{\nabla }}\mathcal{L}-di{v}^{{\tilde{\nabla }}^{*}}{\mathcal{L}}^{*}\right).\hfill \end{array}$$

(40)

Proof. 

Using the definition of ${\mathcal{L}}^{\left(\mathcal{F}\right)}$, we can write

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)={\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\left((1-\mathcal{F})\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)\right)\hfill \\ \hfill =& {\mathcal{V}}_1\left(\mathcal{F}\right)({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3))+(1-\mathcal{F}){\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}{\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3).\hfill \end{array}$$

Thus from the above equation, it follows that

$$\begin{array}{c}\hfill ({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)=(1-\mathcal{F})h({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4)+\mathcal{F}h({\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{V}}_4),\end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4\in \Gamma \left(TN\right)$. Considering ${\mathcal{V}}_1={\mathcal{V}}_4=e_s$ and summing over s in the last equation, where $\{e_1,\cdots ,e_m\}$ is an orthonormal basis on N, we obtain (38). Using (37), we get

$$\begin{array}{cc}\hfill & di{v}^{\nabla }\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)=di{v}^{\widehat{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-{\displaystyle \frac{1}{2}}(di{v}^{{\nabla }^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-di{v}^{\nabla }\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)),\hfill \\ & di{v}^{{\nabla }^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)=di{v}^{\widehat{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{2}}(di{v}^{{\nabla }^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-di{v}^{\nabla }\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)).\hfill \end{array}$$

Then the above equations imply

$$\begin{array}{cc}\hfill & (1-\mathcal{F})di{v}^{\nabla }\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}di{v}^{{\nabla }^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill =& di{v}^{\widehat{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-{\displaystyle \frac{1-2\mathcal{F}}{2}}(di{v}^{{\nabla }^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-di{v}^{\nabla }\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3))\hfill \\ \hfill =& di{v}^{{\nabla }^{\left(\mathcal{F}\right)}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3).\hfill \end{array}$$

Thus (39) holds. Applying the above equation in (38), we have

$$\begin{array}{cc}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)=& (1-\mathcal{F})((1-\mathcal{F})di{v}^{\tilde{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}di{v}^{{\tilde{\nabla }}^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3))\hfill \\ \hfill & +\mathcal{F}((1-\mathcal{F})di{v}^{\tilde{\nabla }}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}di{v}^{{\tilde{\nabla }}^{*}}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)).\hfill \end{array}$$

Similarly

$$\begin{array}{cc}\hfill di{v}^{{\tilde{\nabla }}^{(1-\mathcal{F})}}{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_2,{\mathcal{V}}_3)=& \mathcal{F}(\mathcal{F}di{v}^{\tilde{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+(1-\mathcal{F})di{v}^{{\tilde{\nabla }}^{*}}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3))\hfill \\ \hfill & +(1-\mathcal{F})(\mathcal{F}di{v}^{\tilde{\nabla }}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)+(1-\mathcal{F})di{v}^{{\tilde{\nabla }}^{*}}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)).\hfill \end{array}$$

Subtracting the above two equations, we have (40). □

Lemma 4.

Let N be an m-dimensional submanifold with a statistical structure $(\nabla ,h)$ of an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$. Then

(i)

$h({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{H}^{(1-\mathcal{F})}-{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{H}^{\left(\mathcal{F}\right)},{\mathcal{V}}_3)=\left(1-2\mathcal{F}\right)h({\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}{H}^{*}-{\tilde{\nabla }}_{{\mathcal{V}}_2}H,{\mathcal{V}}_3),\forall {\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right),$

(ii)

$di{v}^{{\tilde{\nabla }}^{(1-\mathcal{F})}}{H}^{(1-\mathcal{F})}=di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{H}^{\left(\mathcal{F}\right)},$

(iii)

$di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{H}^{\left(\mathcal{F}\right)}=-4m\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2+{(1-2\mathcal{F})}^{2}di{v}^{\tilde{\nabla }}H.$

Proof. 

Assume that $\{e_1,\cdots ,e_m\}$ is an orthonormal basis on N. Setting ${\mathcal{V}}_1={\mathcal{V}}_4=e_s$ in (36) and summing over s, we have

$$\begin{array}{c}\hfill \sum _{s=1}^{m}h({\tilde{\nabla }}_{e_s}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3),e_s)=\sum _{s=1}^{m}h({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{\mathcal{L}}^{(1-\mathcal{F})}(e_s,e_s),{\mathcal{V}}_3),\end{array}$$

which gives us

$$\begin{array}{c}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)=mh({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{H}^{(1-\mathcal{F})},{\mathcal{V}}_3).\end{array}$$

(41)

The above equation leads to

$$\begin{array}{c}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{\mathcal{L}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)-di{v}^{{\tilde{\nabla }}^{(1-\mathcal{F})}}{\mathcal{L}}^{(1-\mathcal{F})}({\mathcal{V}}_2,{\mathcal{V}}_3)=mh({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{H}^{(1-\mathcal{F})}-{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{H}^{\left(\mathcal{F}\right)},{\mathcal{V}}_3).\end{array}$$

(40) and the last equation imply

$$\begin{array}{c}\hfill \tilde{h}({\tilde{\nabla }}_{{\mathcal{V}}_2}^{(1-\mathcal{F})}{H}^{(1-\mathcal{F})}-{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{H}^{\left(\mathcal{F}\right)},{\mathcal{V}}_3)={\displaystyle \frac{\left(1-2\mathcal{F}\right)}{m}}\left(di{v}^{\tilde{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-di{v}^{{\tilde{\nabla }}^{*}}{h}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)\right).\end{array}$$

(42)

Setting $\mathcal{F}=0$ and $\mathcal{F}=1$ in (41), it follows

$$\begin{array}{c}\hfill di{v}^{\tilde{\nabla }}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)=mh({\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}{H}^{*},{\mathcal{V}}_3),di{v}^{{\tilde{\nabla }}^{*}}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)=mh({\tilde{\nabla }}_{{\mathcal{V}}_2}H,{\mathcal{V}}_3).\end{array}$$

Applying the last equations in (42), we see that (i) holds. The part (i) gives

$$di{v}^{{\tilde{\nabla }}^{(1-\mathcal{F})}}{H}^{(1-\mathcal{F})}-di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{H}^{\left(\mathcal{F}\right)}=(1-2\mathcal{F})(di{v}^{{\tilde{\nabla }}^{*}}{H}^{*}-di{v}^{\tilde{\nabla }}H).$$

On the other hand, $di{v}^{{\tilde{\nabla }}^{*}}{H}^{*}=di{v}^{\tilde{\nabla }}H$ (see [15]), hence (ii) is obtained. Using (27) and (35), we can write

$$\begin{array}{cc}\hfill \tilde{h}(H^{\left(\mathcal{F}\right)},H^{(1-\mathcal{F})})& ={\displaystyle \frac{1}{m^2}}\tilde{h}(\sum _{s=1}^m{\mathcal{L}}^{\left(\mathcal{F}\right)}(e_s,e_s),\sum _{l=1}^m{\mathcal{L}}^{(1-\mathcal{F})}(e_l,e_l))\hfill \\ \hfill & =-{\displaystyle \frac{1}{m^2}}\sum _{s,l=1}^{m}h({\tilde{\nabla }}_{e_l}^{\left(\mathcal{F}\right)}{\mathcal{L}}^{\left(\mathcal{F}\right)}(e_s,e_s),e_l)\hfill \\ \hfill & =-{\displaystyle \frac{1}{m}}\sum _{l=1}^{m}h({\tilde{\nabla }}_{e_l}^{\left(\mathcal{F}\right)}{H}^{\left(\mathcal{F}\right)},e_l)=-{\displaystyle \frac{1}{m}}di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{H}^{\left(\mathcal{F}\right)}.\hfill \end{array}$$

In particular, for $\mathcal{F}=0$, it follows

$$\begin{array}{c}\hfill -m\tilde{h}(H,H^{*})=di{v}^{\tilde{\nabla }}H.\end{array}$$

(43)

On the other hand, using (28) and (30), we get

$$\begin{array}{cc}\hfill \tilde{h}(H^{\left(\mathcal{F}\right)},H^{(1-\mathcal{F})})& =4\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2+{(1-2\mathcal{F})}^2\tilde{h}(H,H^{*}).\hfill \end{array}$$

The last three equations give (iii). □

Proposition 4.

Let $(N,\nabla ,h)$ be a statistical submanifold in a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$. Then we have

1. 

If N is ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$- (or ${\tilde{\nabla }}^{(1-\mathcal{F})}$-) minimal, then

$$\begin{array}{c}\hfill {\displaystyle \frac{{(1-2\mathcal{F})}^2}{m}}di{v}^{\tilde{\nabla }}H=4\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2.\end{array}$$

2. 

If N is $\tilde{\nabla }$- (or ${\tilde{\nabla }}^{*}$-) minimal, then

$$\begin{array}{c}\hfill di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{H}^{\left(\mathcal{F}\right)}=-4m\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2.\end{array}$$

Proof. 

According to Definition 2 and the part (iii) of Lemma 4, we have (1) and (2). □

Applying (30) and Proposition 4, we conclude:

Corollary 2.

The following conditions are equivalent:

1. 

$di{v}^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}{H}^{\left(\mathcal{F}\right)}=0$;

2. 

$di{v}^{{\tilde{\nabla }}^{(1-\mathcal{F})}}{H}^{\left(\mathcal{F}\right)}=0$;

3. 

$H^{\left(\mathcal{F}\right)}\perp H^{(1-\mathcal{F})}$;

4. 

$4\left|\right|\widehat{H}{\left|\right|}^2=\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2$;

5. 

${(1-2\mathcal{F})}^{2}di{v}^{\tilde{\nabla }}H=4m\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2$;

6. 

$4\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2=-{(1-2\mathcal{F})}^2\tilde{h}(H,H^{*})$.

4. The Gauss, Codazzi and Ricci Equations with Respect to the $\mathcal{F}$-Statistical Connection

In this section, suppose that $(N,\nabla ,h)$ is a statistical submanifold in a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$. We consider the statistical structures $({\nabla }^{\left(\mathcal{F}\right)},h)$ and $({\tilde{\nabla }}^{\left(\mathcal{F}\right)},\tilde{h})$, respectively on N and $\tilde{N}$. For simplicity, we denote by ${\tilde{R}}^{\left(\mathcal{F}\right)}:=R^{{\tilde{\nabla }}^{\left(\mathcal{F}\right)}}$, $R^{\left(\mathcal{F}\right)}:=R^{{\nabla }^{\left(\mathcal{F}\right)}}$ and $R^{\left(\mathcal{F}\right)\perp }:=R^{D^{\left(\mathcal{F}\right)}}$, the $\mathcal{F}$-curvature tensor fields of the connections ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$, ${\nabla }^{\left(\mathcal{F}\right)}$ and $D^{\left(\mathcal{F}\right)}$, respectively. Moreover, if ${\tilde{R}}^{\left(\mathcal{F}\right)}=0$ ($R^{\left(\mathcal{F}\right)}=0$), $\tilde{N}$ (N) is called $\mathcal{F}$-flat.

Proposition 5.

On a statistical submanifold $(N,\nabla ,h)$ in a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$, the curvature tensor $R^{\left(\mathcal{F}\right)}$ satisfies the following

$$\begin{array}{cc}\hfill R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3=& (1-\mathcal{F})R({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+\mathcal{F}{R}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+\mathcal{F}(1-\mathcal{F})[K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_1}]{\mathcal{V}}_3\hfill \\ \hfill & +{\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3\hfill \\ \hfill =& R^h({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+{\displaystyle \frac{1-2\mathcal{F}}{2}}(\left({\nabla }_{{\mathcal{V}}_2}^{h}K\right)({\mathcal{V}}_1,{\mathcal{V}}_3)-\left({\nabla }_{{\mathcal{V}}_1}^{h}K\right)({\mathcal{V}}_2,{\mathcal{V}}_3))\hfill \\ \hfill & +{\left({\displaystyle \frac{1-2\mathcal{F}}{2}}\right)}^2[K_{{\mathcal{V}}_1},K_{{\mathcal{V}}_2}]{\mathcal{V}}_3+{\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3,\hfill \end{array}$$

(44)

where $R:=R^{\left(0\right)}$ and $R^{*}:=R^{\left(1\right)}$, for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$.

Proof. 

Since ${\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_2=(1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2+\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2$, for any ${\mathcal{V}}_1,{\mathcal{V}}_2\in \Gamma \left(TN\right)$, we obtain

$$\begin{array}{cc}\hfill R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3=& {\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\nabla }_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3\hfill \\ \hfill =& {\nabla }_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\left((1-\mathcal{F}){\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3+\mathcal{F}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3\right)\hfill \\ \hfill & -{\nabla }_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}\left((1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3+\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3\right)-(1-\mathcal{F}){\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}{\mathcal{V}}_3-\mathcal{F}{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{*}{\mathcal{V}}_3.\hfill \end{array}$$

The above equation and (24) imply

$$\begin{array}{cc}\hfill R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3=& {(1-\mathcal{F})}^{2}R({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+{\mathcal{F}}^2{R}^{*}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+{\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3\hfill \\ \hfill & -{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3+\mathcal{F}(1-\mathcal{F})({\nabla }_{{\mathcal{V}}_1}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3\hfill \\ \hfill & +{\nabla }_{{\mathcal{V}}_1}^{*}{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}^{*}{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{*}{\mathcal{V}}_3).\hfill \end{array}$$

On the other hand, we get

$$\begin{array}{cc}\hfill & {\nabla }_{{\mathcal{V}}_1}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3+{\nabla }_{{\mathcal{V}}_1}^{*}{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}^{*}{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{*}{\mathcal{V}}_3\hfill \\ \hfill & =R({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+R^{*}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+[K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_1}]{\mathcal{V}}_3.\hfill \end{array}$$

From the above two equations, the first formula follows. Similarly, the second formula follows from (25). □

In the sequel, we review the following proposition:

Proposition 6

([40]). Let $(N,\nabla ,h)$ be a statistical submanifold of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$. The equations of Gauss, Codazzi, and Ricci are given by

$$\begin{array}{cc}\hfill {\left\{\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }=& R({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1,\hfill \\ \hfill {\left\{\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\perp }=& \left({\tilde{\nabla }}_{{\mathcal{V}}_1}\mathcal{L}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)-\left({\tilde{\nabla }}_{{\mathcal{V}}_2}\mathcal{L}\right)({\mathcal{V}}_1,{\mathcal{V}}_3),\hfill \\ \hfill {\left\{\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1\right\}}^{\top }=& {\left({\tilde{\nabla }}_{{\mathcal{V}}_2}A\right)}_{{\zeta }_1}{\mathcal{V}}_1-{\left({\tilde{\nabla }}_{{\mathcal{V}}_1}A\right)}_{{\zeta }_1}{\mathcal{V}}_2,\hfill \\ \hfill {\left\{\tilde{R}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1\right\}}^{\perp }=& R^{\perp }({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1+\mathcal{L}({\mathcal{V}}_2,A_{{\zeta }_1}{\mathcal{V}}_1)-\mathcal{L}({\mathcal{V}}_1,A_{{\zeta }_1}{\mathcal{V}}_2),\hfill \end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$ and ${\zeta }_1\in \Gamma \left(T^{\perp }N\right)$, where

$$\begin{array}{cc}\hfill \left({\tilde{\nabla }}_{{\mathcal{V}}_1}\mathcal{L}\right)({\mathcal{V}}_2,{\mathcal{V}}_3):=& D_{{\mathcal{V}}_1}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{L}({\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{L}({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3),\hfill \\ \hfill {\left({\tilde{\nabla }}_{{\mathcal{V}}_1}A\right)}_{{\zeta }_1}{\mathcal{V}}_2:=& {\nabla }_{{\mathcal{V}}_1}\left(A_{{\zeta }_1}{\mathcal{V}}_2\right)-A_{D_{{\mathcal{V}}_1}{\zeta }_1}{\mathcal{V}}_2-A_{{\zeta }_1}{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_2.\hfill \end{array}$$

Similarly, one can see equations of Gauss, Codazzi, and Ricci for the curvature tensors ${\tilde{R}}^{*}$ and ${\tilde{R}}^{\tilde{h}}$ with respect to the connections ${\tilde{\nabla }}^{*}$ and ${\tilde{\nabla }}^{\tilde{h}}$, respectively.

Now we get the $\mathcal{F}$-curvature tensor with respect to the $\mathcal{F}$-statistical connection ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$, which is the counterpart of the above proposition in $\mathcal{F}$-statistical submanifolds.

Theorem 1.

For a statistical submanifold $(N,\nabla ,h)$ of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ with $\mathcal{F}\in C^{\infty }\left(N\right)$, the equations of Gauss, Codazzi, and Ricci for the $\mathcal{F}$-curvature tensor ${\tilde{R}}^{\left(\mathcal{F}\right)}$ with respect to ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ satisfy the following

$$\begin{array}{cc}\hfill {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }& \\ \hfill =R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3& +\mathcal{F}(1-\mathcal{F})\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1\right)\hfill \\ \hfill +{(1-\mathcal{F})}^2& \left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1\right)+{\mathcal{F}}^2\left(A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2-A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1\right),\hfill \\ \\ \hfill {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\perp }& \\ \hfill ={\mathcal{V}}_1\left(\mathcal{F}\right){\left({\tilde{K}}_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)}^{\perp }& -{\mathcal{V}}_2\left(\mathcal{F}\right){\left({\tilde{K}}_{{\mathcal{V}}_1}{\mathcal{V}}_3\right)}^{\perp }+{(1-\mathcal{F})}^2\left(\left({\tilde{\nabla }}_{{\mathcal{V}}_1}\mathcal{L}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)-\left({\tilde{\nabla }}_{{\mathcal{V}}_2}\mathcal{L}\right)({\mathcal{V}}_1,{\mathcal{V}}_3)\right)\hfill \\ \hfill +\mathcal{F}(1-\mathcal{F})(\left({\tilde{\nabla }}_{{\mathcal{V}}_1}{\mathcal{L}}^{*}\right)& ({\mathcal{V}}_2,{\mathcal{V}}_3)-\left({\tilde{\nabla }}_{{\mathcal{V}}_2}{\mathcal{L}}^{*}\right)({\mathcal{V}}_1,{\mathcal{V}}_3)+\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}\mathcal{L}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)-\left({\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}\mathcal{L}\right)({\mathcal{V}}_1,{\mathcal{V}}_3))\hfill \\ \hfill & +{\mathcal{F}}^2\left(\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{\mathcal{L}}^{*}\right)({\mathcal{V}}_2,{\mathcal{V}}_3)-\left({\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}{\mathcal{L}}^{*}\right)({\mathcal{V}}_1,{\mathcal{V}}_3)\right),\hfill \\ \\ \hfill {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1\right\}}^{\top }& ={\mathcal{V}}_1\left(\mathcal{F}\right){\left({\tilde{K}}_{{\mathcal{V}}_2}{\zeta }_1\right)}^{\top }-{\mathcal{V}}_2\left(\mathcal{F}\right){\left({\tilde{K}}_{{\mathcal{V}}_1}{\zeta }_1\right)}^{\top }+{(1-\mathcal{F})}^2\left({\left({\tilde{\nabla }}_{{\mathcal{V}}_2}A\right)}_{{\zeta }_1}{\mathcal{V}}_1-{\left({\tilde{\nabla }}_{{\mathcal{V}}_1}A\right)}_{{\zeta }_1}{\mathcal{V}}_2\right)\hfill \\ \hfill +& \mathcal{F}(1-\mathcal{F})\left({\left({\tilde{\nabla }}_{{\mathcal{V}}_2}{A}^{*}\right)}_{{\zeta }_1}{\mathcal{V}}_1-{\left({\tilde{\nabla }}_{{\mathcal{V}}_1}{A}^{*}\right)}_{{\zeta }_1}{\mathcal{V}}_2+{\left({\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}A\right)}_{{\zeta }_1}{\mathcal{V}}_1-{\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}A\right)}_{{\zeta }_1}{\mathcal{V}}_2\right)\hfill \\ \hfill & +{\mathcal{F}}^2\left({\left({\tilde{\nabla }}_{{\mathcal{V}}_2}^{*}{A}^{*}\right)}_{{\zeta }_1}{\mathcal{V}}_1-{\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}{A}^{*}\right)}_{{\zeta }_1}{\mathcal{V}}_2\right),\hfill \\ \\ \hfill {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1\right\}}^{\perp }=& R^{\perp \left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1+\mathcal{F}(1-\mathcal{F})\left({\mathcal{L}}^{*}({\mathcal{V}}_2,A_{{\zeta }_1}{\mathcal{V}}_1)-{\mathcal{L}}^{*}({\mathcal{V}}_1,A_{{\zeta }_1}{\mathcal{V}}_2)+\mathcal{L}({\mathcal{V}}_2,A_{{\zeta }_1}^{*}{\mathcal{V}}_1)-\mathcal{L}({\mathcal{V}}_1,A_{{\zeta }_1}^{*}{\mathcal{V}}_2)\right)\hfill \\ \hfill & +{(1-\mathcal{F})}^2\left(\mathcal{L}({\mathcal{V}}_2,A_{{\zeta }_1}{\mathcal{V}}_1)-\mathcal{L}({\mathcal{V}}_1,A_{{\zeta }_1}{\mathcal{V}}_2)\right)+{\mathcal{F}}^2\left({\mathcal{L}}^{*}({\mathcal{V}}_2,A_{{\zeta }_1}^{*}{\mathcal{V}}_1)-{\mathcal{L}}^{*}({\mathcal{V}}_1,A_{{\zeta }_1}^{*}{\mathcal{V}}_2)\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}\mathcal{L}\right)({\mathcal{V}}_2,{\mathcal{V}}_3):=& D_{{\mathcal{V}}_1}^{*}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{L}({\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{L}({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3),\hfill \\ \hfill {\left({\tilde{\nabla }}_{{\mathcal{V}}_1}^{*}A\right)}_{{\zeta }_1}{\mathcal{V}}_2:=& {\nabla }_{{\mathcal{V}}_1}^{*}\left(A_{{\zeta }_1}{\mathcal{V}}_2\right)-A_{D_{{\mathcal{V}}_1}^{*}{\zeta }_1}{\mathcal{V}}_2-A_{{\zeta }_1}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_2,\hfill \end{array}$$

in the same setting we have $\tilde{\nabla }{\mathcal{L}}^{*},{\tilde{\nabla }}^{*}{\mathcal{L}}^{*},\tilde{\nabla }{A}^{*}$ and ${\tilde{\nabla }}^{*}{A}^{*}$ for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$ and ${\zeta }_1\in \Gamma \left(T^{\perp }N\right)$.

Proof. 

Using (20), we can write

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3=& {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}\left((1-\mathcal{F}){\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3+\mathcal{F}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)\right).\hfill \end{array}$$

Applying (20) and (21) in the above equation, it follows

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3\hfill \\ \hfill =& {\mathcal{V}}_1\left(\mathcal{F}\right)({\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3+{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3))\hfill \\ \hfill & +(1-\mathcal{F})\left(\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3+(1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3)+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_1,{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3)\right)\hfill \\ \hfill & +\mathcal{F}\left(\mathcal{F}{\nabla }_{{\mathcal{V}}_1}^{*}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3+(1-\mathcal{F}){\nabla }_{{\mathcal{V}}_1}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3)+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_1,{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3)\right)\hfill \\ \hfill & +(1-\mathcal{F})\left(-\mathcal{F}{A}_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1-(1-\mathcal{F})A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1+\mathcal{F}{D}_{{\mathcal{V}}_1}^{*}h({\mathcal{V}}_2,{\mathcal{V}}_3)+(1-\mathcal{F})D_{{\mathcal{V}}_1}\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)\right)\hfill \\ \hfill & +\mathcal{F}\left(-\mathcal{F}{A}_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1-(1-\mathcal{F})A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1+\mathcal{F}{D}_{{\mathcal{V}}_1}^{*}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)+(1-\mathcal{F})D_{{\mathcal{V}}_1}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)\right).\hfill \end{array}$$

By interchanging ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$ in the last equation, we have

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{{\mathcal{V}}_2}^{\left(\mathcal{F}\right)}{\tilde{\nabla }}_{{\mathcal{V}}_1}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3\hfill \\ \hfill =& {\mathcal{V}}_2\left(\mathcal{F}\right)({\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3+{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)-\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3))\hfill \\ \hfill & +(1-\mathcal{F})\left((1-\mathcal{F}){\nabla }_{{\mathcal{V}}_2}{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3+\mathcal{F}{\nabla }_{{\mathcal{V}}_2}^{*}{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3)\right)\hfill \\ \hfill & +\mathcal{F}\left((1-\mathcal{F}){\nabla }_{{\mathcal{V}}_2}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3+\mathcal{F}{\nabla }_{{\mathcal{V}}_2}^{*}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3+(1-\mathcal{F})\mathcal{L}({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3)+\mathcal{F}{\mathcal{L}}^{*}({\mathcal{V}}_2,{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3)\right)\hfill \\ \hfill & +(1-\mathcal{F})\left(-(1-\mathcal{F})A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-\mathcal{F}{A}_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+(1-\mathcal{F})D_{{\mathcal{V}}_2}\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)+\mathcal{F}{D}_{{\mathcal{V}}_2}^{*}\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)\right)\hfill \\ \hfill & +\mathcal{F}\left(-(1-\mathcal{F})A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-\mathcal{F}{A}_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+(1-\mathcal{F})D_{{\mathcal{V}}_2}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)+\mathcal{F}{D}_{{\mathcal{V}}_2}^{*}{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)\right).\hfill \end{array}$$

Again, using (20) we obtain

$${\tilde{\nabla }}_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{\left(\mathcal{F}\right)}{\mathcal{V}}_3=\mathcal{F}{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{*}{\mathcal{V}}_3+(1-\mathcal{F}){\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}{\mathcal{V}}_3+(1-\mathcal{F})\mathcal{L}([{\mathcal{V}}_1,{\mathcal{V}}_2],{\mathcal{V}}_3)+\mathcal{F}{\mathcal{L}}^{*}([{\mathcal{V}}_1,{\mathcal{V}}_2],{\mathcal{V}}_3).$$

Setting the last three equations in the definition of the curvature tensor and using

$$\begin{array}{cc}\hfill R({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+R^{*}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3=& {\nabla }_{{\mathcal{V}}_1}{\nabla }_{{\mathcal{V}}_2}^{*}{\mathcal{V}}_3+{\nabla }_{{\mathcal{V}}_1}^{*}{\nabla }_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}{\mathcal{V}}_3-{\nabla }_{{\mathcal{V}}_2}{\nabla }_{{\mathcal{V}}_1}^{*}{\mathcal{V}}_3\hfill \\ \hfill & -{\nabla }_{{\mathcal{V}}_2}^{*}{\nabla }_{{\mathcal{V}}_1}{\mathcal{V}}_3-{\nabla }_{[{\mathcal{V}}_1,{\mathcal{V}}_2]}^{*}{\mathcal{V}}_3-[K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_1}]{\mathcal{V}}_3,\hfill \end{array}$$

we obtain the first formula claimed by the theorem. In the same way, the other parts are concluded. □

Remark 1.

Considering $\mathcal{F}=0,\mathcal{F}=1$ and $\mathcal{F}={\displaystyle \frac{1}{2}}$ in the above theorem, one can see the equations of Gauss, Codazzi, and Ricci for the curvature tensors $\tilde{R}$, ${\tilde{R}}^{*}$ and ${\tilde{R}}^{\tilde{h}}$ with respect to the connections $\tilde{\nabla }$, ${\tilde{\nabla }}^{*}$ and ${\tilde{\nabla }}^{\tilde{h}}$, respectively in Proposition 6.

According to the above theorem, we deduce the following corollaries:

Corollary 3.

Let $(\tilde{N},\tilde{\nabla },\tilde{h})$ be a statistical manifold and $(N,\nabla ,h)$ be a doubly autoparallel statistical submanifold in $\tilde{N}$. If $\tilde{N}$ is $\mathcal{F}$-flat where $\mathcal{F}\in C^{\infty }\left(N\right)$, then M is $\mathcal{F}$-flat.

Corollary 4.

The following formulas hold

$$\begin{array}{cc}\hfill h({\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top },{\mathcal{V}}_4)=& -h({\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_1){\mathcal{V}}_3\right\}}^{\top },{\mathcal{V}}_4),\hfill \\ \hfill h({\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top },{\mathcal{V}}_4)=& -h({\left\{{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_4\right\}}^{\top },{\mathcal{V}}_3),\hfill \\ \hfill \tilde{h}({\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_2\right\}}^{\perp },{\zeta }_1)=& -\tilde{h}({\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_1){\zeta }_2\right\}}^{\perp },{\zeta }_1),\hfill \\ \hfill \tilde{h}({\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_1\right\}}^{\perp },{\zeta }_2)=& -\tilde{h}({\left\{{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\zeta }_2\right\}}^{\perp },{\zeta }_1),\hfill \end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4\in \Gamma \left(TN\right)$ and ${\zeta }_2,{\zeta }_1\in \Gamma \left(T^{\perp }N\right)$.

Lemma 5.

The $\mathcal{F}$-curvature tensors ${\tilde{R}}^{\left(\mathcal{F}\right)}$ and ${\tilde{R}}^{(1-\mathcal{F})}$ satisfy the following equations:

$${\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }-{\left\{{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }$$

$$=(1-2\mathcal{F})\left(R({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3-R^{*}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right)+2{\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3-2{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3$$

$$+(1-2\mathcal{F})\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1\right),$$

$${\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }+{\left\{{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }$$

$$=2R^h({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+{\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}[K_{{\mathcal{V}}_1},K_{{\mathcal{V}}_2}]{\mathcal{V}}_3$$

$$+2\mathcal{F}(1-\mathcal{F})\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1\right)$$

$$+(1-2\mathcal{F}+2{\mathcal{F}}^2)\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1+A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2-A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1\right),$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$.

Proof. 

According to Theorem 1, we can write

$$\begin{array}{cc}\hfill & {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }-{\left\{{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill =& R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3-R^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+(1-2\mathcal{F})(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2\hfill \\ \hfill & -A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1).\hfill \end{array}$$

(46)

Applying Proposition 5 in the above equation, we get the first part. Similarly, the second part follows. □

Corollary 5.

Let $(N,\nabla ,h)$ be a doubly autoparallel statistical submanifold in a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$. Then $R^{\left(\mathcal{F}\right)}=R^{(1-\mathcal{F})}$ if and only if ${\left({\tilde{R}}^{\left(\mathcal{F}\right)}\right)}^{\top }={\left({\tilde{R}}^{(1-\mathcal{F})}\right)}^{\top }$.

Proof. 

As $\mathcal{L}=0={\mathcal{L}}^{*}$, from (45) we get the assertion. □

Lemma 6.

If $(N,\nabla ,h)$ is a conjugate statistical submanifold in a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$, then

$$\begin{array}{cc}\hfill & {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }-{\left\{{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right\}}^{\top }\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill =& 2{\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3-2{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3\hfill \\ \hfill +& (1-2\mathcal{F})\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1\right),\hfill \end{array}$$

(48)

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$. Moreover, if $\tilde{N}$ is $\mathcal{F}$-flat, then the following holds

$$\begin{array}{cc}\hfill & {\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3-{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill =& {\displaystyle \frac{(2\mathcal{F}-1)}{2}}\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1\right).\hfill \end{array}$$

(50)

Proof. 

As N is a conjugate statistical submanifold, so $R=R^{*}$. Hence, Lemma 5 implies (47). Moreover, if ${\tilde{R}}^{\left(\mathcal{F}\right)}=0$, we get (49). □

From Lemmas 5 and 6, we conclude the following proposition:

Proposition 7.

On a doubly autoparallel statistical submanifold $(N,\nabla ,h)$ of a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$, if $(h,\nabla )$ is a conjugate statistical structure on N, then

$$\begin{array}{cc}\hfill {\{{\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3-{\tilde{R}}^{(1-\mathcal{F})}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\}}^{\top }=& 2{\mathcal{V}}_1\left(\mathcal{F}\right)K_{{\mathcal{V}}_2}{\mathcal{V}}_3-2{\mathcal{V}}_2\left(\mathcal{F}\right)K_{{\mathcal{V}}_1}{\mathcal{V}}_3,\hfill \end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$. Moreover, if $\mathcal{F}$ is constant, then we have

$$\begin{array}{c}\hfill {\left\{{\tilde{R}}^{\left(\mathcal{F}\right)}\right\}}^{\top }={\left\{{\tilde{R}}^{(1-\mathcal{F})}\right\}}^{\top }.\end{array}$$

In a statistical submanifold $(N,\nabla ,h)$, the Ricci curvature tensor $Ri{c}^{\left(\mathcal{F}\right)}$ of the $\mathcal{F}$-connection ${\nabla }^{\left(\mathcal{F}\right)}$ is defined by

$$\begin{array}{c}\hfill Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)=tr\{{\mathcal{V}}_1\to R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\}.\end{array}$$

(51)

Similarly, $Ri{c}^{(1-\mathcal{F})}$ of ${\nabla }^{(1-\mathcal{F})}$ can be described analogously.

Theorem 2.

Let $(\tilde{N},\tilde{\nabla },\tilde{h})$ be an $(m+n)$-dimensional statistical manifold and $(N,\nabla ,h)$ be an m-dimensional statistical submanifold of $\tilde{N}$. Let the statistical connection ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ on $(\tilde{N},\tilde{h})$ has the constant curvature c, where $\mathcal{F}\in C^{\infty }\left(N\right)$. If $\{e_1,\cdots ,e_m\}$ and $\{e_{m+1},\cdots ,e_{m+n}\}$ are orthonormal tangent and normal frames, respectively on N, then the Ricci tensor $Ri{c}^{\left(\mathcal{F}\right)}$ of N satisfies

$$\begin{array}{cc}\hfill & Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill =& c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)+\sum _{s=1}^n\{-\mathcal{F}(1-\mathcal{F})(h(A_{e_{m+i}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3)+h(A_{e_{m+s}}{\mathcal{V}}_2,A_{e_{m+s}}{\mathcal{V}}_3)\hfill \\ \hfill & -h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}-h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}})-{(1-\mathcal{F})}^2(h(A_{e_{m+s}}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3)\hfill \\ \hfill & -h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}})-{\mathcal{F}}^2(h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}{\mathcal{V}}_3)-h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*})\},\hfill \end{array}$$

(52)

for any ${\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$.

Proof. 

Applying (18) in Gauss equation of Theorem 1, we can write

$$\begin{array}{cc}\hfill & \tilde{h}({\tilde{R}}^f({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3,{\mathcal{V}}_4)\hfill \\ & =h(R^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3,{\mathcal{V}}_4)+\mathcal{F}(1-\mathcal{F})(\tilde{h}(\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_4)+\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_4))\hfill \\ \hfill & -\tilde{h}(h({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_4))-\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_4)))+{(1-\mathcal{F})}^2(\tilde{h}(\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_4))\hfill \\ \hfill & -\tilde{h}(h({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_4)))+{\mathcal{F}}^2\left(\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_4))-\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_4))\right),\hfill \end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4\in \Gamma \left(TN\right)$. As

$${\tilde{R}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3=c\{h({\mathcal{V}}_2,{\mathcal{V}}_3){\mathcal{V}}_1-h({\mathcal{V}}_1,{\mathcal{V}}_3){\mathcal{V}}_2\},$$

thus from the above two equations and using (51), it follows that

$$\begin{array}{cc}\hfill & Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill =& c\sum _{l=1}^m\{h({\mathcal{V}}_2,{\mathcal{V}}_3)h(e_l,e_l)-h({\mathcal{V}}_2,e_l)h({\mathcal{V}}_3,e_l)\}+\sum _{l=1}^m\{-\mathcal{F}(1-\mathcal{F})(\tilde{h}(\mathcal{L}(e_l,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_2,e_l))\hfill \\ \hfill & +\tilde{h}({\mathcal{L}}^{*}(e_l,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_2,e_l))-\tilde{h}(\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}(e_l,e_l))-\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}(e_l,e_l)))\hfill \\ \hfill & -{(1-\mathcal{F})}^2\left(\tilde{h}(\mathcal{L}(e_l,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_2,e_l))-\tilde{h}(\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}(e_l,e_l))\right)\hfill \\ \hfill & -{\mathcal{F}}^2\left(\tilde{h}({\mathcal{L}}^{*}(e_l,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_2,e_l))-\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}(e_l,e_l))\right)\}.\hfill \end{array}$$

(53)

On the other hand, we have

$$\begin{array}{cc}\hfill \sum _{l=1}^m\{h({\mathcal{V}}_2,{\mathcal{V}}_3)h(e_l,e_l)-h({\mathcal{V}}_2& ,e_l\left)h({\mathcal{V}}_3,e_l)\right\}=(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3),\hfill \\ \hfill \sum _{l=1}^m\tilde{h}(\mathcal{L}({\mathcal{V}}_2,e_l),\mathcal{L}({\mathcal{V}}_3,e_l))=& \sum _{s=1}^{n}h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3),\hfill \\ \hfill \sum _{l=1}^m\tilde{h}(\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}(e_l,e_l))=& \sum _{s=1}^{n}h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*},\hfill \\ \hfill \sum _{l=1}^m\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}(e_l,e_l))=& \sum _{s=1}^{n}h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}},\hfill \\ \hfill \sum _{l=1}^m\tilde{h}(\mathcal{L}({\mathcal{V}}_2,e_l),{\mathcal{L}}^{*}({\mathcal{V}}_3,e_l))=& \sum _{s=1}^{n}h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}{\mathcal{V}}_3).\hfill \end{array}$$

Setting the above equations in (53), we find the assertion. □

Theorem 2 leads to

$$Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)+Ri{c}^{(1-\mathcal{F})}({\mathcal{V}}_2,{\mathcal{V}}_3)$$

(54)

$$=2c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)-2\mathcal{F}(1-\mathcal{F}){\Delta }_1+(2\mathcal{F}(1-\mathcal{F})-1){\Delta }_2,$$

(55)

where

$$\begin{array}{cc}\hfill {\Delta }_1:=& \sum _{s=1}^n(h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3)+h(A_{e_{m+s}}{\mathcal{V}}_2,A_{e_{m+s}}{\mathcal{V}}_3)\hfill \\ & -h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}-h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}),\hfill \\ \hfill {\Delta }_2:=& \sum _{s=1}^n(h(A_{e_{m+s}}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3)-h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}\hfill \\ \hfill +& h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}{\mathcal{V}}_3)-h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}).\hfill \end{array}$$

Moreover, one see that

$$\begin{array}{cc}\hfill & Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{(1-\mathcal{F})}({\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill =& (-1+2\mathcal{F})\sum _{s=1}^n(h([A_{e_{m+s}}^{*},A_{e_{m+s}}]{\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill & +h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}-h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}),\hfill \end{array}$$

(56)

where $[A_{e_{m+s}}^{*},A_{e_{m+s}}]=A_{e_{m+s}}^{*}{A}_{e_{m+s}}-A_{e_{m+s}}{A}_{e_{m+s}}^{*}$.

Proposition 8.

According to the hypothesis of Theorem 2, for any ${\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$ we have

1. 

If N is conjugate symmetric with respect to the statistical connection ∇, then

$$\begin{array}{cc}\hfill (-1+2\mathcal{F})\sum _{s=1}^n(h([A_{e_{m+s}}^{*},A_{e_{m+s}}]{\mathcal{V}}_2,{\mathcal{V}}_3)+h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}& \\ \hfill -h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}})=2K_{{\mathcal{V}}_2}{\mathcal{V}}_3\left(\mathcal{F}\right)-2{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right),\end{array}$$

where ${\tau }_h\left({\mathcal{V}}_3\right)=tr{K}_{{\mathcal{V}}_3}$.

2. 

If the function $\mathcal{F}$ on N is constant, then the following hold

$$\begin{array}{cc}\hfill & \sum _{s=1}^n(h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}-h(A_{e_{m+s}}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3))\hfill \\ \hfill =& Ri{c}^h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{2}}\left(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left(di{v}^{{\nabla }^h}K\right)({\mathcal{V}}_2,{\mathcal{V}}_3)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\left({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})\right)-c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3),\hfill \end{array}$$

(57)

and

$$\begin{array}{cc}\hfill & \sum _{s=1}^n(h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}-h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,A_{e_{m+s}}{\mathcal{V}}_3))\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill =& Ri{c}^h({\mathcal{V}}_2,{\mathcal{V}}_3)-{\displaystyle \frac{1}{2}}\left(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left(di{v}^{{\nabla }^h}K\right)({\mathcal{V}}_2,{\mathcal{V}}_3)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\left({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})\right)-c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3).\hfill \end{array}$$

(59)

Proof. 

(44) gives

$$\begin{array}{cc}\hfill Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)=& (1-\mathcal{F})Ric({\mathcal{V}}_2,{\mathcal{V}}_3)+\mathcal{F}Ri{c}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)-\mathcal{F}(1-\mathcal{F})({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)\hfill \\ \hfill -& h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3}))+K_{{\mathcal{V}}_2}{\mathcal{V}}_3\left(\mathcal{F}\right)-{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right),\hfill \end{array}$$

(60)

which gives

$$\begin{array}{cc}\hfill & Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{(1-\mathcal{F})}({\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill =& (1-2\mathcal{F})\left(Ric({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)\right)+2K_{{\mathcal{V}}_2}{\mathcal{V}}_3\left(\mathcal{F}\right)-2{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right),\hfill \end{array}$$

where $Ric$ and $Ri{c}^{*}$ are the Ricci tensors associated with the statistical connections ∇ and ${\nabla }^{*}$. The above equation and (56) imply

$$\begin{array}{cc}\hfill & (-1+2\mathcal{F})\sum _{s=1}^n(h([A_{e_{m+s}}^{*},A_{e_{m+s}}]{\mathcal{V}}_2,{\mathcal{V}}_3)+h(A_{e_{m+s}}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}^{*}\hfill \\ & -h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}})\hfill \\ \hfill & =(1-2\mathcal{F})\left(Ric({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)\right)+2K_{{\mathcal{V}}_2}{\mathcal{V}}_3\left(\mathcal{F}\right)-2{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right).\hfill \end{array}$$

(61)

If N is conjugate symmetric, we have $R=R^{*}$ and $Ric=Ri{c}^{*}$. Hence (61) implies (1). Using (44) and (51), we get

$$\begin{array}{cc}\hfill & Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)={Ric}^h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{(1-2\mathcal{F})}{2}}\left(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left(di{v}^{{\nabla }^h}K\right)({\mathcal{V}}_2,{\mathcal{V}}_3)\right)\hfill \\ \hfill & +{\left({\displaystyle \frac{1-2\mathcal{F}}{2}}\right)}^2\left({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})\right)+K_{{\mathcal{V}}_2}{\mathcal{V}}_3\left(\mathcal{F}\right)-{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right).\hfill \end{array}$$

(62)

Setting $\mathcal{F}=0$ in the above equation and (52), respectively, it follows

$$Ric({\mathcal{V}}_2,{\mathcal{V}}_3)={Ric}^h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{2}}\left(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left(di{v}^{{\nabla }^h}K\right)({\mathcal{V}}_2,{\mathcal{V}}_3)\right)+{\displaystyle \frac{1}{4}}\left({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})\right),$$

and

$$Ric({\mathcal{V}}_2,{\mathcal{V}}_3)=c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)-\sum _{s=1}^n(h(A_{e_{m+s}}{\mathcal{V}}_2,A_{e_{m+s}}^{*}{\mathcal{V}}_3)-h(A_{e_{m+s}}^{*}{\mathcal{V}}_2,{\mathcal{V}}_3)tr{A}_{e_{m+s}}).$$

The above two equations imply (57). From (62), we have

$$Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)+Ri{c}^{(1-\mathcal{F})}({\mathcal{V}}_2,{\mathcal{V}}_3)=2{Ric}^h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}\left({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})\right).$$

For $\mathcal{F}=0$, this gives

$$\begin{array}{c}\hfill Ric({\mathcal{V}}_2,{\mathcal{V}}_3)+Ri{c}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)=2{Ric}^h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{2}}\left({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})\right).\end{array}$$

(63)

On the other hand, (54) yields

$$\begin{array}{cc}\hfill Ric({\mathcal{V}}_2,{\mathcal{V}}_3)+Ri{c}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)=& 2c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)-{\Delta }_2.\hfill \end{array}$$

Moreover, considering $\mathcal{F}={\displaystyle \frac{1}{2}}$ in (54) we get

$$\begin{array}{cc}\hfill 2Ri{c}^h({\mathcal{V}}_2,{\mathcal{V}}_3)=& 2c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)-{\displaystyle \frac{1}{2}}({\Delta }_1+{\Delta }_2).\hfill \end{array}$$

(64)

The last three equations give

$$\begin{array}{c}\hfill {\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})={\Delta }_1-{\Delta }_2.\end{array}$$

(65)

Putting the last equation in (64), it follows

$$\begin{array}{cc}\hfill Ri{c}^h({\mathcal{V}}_2,{\mathcal{V}}_3)=& c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{4}}(h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})-{\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right))-{\displaystyle \frac{1}{2}}{\Delta }_2.\hfill \end{array}$$

(66)

From (57) and the above equation, we get (58). □

Corollary 6.

We have

$$\begin{array}{cc}\hfill {\Delta }_1=& -2Ri{c}^h({\mathcal{V}}_2,{\mathcal{V}}_3)+2c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{2}}({\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right)-h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})),\hfill \\ \hfill {\Delta }_2=& 2c(m-1)h({\mathcal{V}}_2,{\mathcal{V}}_3)+{\displaystyle \frac{1}{2}}(h(K_{{\mathcal{V}}_2},K_{{\mathcal{V}}_3})-2{\tau }_h\left(K_{{\mathcal{V}}_2}{\mathcal{V}}_3\right))-2Ri{c}^h({\mathcal{V}}_2,{\mathcal{V}}_3),\hfill \end{array}$$

for any ${\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$.

Proof. 

Applying (57) and (58) in (65), we have ${\Delta }_1$. From (66), ${\Delta }_2$ follows. □

Theorem 3.

Let $(N,\nabla ,h)$ be an m-dimensional statistical submanifold of an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$. Let the statistical connection ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}$ on $(\tilde{N},\tilde{h})$ has the constant curvature c. If least one of the following hold

1. 

the Ricci tensor $Ric$ is symmetric;

2. 

the Ricci tensor $Ri{c}^{*}$ is symmetric;

3. 

$\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3=\left({\nabla }_{{\mathcal{V}}_3}^h{\tau }_h\right){\mathcal{V}}_2$,

then

$$\begin{array}{c}\hfill (-1+2\mathcal{F})\sum _{s=1}^{n}h([A_{e_{m+s}}^{*},A_{e_{m+s}}]{\mathcal{V}}_2,{\mathcal{V}}_3)={\mathcal{V}}_3\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_2\right)-{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right),\end{array}$$

(67)

for any ${\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$. Moreover, if $\mathcal{F}$ is constant, the Ricci tensor $Ri{c}^{\left(\mathcal{F}\right)}$ is symmetric.

Proof. 

According to (52), it follows

$$\begin{array}{cc}\hfill Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_3,{\mathcal{V}}_2)=& (-1+2\mathcal{F})\sum _{s=1}^{n}g([A_{e_{m+s}}^{*},A_{e_{m+s}}]{\mathcal{V}}_2,{\mathcal{V}}_3).\hfill \end{array}$$

On the other hand, (62) implies

$$\begin{array}{cc}\hfill & Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{\left(\mathcal{F}\right)}({\mathcal{V}}_3,{\mathcal{V}}_2)\hfill \\ \hfill =& {\displaystyle \frac{(1-2\mathcal{F})}{2}}(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left({\nabla }_{{\mathcal{V}}_3}^h{\tau }_h\right){\mathcal{V}}_2)+{\mathcal{V}}_3\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_2\right)-{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right).\hfill \end{array}$$

(68)

From the above equations, we have

$$\begin{array}{cc}\hfill & (-1+2\mathcal{F})\sum _{s=1}^{n}h([A_{e_{m+s}}^{*},A_{e_{m+s}}]{\mathcal{V}}_2,{\mathcal{V}}_3)\hfill \\ \hfill =& {\displaystyle \frac{(1-2\mathcal{F})}{2}}(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left({\nabla }_{{\mathcal{V}}_3}^h{\tau }_h\right){\mathcal{V}}_2)+{\mathcal{V}}_3\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_2\right)-{\mathcal{V}}_2\left(\mathcal{F}\right){\tau }_h\left({\mathcal{V}}_3\right).\hfill \end{array}$$

Moreover, from (68) for $\mathcal{F}=0,1$, we see that

$$\begin{array}{cc}\hfill Ric({\mathcal{V}}_2,{\mathcal{V}}_3)-Ric({\mathcal{V}}_3,{\mathcal{V}}_2)=& {\displaystyle \frac{1}{2}}(\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3-\left({\nabla }_{{\mathcal{V}}_3}^h{\tau }_h\right){\mathcal{V}}_2),\hfill \\ \hfill Ri{c}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)-Ri{c}^{*}({\mathcal{V}}_3,{\mathcal{V}}_2)=& {\displaystyle \frac{1}{2}}(\left({\nabla }_{{\mathcal{V}}_3}^h{\tau }_h\right){\mathcal{V}}_2-\left({\nabla }_{{\mathcal{V}}_2}^h{\tau }_h\right){\mathcal{V}}_3).\hfill \end{array}$$

Hence, (67) is obtained and implies ${\sum }_{s=1}^n[A_{e_{m+s}}^{*},A_{e_{m+s}}]=0$ if $\mathcal{F}$ is constant. □

Theorem 4.

Let $(\tilde{N},\tilde{\nabla },\tilde{h})$ be an $(m+n)$-dimensional statistical manifold of constant curvature c and $(N,{\nabla }^{\left(\mathcal{F}\right)},h)$ be a statistical manifold of an m-dimensional statistical submanifold $(N,\nabla ,h)$ in $\tilde{N}$. Then

$$\begin{array}{cc}\hfill {\sigma }^{\left(\mathcal{F}\right)}\ge & m(m-1)c-{\mathcal{F}(1-\mathcal{F})\left(\right|\left|\mathcal{L}\right||}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2\left)\right)\hfill \\ \hfill & +(-1-2{\mathcal{F}}^2+2\mathcal{F})\left(\right|\left|\mathcal{L}\right|\left|\right||{\mathcal{L}}^{*}\left|\right|+mdi{v}^{\tilde{\nabla }}H),\hfill \end{array}$$

where ${\sigma }^{\left(\mathcal{F}\right)}$ is the scalar curvature of $(N,{\nabla }^{\left(\mathcal{F}\right)},h)$, i.e., ${\sigma }^{\left(\mathcal{F}\right)}={\sum }_{s,l=1}^{m}h(R^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)$. Moreover, the equality holds if and only if $\mathcal{L}$ perpendicular to ${\mathcal{L}}^{*}$.

Proof. 

Setting ${\mathcal{V}}_2={\mathcal{V}}_3=e_s$ in (53) and by summing over $1\le s,l\le m$, we have

$$\begin{array}{cc}\hfill {\sigma }^{\left(\mathcal{F}\right)}=& m(m-1)c+\sum _{s,l=1}^m\sum _{\gamma =1}^n\{-\mathcal{F}(1-\mathcal{F})\left({\left({\mathcal{L}}_{sl}^{\gamma }\right)}^2+{\left({\mathcal{L}}_{sl}^{*\gamma }\right)}^2-{\mathcal{L}}_{ss}^{\gamma }{\mathcal{L}}_{ll}^{\gamma }-{\mathcal{L}}_{ss}^{*\gamma }{\mathcal{L}}_{ll}^{*\gamma }\right)\hfill \\ \hfill & -{(1-\mathcal{F})}^2\left({\mathcal{L}}_{sl}^{\gamma }{\mathcal{L}}_{sl}^{*\gamma }-{\mathcal{L}}_{ss}^{\gamma }{\mathcal{L}}_{ll}^{*\gamma }\right)-{\mathcal{F}}^2\left({\mathcal{L}}_{sl}^{*\gamma }{\mathcal{L}}_{sl}^{\gamma }-{\mathcal{L}}_{ss}^{*\gamma }{\mathcal{L}}_{ll}^{\gamma }\right)\}.\hfill \end{array}$$

As ${\left|\right|\mathcal{L}\left|\right|}^2={\sum }_{s,l=1}^m{\sum }_{\gamma =1}^n{\left({\mathcal{L}}_{sl}^{\gamma }\right)}^2$ and similarly $\left|\right|{\mathcal{L}}^{*}\left|\right|$, so it follows

$$\begin{array}{cc}\hfill {\sigma }^{\left(\mathcal{F}\right)}=& m(m-1)c-\mathcal{F}(1-\mathcal{F})\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2)\right)\hfill \\ \hfill & +(-1-2{\mathcal{F}}^2+2\mathcal{F})(\sum _{s,l=1}^m\sum _{\gamma =1}^n{\mathcal{L}}_{sl}^{\gamma }{\mathcal{L}}_{sl}^{*\gamma }-m^2\tilde{h}(H,H^{*})).\hfill \end{array}$$

(69)

As $(-1-2{\mathcal{F}}^2+2\mathcal{F})\le 0$, the last equation implies

$$\begin{array}{cc}\hfill {\sigma }^{\left(\mathcal{F}\right)}& \ge m(m-1)c-\mathcal{F}(1-\mathcal{F})\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2)\right)\hfill \\ \hfill & +(-1-2{\mathcal{F}}^2+2\mathcal{F})\left(\left|\right|\mathcal{L}\left|\right|\left|\right|{\mathcal{L}}^{*}\left|\right|-m^2\tilde{h}(H,H^{*})\right).\hfill \end{array}$$

This and (43) give us the assertion. □

Corollary 7.

We have

$${\sigma }^{\left(\mathcal{F}\right)}={\sigma }^{(1-\mathcal{F})}.$$

Proof. 

From (69), it follows

$$\begin{array}{cc}\hfill {\sigma }^{\left(\mathcal{F}\right)}=& m(m-1)c-\mathcal{F}(1-\mathcal{F})\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2)\right)\hfill \\ \hfill & +(-1-2{\mathcal{F}}^2+2\mathcal{F})\left(\sum _{s,l=1}^m\sum _{\gamma =1}^n{\mathcal{L}}_{sl}^{\gamma }{\mathcal{L}}_{sl}^{*\gamma }-m^2\tilde{h}(H,H^{*})\right)={\sigma }^{(1-\mathcal{F})}.\hfill \end{array}$$

(70)

□

Proposition 9.

For an N-dimensional statistical submanifold $(N,\nabla ,h)$ in an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ of the constant curvature c, if

(i)

N has the property that $\mathcal{L}$ and ${\mathcal{L}}^{*}$ are perpendicular;

(ii)

N is ∇- and ${\nabla }^{*}$-minimal;

then we have

$$\begin{array}{cc}\hfill & {\mathcal{F}(1-\mathcal{F})\left(\right|\left|\mathcal{L}\right||}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2)\hfill \\ \hfill =& \mathcal{F}(1-\mathcal{F})\sum _{s,l,l=1}^m(K_{ss}^l{K}_{ll}^l-K_{sl}^l{K}_{sl}^l)-\sum _{s,l=1}^m{K}_{ss}^l{\partial }_l\left(\mathcal{F}\right)+\sum _{s,l=1}^m{\partial }_s\left(\mathcal{F}\right)K_{rs}^r.\hfill \end{array}$$

(71)

Moreover, if $\mathcal{F}$ is constant, it follows

$$\begin{array}{c}\hfill {\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2=\sum _{s,l,l=1}^m(K_{ss}^l{K}_{ll}^l-K_{sl}^l{K}_{sl}^l),\end{array}$$

(72)

where $\mathcal{F}\in C^{\infty }\left(N\right)$.

Proof. 

Using (60), we get

$$\begin{array}{c}\hfill {\sigma }^{\left(\mathcal{F}\right)}=\sigma -\mathcal{F}(1-\mathcal{F})\sum _{s,l,l=1}^m(K_{ss}^l{K}_{ll}^l-K_{sl}^l{K}_{sl}^l)+\sum _{s,l=1}^m{K}_{ss}^l{\partial }_l\left(\mathcal{F}\right)-\sum _{s,l=1}^m{\partial }_s\left(\mathcal{F}\right)K_{ls}^l.\end{array}$$

On the other hand, (70) yields

$$\begin{array}{cc}\hfill {\sigma }^{\left(\mathcal{F}\right)}=& m(m-1)c-\mathcal{F}(1-\mathcal{F})\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2\right).\hfill \end{array}$$

Considering $\mathcal{F}=0$ in the last equation, one see that $\sigma =m(m-1)c$. Hence, from the above two equations, (71) follows. (72) is obtained from (71). □

Let $(N,\nabla ,h)$ be an m-dimensional statistical submanifold in an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$. For the statistical submanifold $(N,{\nabla }^{\left(\mathcal{F}\right)},h)$ induced by $(N,\nabla ,h)$, we define tensor fields ${\tilde{S}}^{\left(\mathcal{F}\right)}$ and $S^{\left(\mathcal{F}\right)}$ of type $(1,3)$ on $\tilde{N}$ and N, respectively by

$$\begin{array}{cc}\hfill {\tilde{S}}^{\left(\mathcal{F}\right)}=& {\displaystyle \frac{1}{2}}\{{\tilde{R}}^{\left(\mathcal{F}\right)}+{\tilde{R}}^{(1-\mathcal{F})}\},\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill S^{\left(\mathcal{F}\right)}=& {\displaystyle \frac{1}{2}}\{R^{\left(\mathcal{F}\right)}+R^{(1-\mathcal{F})}\}.\hfill \end{array}$$

(74)

The tensor fields ${\tilde{S}}^{\left(\mathcal{F}\right)}$ and $S^{\left(\mathcal{F}\right)}$ are called the statistical curvature tensor fields of $({\tilde{\nabla }}^{\left(\mathcal{F}\right)},\tilde{h})$ and $(h,{\nabla }^{\left(\mathcal{F}\right)})$, respectively. The statistical Ricci curvature tensor and the statistical scalar curvature of $S^{\left(\mathcal{F}\right)}$ are described by

$$\begin{array}{cc}\hfill L^{\left(\mathcal{F}\right)}({\mathcal{V}}_2,{\mathcal{V}}_3)=& tr\{{\mathcal{V}}_1\to S^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\},\hfill \\ \hfill {\rho }^{\left(\mathcal{F}\right)}=& tr{L}^{\left(\mathcal{F}\right)},\hfill \end{array}$$

respectively. Now, let $\{e_1,\cdots ,e_m\}$ and be an orthonormal frame on N. We describe $\mathcal{F}$-statistical sectional curvature of $(N,h,{\nabla }^{\left(\mathcal{F}\right)})$ for $e_s\wedge e_l$, as

$$\begin{array}{c}\hfill {\mathcal{K}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)=h(S^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s).\end{array}$$

Shortly, we denote $L^{\left(0\right)}$, $L^{\left(1\right)}$, $L^{\left(\frac{1}{2}\right)}$, ${\rho }^{\left(0\right)}$, ${\rho }^{\left(1\right)}$ and ${\rho }^{\left(\frac{1}{2}\right)}$, respectively by L, $L^{*}$, $L^h$, $\rho$, ${\rho }^{*}$ and ${\rho }^h$. Similarly, ${\mathcal{K}}^{\left(0\right)}$, ${\mathcal{K}}^{\left(1\right)}$ and ${\mathcal{K}}^{\left(\frac{1}{2}\right)}$ are denoted by $\mathcal{K}$, ${\mathcal{K}}^{*}$ and ${\mathcal{K}}^h$, respectively.

Proposition 10.

If $(N,\nabla ,h)$ is a statistical submanifold in a statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$ and $\mathcal{F}\in C^{\infty }\left(N\right)$, then for the statistical submanifold $(N,{\nabla }^{\left(\mathcal{F}\right)},h)$ we get

$$\begin{array}{cc}\hfill & {\left({\tilde{S}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3\right)}^{\top }\hfill \\ \hfill =& S^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3+\mathcal{F}(1-\mathcal{F})\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2+A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1-A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1\right)\hfill \\ \hfill & +{\displaystyle \frac{1-2\mathcal{F}(1-\mathcal{F})}{2}}\left(A_{\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3)}{\mathcal{V}}_2-A_{\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3)}{\mathcal{V}}_1+A_{{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_2-A_{{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3)}^{*}{\mathcal{V}}_1\right),\hfill \end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3\in \Gamma \left(TN\right)$.

Proof. 

From ${\tilde{S}}^{\left(\mathcal{F}\right)}={\tilde{S}}^{\left(\mathcal{F}\right)\top }+{\tilde{S}}^{\left(\mathcal{F}\right)\perp }$, (73) and using the equations of Gauss in Theorem 1, we obtain the assertion. □

Lemma 7.

Let $(N,\nabla ,h)$ be an m-dimensional statistical submanifold in an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$. If $\mathcal{F}\in C^{\infty }\left(N\right)$, the statistical scalar curvature ${\rho }^{\left(\mathcal{F}\right)}$ of $(N,h,{\nabla }^{\left(\mathcal{F}\right)})$ satisfies the following

$$\begin{array}{cc}\hfill {\rho }^{\left(\mathcal{F}\right)}=& 2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)+{\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left|\right|H\left|\right|}^2-m^2\left|\right|H^{*}{\left|\right|}^2\right)\hfill \\ \hfill & +(1+{(1-2\mathcal{F})}^2)\left({\rho }^h-2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\tilde{h}}(e_s\wedge e_l)\right),\hfill \end{array}$$

where ${\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}$ is the statistical sectional curvature of $(\tilde{N},{\tilde{\nabla }}^{\left(\mathcal{F}\right)},\tilde{h})$.

Proof. 

Proposition 10 and (18) lead to

$$\begin{array}{cc}\hfill & h({\tilde{S}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3,{\mathcal{V}}_4)\hfill \\ \hfill =& h(S^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_2){\mathcal{V}}_3,{\mathcal{V}}_4)+\mathcal{F}(1-\mathcal{F})(\tilde{h}(\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_4))+\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_4))\hfill \\ \hfill & -\tilde{h}(h({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_4))-\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_4)))\hfill \\ \hfill & +{\displaystyle \frac{1-2\mathcal{F}(1-\mathcal{F})}{2}}(\tilde{h}(\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_4))-\tilde{h}(\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_3),{\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_4))\hfill \\ \hfill & +\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_1,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_2,{\mathcal{V}}_4))-\tilde{h}({\mathcal{L}}^{*}({\mathcal{V}}_2,{\mathcal{V}}_3),\mathcal{L}({\mathcal{V}}_1,{\mathcal{V}}_4))),\hfill \end{array}$$

for any ${\mathcal{V}}_1,{\mathcal{V}}_2,{\mathcal{V}}_3,{\mathcal{V}}_4\in \Gamma \left(TN\right)$. Considering $\{e_1,\cdots ,e_m\}$ and $\{e_{m+1},\cdots ,e_{n+m}\}$ as orthonormal tangent and normal frames, respectively on N and setting ${\mathcal{V}}_1={\mathcal{V}}_4=e_i$ and ${\mathcal{V}}_2={\mathcal{V}}_3=e_l$ in the above equation and summing over $i,l$, provides

$$\begin{array}{cc}\hfill {\rho }^{\left(\mathcal{F}\right)}=& 2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)-\mathcal{F}(1-\mathcal{F})\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left|\right|H\left|\right|}^2-m^2\left|\right|H^{*}{\left|\right|}^2\right)\hfill \\ \hfill & +(1-2\mathcal{F}(1-\mathcal{F}))\left(-\tilde{h}(\mathcal{L},{\mathcal{L}}^{*})+m^2\tilde{h}(H,H^{*})\right).\hfill \end{array}$$

(75)

Setting (31) and (32) in (75), it follows that

$$\begin{array}{cc}\hfill & {\rho }^{\left(\mathcal{F}\right)}=2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)+{\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left|\right|H\left|\right|}^2-m^2\left|\right|H^{*}{\left|\right|}^2\right)\hfill \\ \hfill & +(1+{(1-2\mathcal{F})}^2)(-||\widehat{\mathcal{L}}{\left|\right|}^2+m^2\left|\right|\widehat{H}{\left|\right|}^2).\hfill \end{array}$$

On the other hand, setting $\mathcal{F}={\displaystyle \frac{1}{2}}$ in the last equation we get

$$\begin{array}{cc}\hfill {\rho }^h=& 2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\tilde{h}}(e_s\wedge e_l)-\left|\right|\widehat{\mathcal{L}}{\left|\right|}^2+m^2\left|\right|\widehat{H}{\left|\right|}^2.\hfill \end{array}$$

Hence the assertion is obtained from the above two relations. □

Proposition 11.

We have

$$\begin{array}{cc}\hfill & {(1-2\mathcal{F})}^{2}di{v}^{\tilde{\nabla }}H\hfill \\ \hfill =& -{\displaystyle \frac{1}{m}}\left\{2\right||\widehat{\mathcal{L}}{\left|\right|}^2-{\displaystyle \frac{1}{2}}\left(\right||{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2)+{\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left|\right|H\left|\right|}^2-m^2\left|\right|H^{*}{\left|\right|}^2\right)\hfill \\ \hfill & +(1+{(1-2\mathcal{F})}^2)({\rho }^h-2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\tilde{h}}(e_s\wedge e_l))\}+4m\mathcal{F}(1-\mathcal{F})||\widehat{H}{\left|\right|}^2.\hfill \end{array}$$

Proof. 

(29) and (30) imply

$$\begin{array}{cc}\hfill & m^2\tilde{h}(H^{\left(\mathcal{F}\right)},H^{(1-\mathcal{F})})-\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)},{\mathcal{L}}^{(1-\mathcal{F})})\hfill \\ \hfill =& {\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}\left({\left|\right|\mathcal{L}\left|\right|}^2+\left|\right|{\mathcal{L}}^{*}{\left|\right|}^2-m^2{\left|\right|H\left|\right|}^2-m^2\left|\right|H^{*}{\left|\right|}^2\right)\hfill \\ \hfill & +(1+{(1-2\mathcal{F})}^2)\left({\rho }^h-2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\tilde{h}}(e_s\wedge e_l)\right).\hfill \end{array}$$

Applying Lemma 4 in the above equation, the assertion follows. □

From Lemma 7 and the last equation, we have the following corollary:

Corollary 8.

The following holds

$$\begin{array}{cc}\hfill {\rho }^{\left(\mathcal{F}\right)}=& 2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)+m^2\tilde{h}(H^{\left(\mathcal{F}\right)},H^{(1-\mathcal{F})})-\tilde{h}({\mathcal{L}}^{\left(\mathcal{F}\right)},{\mathcal{L}}^{(1-\mathcal{F})}).\hfill \end{array}$$

Lemma 8.

The tensor field ${\mathcal{L}}^{\left(\mathcal{F}\right)}$ satisfy in the following

$$\begin{array}{c}\hfill \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2\ge {\displaystyle \frac{m^2}{2}}\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+2\sum _{\gamma =1}^n\sum _{2\le s<l\le m}({\left({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }\right)}^2-{\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma }).\end{array}$$

Proof. 

As $\left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2={\sum }_{\gamma =1}^n{\sum }_{s,l=1}^m{\left({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }\right)}^2$, one can see that

$$\begin{array}{cc}\hfill & \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2={\displaystyle \frac{1}{2}}\sum _{\gamma =1}^n\{{({\mathcal{L}}_{11}^{\left(\mathcal{F}\right)\gamma }+({\mathcal{L}}_{22}^{\left(\mathcal{F}\right)\gamma }+\cdots +{\mathcal{L}}_{mm}^{\left(\mathcal{F}\right)\gamma }))}^2+{({\mathcal{L}}_{11}^{\left(\mathcal{F}\right)\gamma }-({\mathcal{L}}_{22}^{\left(\mathcal{F}\right)\gamma }+\cdots +{\mathcal{L}}_{mm}^{\left(\mathcal{F}\right)\gamma }))}^2\}\hfill \\ \hfill & +2\sum _{\gamma =1}^n\{\sum _{l=2}^m{\left({\mathcal{L}}_{1l}^{\left(\mathcal{F}\right)\gamma }\right)}^2+\sum _{2\le s<l\le m}({\left({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }\right)}^2-{\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma })\},\hfill \end{array}$$

which gives

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2\ge & {\displaystyle \frac{1}{2}}\sum _{\gamma =1}^n{({\mathcal{L}}_{11}^{\left(\mathcal{F}\right)\gamma }+{\mathcal{L}}_{22}^{\left(\mathcal{F}\right)\gamma }+\cdots +{\mathcal{L}}_{mm}^{\left(\mathcal{F}\right)\gamma })}^2\hfill \\ \hfill & +2\sum _{\gamma =1}^n\sum _{2\le s<l\le m}({\left({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }\right)}^2-{\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma })\hfill \\ & ={\displaystyle \frac{m^2}{2}}\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+2\sum _{\gamma =1}^n\sum _{2\le s<l\le m}({\left({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }\right)}^2-{\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma }).\hfill \end{array}$$

□

Lemma 9.

We have

$$\begin{array}{cc}\hfill & \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2\hfill \\ \hfill \ge & {\displaystyle \frac{m^2}{2}}\left(\right(\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2)+8\sum _{2\le s<l\le m}\left(\right||{\widehat{\mathcal{L}}}_{sl}{\left|\right|}^2-\tilde{h}({\widehat{\mathcal{L}}}_{ss},{\widehat{\mathcal{L}}}_{ll}))\hfill \\ \hfill & -4\sum _{2\le s<l\le m}(\tilde{h}({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)},{\mathcal{L}}_{sl}^{(1-\mathcal{F})})-\tilde{h}({\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)},{\mathcal{L}}_{ll}^{(1-\mathcal{F})})).\hfill \end{array}$$

Proof. 

Considering $(1-\mathcal{F})$ instead of $\mathcal{F}$ in Lemma 8, it follows

$$\begin{array}{c}\hfill \left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2\ge {\displaystyle \frac{m^2}{2}}\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2+2\sum _{\gamma =1}^p\sum _{2\le s<l\le m}({\left({\mathcal{L}}_{sl}^{(1-\mathcal{F})\gamma }\right)}^2-{\mathcal{L}}_{ss}^{(1-\mathcal{F})\gamma }{\mathcal{L}}_{ll}^{(1-\mathcal{F})\gamma }).\end{array}$$

Hence, we obtain

$$\begin{array}{cc}\hfill & \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2\hfill \\ \hfill \ge & {\displaystyle \frac{m^2}{2}}\left(\right(\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2)\hfill \\ \hfill & +2\sum _{\gamma =1}^p\sum _{2\le s<l\le m}({\left({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }\right)}^2-{\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma }+{\left({\mathcal{L}}_{sl}^{(1-\mathcal{F})\gamma }\right)}^2-{\mathcal{L}}_{ss}^{(1-\mathcal{F})\gamma }{\mathcal{L}}_{ll}^{(1-\mathcal{F})\gamma })\hfill \\ \hfill & ={\displaystyle \frac{m^2}{2}}\left(\right(\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2)\hfill \\ \hfill & +2\sum _{\gamma =1}^p\sum _{2\le s<l\le m}({({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }+{\mathcal{L}}_{sl}^{(1-\mathcal{F})\gamma })}^2-({\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }+{\mathcal{L}}_{ss}^{(1-\mathcal{F})\gamma })({\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma }+{\mathcal{L}}_{ll}^{(1-\mathcal{F})\gamma })\hfill \\ \hfill & +({\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{ll}^{(1-\mathcal{F})\gamma }-{\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{sl}^{(1-\mathcal{F})\gamma })+({\mathcal{L}}_{ss}^{(1-\mathcal{F})\gamma }{\mathcal{L}}_{ll}^{\left(\mathcal{F}\right)\gamma }-{\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)\gamma }{\mathcal{L}}_{sl}^{(1-\mathcal{F})\gamma }).\hfill \end{array}$$

Applying (29) in the last relation, we get the assertion. □

Theorem 5.

Let $(N,\nabla ,h)$ be an m-dimensional statistical submanifold in an $(m+n)$-dimensional statistical manifold $(\tilde{N},\tilde{\nabla },\tilde{h})$. Considering $\mathcal{F}\in C^{\infty }\left(N\right)$, for ${\mathcal{V}}_1\in T_{p}N$, $p\in N$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}{\left(\right|\left|H\right||}^2+\left|\right|H^{*}{\left|\right|}^2)\ge & {\displaystyle \frac{4}{m^2}}(2Ri{c}^h({\mathcal{V}}_1,{\mathcal{V}}_1)-L^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_1)-2\sum _{l=2}^m{\tilde{\mathcal{K}}}^{\tilde{h}}({\mathcal{V}}_1\wedge e_l)\hfill \\ \hfill & +\sum _{l=2}^m{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1\wedge e_l)-\mathcal{F}(1-\mathcal{F})m^2\left|\right|\widehat{H}{\left|\right|}^2),\hfill \end{array}$$

where $\{{\mathcal{V}}_1=e_1,e_2,\cdots ,e_m\}$ is an orthonormal basis of $T_{p}N$.

Proof. 

We have

$$h({\tilde{S}}^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)=h(S^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)+\tilde{h}({\mathcal{L}}_{sl}^{\left(\mathcal{F}\right)},{\mathcal{L}}_{sl}^{(1-\mathcal{F})})-\tilde{h}({\mathcal{L}}_{ss}^{\left(\mathcal{F}\right)},{\mathcal{L}}_{ll}^{(1-\mathcal{F})}).$$

The above equation and Lemma 9 imply

$$\begin{array}{cc}\hfill \left|\right|{\mathcal{L}}^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|{\mathcal{L}}^{(1-\mathcal{F})}{\left|\right|}^2\ge & {\displaystyle \frac{m^2}{2}}\left(\right(\left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}\left|\right|)\hfill \\ & -8\sum _{2\le s<l\le m}(h(R^h(e_s,e_l)e_l,e_s)-h({\tilde{R}}^{\tilde{h}}(e_s,e_l)e_l,e_s))\hfill \\ \hfill & +4\sum _{2\le s<l\le m}(h(S^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)-h({\tilde{S}}^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)).\hfill \end{array}$$

Setting (33) in the last relation, it follows that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}{\left(\right|\left|\mathcal{L}\right||}^2+\left|\right|{\mathcal{L}}^{*}\left|\right|)\hfill \\ \hfill \ge & -4\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{\mathcal{L}}{\left|\right|}^2+{\displaystyle \frac{m^2}{2}}({\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}{\left(\right|\left|H\right||}^2+\left|\right|H^{*}\left|\right|)+4\mathcal{F}(1-\mathcal{F})\left|\right|\widehat{H}{\left|\right|}^2)\hfill \\ \hfill & -4\sum _{2\le s<l\le m}(h(R^h(e_s,e_l)e_l,e_s)-h({\tilde{R}}^{\tilde{h}}(e_s,e_l)e_l,e_s))\hfill \\ \hfill & +2\sum _{2\le s<l\le m}(h(S^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)-h({\tilde{S}}^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)).\hfill \end{array}$$

(76)

Using (76) and Lemma 7, we get

$$\begin{array}{cc}\hfill {\rho }^{\left(\mathcal{F}\right)}\ge & 2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)-{\displaystyle \frac{{(1-2\mathcal{F})}^2{m}^2}{4}}\left({\left|\right|H\left|\right|}^2+\left|\right|H^{*}{\left|\right|}^2\right)\hfill \\ \hfill & +(1+{(1-2\mathcal{F})}^2)\left({\rho }^h-2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\tilde{h}}(e_s\wedge e_l)\right)\hfill \\ \hfill & +4\mathcal{F}(1-\mathcal{F})\left({\displaystyle \frac{m^2}{2}}\right|\left|\widehat{H}\right|{|}^2-\left|\right|\widehat{\mathcal{L}}\left|{|}^2\right)\hfill \\ \hfill & -4\sum _{2\le s<l\le m}(h(R^h(e_s,e_l)e_l,e_s)-h({\tilde{R}}^{\tilde{h}}(e_s,e_l)e_l,e_s))\hfill \\ \hfill & +2\sum _{2\le s<l\le m}(h(S^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)-h({\tilde{S}}^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)),\hfill \end{array}$$

which gives us

$$\begin{array}{cc}\hfill {\displaystyle \frac{{(1-2\mathcal{F})}^2}{2}}\left({\left|\right|H\left|\right|}^2+\left|\right|H^{*}{\left|\right|}^2\right)\ge & {\displaystyle \frac{2}{m^2}}\{2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}(e_s\wedge e_l)-{\rho }^{\left(\mathcal{F}\right)}\hfill \\ \hfill & +(1+{(1-2\mathcal{F})}^2)\left({\rho }^h-2\sum _{1\le s<l\le m}{\tilde{\mathcal{K}}}^{\tilde{h}}(e_s\wedge e_l)\right)\hfill \\ \hfill & +4\mathcal{F}(1-\mathcal{F})\left({\displaystyle \frac{m^2}{2}}\right|\left|\widehat{H}\right|{|}^2-\left|\right|\widehat{\mathcal{L}}\left|{|}^2\right)\hfill \\ \hfill & -4\sum _{2\le s<l\le m}(h(R^h(e_s,e_l)e_l,e_s)-h({\tilde{R}}^{\tilde{h}}(e_s,e_l)e_l,e_s))\hfill \\ \hfill & +2\sum _{2\le s<l\le m}(h(S^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s)-h({\tilde{S}}^{\left(\mathcal{F}\right)}(e_s,e_l)e_l,e_s))\}\hfill \\ \hfill & ={\displaystyle \frac{4}{m^2}}(2Ri{c}^h({\mathcal{V}}_1,{\mathcal{V}}_1)-L^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_1)-2\sum _{l=2}^m{\tilde{\mathcal{K}}}^{\tilde{h}}({\mathcal{V}}_1\wedge e_l)\hfill \\ \hfill & +\sum _{l=2}^m{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1\wedge e_l)-\mathcal{F}(1-\mathcal{F})m^2\left|\right|\widehat{H}{\left|\right|}^2).\hfill \end{array}$$

□

Corollary 9.

We have

$$\begin{array}{cc}\hfill \left|\right|H^{\left(\mathcal{F}\right)}{\left|\right|}^2+\left|\right|H^{(1-\mathcal{F})}{\left|\right|}^2\ge & {\displaystyle \frac{8}{m^2}}(2Ri{c}^h({\mathcal{V}}_1,{\mathcal{V}}_1)-L^{\left(\mathcal{F}\right)}({\mathcal{V}}_1,{\mathcal{V}}_1)-2\sum _{l=2}^m{\tilde{\mathcal{K}}}^{\tilde{h}}({\mathcal{V}}_1\wedge e_l)\hfill \\ \hfill & +\sum _{l=2}^m{\tilde{\mathcal{K}}}^{\left(\mathcal{F}\right)}({\mathcal{V}}_1\wedge e_l)).\hfill \end{array}$$

Proof. 

Applying (34) in Theorem 5, the assertion follows. □

5. Conclusions

In this paper we have studied a new family of connections, namely $\mathcal{F}$-statistical connections on statistical submanifolds, which is given by an affine combination of the conjugate connections, i.e., ${\tilde{\nabla }}^{\left(\mathcal{F}\right)}=(1-\mathcal{F})\tilde{\nabla }+\mathcal{F}{\tilde{\nabla }}^{*}$, where $\mathcal{F}$ is a smooth function on the statistical manifold. Then, the Gauss and Weingarten formulas are given for $\mathcal{F}$-statistical connections. There are presented several relevant geometric notions including $\mathcal{F}$-second fundamental forms, $\mathcal{F}$-mean curvatures and properties of statistical submanifolds about the $\mathcal{F}$-doubly auto-parallelism, geodesic and minimality. We have established the concepts of divergence for $\mathcal{F}$-mean curvature vector fields and $\mathcal{F}$-fundamental forms and have studied some of their properties. After that, Gauss equations and Codazzi equations of $\mathcal{F}$-statistical connections are obtained. When the statistical submanifolds are conjugate symmetric, such structures are discussed. It is also shown an inequality for the lower bound of the scalar curvature of a $\mathcal{F}$-statistical connection and an explicit condition for the equality to hold. Finally, we have given an inequality involving statistical Ricci curvature and the squared $\mathcal{F}$-mean curvature of a statistical submanifold of statistical manifolds. These approaches may give further development in statistical manifolds and affine differential geometry in the future.