Lifts of a Semi-Symmetric Metric Connection from Sasakian Statistical Manifolds to Tangent Bundle

Abstract

The lifts of Sasakian statistical manifolds associated with a semi-symmetric metric connection in the tangent bundle are characterized in the current research. The relationship between the complete lifts of a statistical manifold with semi-symmetric metric connections and Sasakian statistical manifolds with a semi-symmetric metric connection in the tangent bundle is investigated. We also discuss the classification of Sasakian statistical manifolds with respect to semi-symmetric metric connections in the tangent bundle. Finally, we derive an example of the lifts of Sasakian statistical manifolds to the tangent bundle.

1. Introduction

Friedmann and Schouten [1] proposed the concept of a semi-symmetric connection (SSC) on a differentiable manifold. If a linear connection satisfies the expression:

$$\ddot{T}(X_0,Y_0)={\omega }_0\left(Y_0\right)X_0-{\omega }_0\left(X_0\right)Y_0,$$

and is not torsion-free, it is referred to as a semi-symmetric connection (SSC), where $\ddot{T}$ is the torsion tensor, ${\omega }_0$ is a one-form, and $X_0,Y_0$ are vector fields. Hayden [2] introduced the notion of the metric connection, which was called the Hayden connection. The semi-symmetric metric connection (SSMC) satisfies the semi-symmetric condition and is known to exist if $\ddot{\nabla }{g}_0=0$; otherwise, it is a semi-symmetric non-metric connection (SSNMC), and it was further studied in [2,3]. Amari [4] was the first to study statistical inference problems in information geometry, which was the concept of statistical structure. Every point on a statistical manifold, a differentiable manifold, depicts a probability distribution. A statistical manifold with infinite dimensions can be found in the collection of all probability measurements. Furuhata et al. investigated the concepts of the Sasakian and Kenmotsu statistical structures [5,6,7]. Kurose [8] studied the concept of the holomorphic statistical structure as a generalization of Kahler’s structure. Kazan and Kazan [9] investigated the SSMC on Sasakian statistical manifolds. In [10,11], the authors investigated connections on statistical manifolds. Numerous geometers have explored the tangent bundle of differential geometry, including Yano and Kobayashi [12], Yano and Ishihara [13], Tani [14], and Pandey and Chaturvedi [15]. Yano and Ishihara [13] established the lifts of the manifold, as well as the connection in the tangent bundle. Different manifolds associated with different connections in the tangent bundle were studied in [16,17,18,19,20,21,22,23,24]. Kumar et al. [25] recently studied the lifts of the semi-symmetric non-metric connection (SSNMC) from statistical manifolds to the tangent bundle.

This Introduction is followed by a section on the preliminary materials. Section 3 investigates the statistical manifold and Sasakian statistical manifolds’ lifts to its tangent bundle, and Section 4 computes the SSMC in the tangent bundle, whereas Section 5 is concerned with the investigation of the lifts of the curvature tensor of a statistical manifold with the SSMC in the tangent bundle. Section 6 investigates the lifts of some curvature tensors of Sasakian statistical manifolds with the SSMC in the tangent bundle and proves some theorems. Finally, in Section 7, an example is provided to demonstrate the lifts of Sasakian statistical manifolds in the tangent bundle.

2. Preliminaries

In a differentiable manifold M, let $T_{0}M={\bigcup }_{p\in M}{T}_{0p}M$ be the tangent bundle, where $T_{0p}M$ is the tangent space at point $p\in M$ and $\pi :T_{0}M\to M$ is the natural bundle structure of $T_{0}M$ over M. For any coordinate system $(Q,x^h)$ in M, where $\left(x^h\right)$ is a local coordinate system in the neighborhood Q, $({\pi }^{-1}\left(Q\right),x^h,y^h)$ is the coordinate system in $T_{0}M$, where $(x^h,y^h)$ is an induced coordinate system in ${\pi }^{-1}\left(Q\right)$ from $\left(x^h\right)$ [13].

2.1. Vertical and Complete Lifts

Let us define a vector field $X_0$, a tensor field $F_0$ of type $(1,1)$, a function $f_0$, a one-form ${\omega }_0$, and affine connection $\ddot{\nabla }$ in M; its vertical and complete lifts are given by $f_0^v,X_0^v,{\omega }_0^v,F_0^v,{\ddot{\nabla }}^v$ and $f_0^c,X_0^c,{\omega }_0^c,F_0^c,{\ddot{\nabla }}^c$, respectively. The following formulas for complete and vertical lifts were defined by [13]:

$$\begin{array}{cc}\hfill {\left(f_0{X}_0\right)}^v& =f_0^v{X}_0^v,{\left(f_0{X}_0\right)}^c=f_0^c{X}_0^v+f_0^v{X}_0^c,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill X_0^v{f}_0^v& =0,X_0^v{f}_0^c=X_0^c{f}_0^v={\left(X_0{f}_0\right)}^v,X_0^c{f}_0^c={\left(X_0{f}_0\right)}^c,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\omega }_0\left(f_0^v\right)& =0,{\omega }_0^v\left(X_0^c\right)={\omega }_0^c\left(X_0^v\right)={\omega }_0{\left(X_0\right)}^v,{\omega }_0^c\left(X_0^c\right)={\omega }_0{\left(X_0\right)}^c,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill F_0^v{X}_0^c& ={\left(F_0{X}_0\right)}^v,F_0^c{X}_0^c={\left(F_0{X}_0\right)}^c,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill [X_0,Y_0{]}^v& =[X_0^c,Y_0^v]=[X_0^v,Y_0^c],{[X_0,Y_0]}^c=[X_0^c,Y_0^c],\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {{\ddot{\nabla }}^c}_{X_0^c}{Y}_0^c& ={\left({\ddot{\nabla }}_{X_0}{Y}_0\right)}^c,{{\ddot{\nabla }}^c}_{X_0^c}{Y}_0^v={\left({\ddot{\nabla }}_{X_0}{Y}_0\right)}^v.\hfill \end{array}$$

(6)

2.2. Statistical Manifold

In an n-dimensional Riemannian manifold $(M,g_0)$ with Riemannian metric $g_0$, we consider $\ddot{\nabla }$ as an affine connection and $\tilde{\nabla }$ as its Levi-Civita connection. The structure $(M,\ddot{\nabla },g_0)$ is known as a statistical manifold if $\ddot{\nabla }$ satisfy an affine and torsion-free connection and $\ddot{\nabla }{g}_0$ satisfies the Codazzi equation:

$$\left({\ddot{\nabla }}_{X_0}{g}_0\right)(Y_0,Z_0)=\left({\ddot{\nabla }}_{Y_0}{g}_0\right)(X_0,Z_0),$$

(7)

for all $X_0,Y_0,Z_0\in {\chi }_0\left(M\right)$, where ${\chi }_0\left(M\right)$ is the set of all tangent vector fields on M. We know that there exists an affine connection ${\ddot{\nabla }}^{\ast }$, which is the dual of $\ddot{\nabla }$ with respect to $g_0$ such that

$$X_0{g}_0(Y_0,Z_0)=g_0({\ddot{\nabla }}_{X_0},Z_0)+g_0(Y_0,{\ddot{\nabla }}_{X_0}^{\ast }{Z}_0).$$

(8)

Also, the pair of connections $\ddot{\nabla }$ and ${\ddot{\nabla }}^{\ast }$ satisfies ${\left({\ddot{\nabla }}^{\ast }\right)}^{\ast }=\ddot{\nabla };$ one can obtain

$$\tilde{\nabla }=\frac{1}{2}(\ddot{\nabla }+{\ddot{\nabla }}^{\ast }).$$

(9)

The tensor field $K_0$ of type $(1,2)$ on $(M,\ddot{\nabla },g_0)$ is defined by

$$K_0(X_0,Y_0)={\ddot{\nabla }}_{X_0}{Y}_0-{\tilde{\nabla }}_{X_0}{Y}_0,K_0(X_0,Y_0)=\frac{1}{2}\left({\ddot{\nabla }}_{X_0}{Y}_0-{\ddot{\nabla }}_{X_0}^{\ast }{Y}_0\right),$$

(10)

and $K_0$ is symmetric, which gives

$$g_0(K_0(X_0,Y_0),Z_0)=g_0(K_0(X_0,Z_0),Y_0),K_0(X_0,Y_0)=K_0(Y_0,X_0).$$

(11)

The statistical curvature tensor field associated with $\ddot{\nabla }$ is defined as

$$\ddot{R}(X_0,Y_0)Z_0={\ddot{\nabla }}_{X_0}{\ddot{\nabla }}_{Y_0}{Z}_0-{\ddot{\nabla }}_{Y_0}{\ddot{\nabla }}_{X_0}{Z}_0-{\ddot{\nabla }}_{[X_0,Y_0]}{Z}_0.$$

(12)

By replacing $\ddot{\nabla }$ with ${\ddot{\nabla }}^{\ast }$, we can obtain the statistical curvature tensor field ${\ddot{R}}^{\ast }$. The curvature tensor fields $\ddot{R}$ and ${\ddot{R}}^{\ast }$ satisfy

$$\begin{array}{c}\hfill \ddot{R}(X_0,Y_0)Z_0=-\ddot{R}(Y_0,X_0)Z_0,{\ddot{R}}^{\ast }(X_0,Y_0)Z_0=-{\ddot{R}}^{\ast }(Y_0,X_0)Z_0,\end{array}$$

(13)

$$\begin{array}{c}\hfill g_0(\ddot{R}(X_0,Y_0)Z_0,W_0)=-g_0({\ddot{R}}^{\ast }(X_0,Y_0)W_0,Z_0),\end{array}$$

(14)

$$\begin{array}{c}\hfill \ddot{R}(X_0,Y_0)Z_0+\ddot{R}(Y_0,Z_0)X_0+\ddot{R}(Z_0,X_0)Y_0=0.\end{array}$$

(15)

2.3. Sasakian Statistical Manifolds

Let M be a $(2n+1)$-dimensional differentiable manifold, and it is known to admit an almost contact Riemannian structure $({\mathrm{\Phi }}_0,{\omega }_0,A,g_0)$, where ${\mathrm{\Phi }}_0$ is a $(1,1)$ tensor field, A is a vector field, ${\omega }_0$ is a one-form, and $g_0$ is a Riemannian metric on M such that

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{0}A=0,{\omega }_0\left(A\right)=1,g(A,X_0)={\omega }_0\left(X_0\right),\end{array}$$

(16)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^2{X}_0=-X_0+{\omega }_0\left(X_0\right)A,\end{array}$$

(17)

$$\begin{array}{c}\hfill g_0({\mathrm{\Phi }}_0{X}_0,{\mathrm{\Phi }}_0{Y}_0)=g_0(X_0,Y_0)-{\omega }_0\left(X_0\right){\omega }_0\left(Y_0\right),\end{array}$$

(18)

for all vector fields $X_0,Y_0$ on M. Also, if $({\mathrm{\Phi }}_0,{\omega }_0,A,g_0)$ satisfy

$$\begin{array}{c}\hfill d{\omega }_0=0,{\tilde{\nabla }}_{X_0}A={\mathrm{\Phi }}_0{X}_0,\end{array}$$

(19)

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{X_0}{\mathrm{\Phi }}_0\right)Y_0={\omega }_0\left(Y_0\right)X_0-g_0(X_0,Y_0)A,\end{array}$$

(20)

then M is called a Sasakian manifold [6,26].

A quadruple $(\ddot{\nabla },g_0,{\mathrm{\Phi }}_0,A)$ is known as a Sasakian statistical structure on M, if $(\ddot{\nabla },g_0)$ is a statistical structure and $(g_0,{\mathrm{\Phi }}_0,A)$ is a Sasakian structure on M and the formula:

$$K_0(X_0,{\mathrm{\Phi }}_0{Y}_0)+{\mathrm{\Phi }}_0{K}_0(X_0,Y_0)=0,$$

(21)

holds for any vector fields $X_0$ and $Y_0$ on M [6]. In a statistical structure $(\ddot{\nabla },g_0)$ and an almost-contact metric structure $(g_0,{\mathrm{\Phi }}_0,A)$ on M, the structure $(\ddot{\nabla },g_0,{\mathrm{\Phi }}_0,A)$ is known as a Sasakian statistical structure if and only if it satisfies the following formulas [9,27]:

$$\begin{array}{c}\hfill {\ddot{\nabla }}_{X_0}{\mathrm{\Phi }}_0{Y}_0-{\mathrm{\Phi }}_0{\ddot{\nabla }}_{X_0}^{\ast }{Y}_0=g_0(Y_0,A)X_0-g_0(Y_0,X_0)A,\end{array}$$

(22)

$$\begin{array}{c}\hfill {\ddot{\nabla }}_{X_0}A={\mathrm{\Phi }}_0{X}_0+g_0({\ddot{\nabla }}_{X_0}A,A)A.\end{array}$$

(23)

3. Statistical Manifold and Sasakian Statistical Manifolds in the Tangent Bundle

In this section, we obtain the complete lifts of the statistical manifolds and Sasakian statistical manifolds to the tangent bundle.

Suppose $T_{0}M$ is the tangent bundle and $X_0=X_0^i\frac{\partial }{\partial x^i}$ is a local vector field on M; then, its vertical and complete lifts in terms of partial differential equations are:

$$\begin{array}{c}\hfill X_0^v=X_0^i\frac{\partial }{\partial y^i},\end{array}$$

(24)

$$\begin{array}{c}\hfill X_0^c=X_0^i\frac{\partial }{\partial x^i}+\frac{\partial X_0^i}{\partial x^j}{y}^j\frac{\partial }{\partial y^i}.\end{array}$$

(25)

Let the tangent bundle be denoted by $T_{0}M$. Then, obtaining the complete lifts on Equations (7)–(15) by mathematical operators, we obtain

$$\begin{array}{c}\hfill \left({\ddot{\nabla }}_{X_0^c}^c{g}_0^c\right)(Y_0^c,Z_0^c)=\left({\ddot{\nabla }}_{Y_0^c}^c{g}_0^c\right)(X_0^c,Z_0^c),\end{array}$$

(26)

$$\begin{array}{c}\hfill X_0^c{g}_0^c(Y_0^c,Z_0^c)=g_0^c({\ddot{\nabla }}_{X_0^c}^c{Y}_0^c,Z_0^c)+g_0^c(Y_0^c,{\ddot{\nabla }}_{X_0^c}^{\ast c}{Z}_0^c),\end{array}$$

(27)

$$\begin{array}{c}\hfill {\tilde{\nabla }}^c=\frac{1}{2}({\ddot{\nabla }}^c+{\nabla }^{\ast c}),\end{array}$$

(28)

$$\begin{array}{c}\hfill K_0^c(X_0^c,Y_0^c)={\ddot{\nabla }}_{X_0^c}^c{Y}_0^c-{\tilde{\nabla }}_{X_0^c}^c{Y}_0^c,\end{array}$$

(29)

$$\begin{array}{c}\hfill K_0^c(X_0^c,Y_0^c)=\frac{1}{2}({\ddot{\nabla }}_{X_0^c}^c{Y}_0^c-{\ddot{\nabla }}_{X_0^c}^{\ast c}{Y}_0^c),\end{array}$$

(30)

$$\begin{array}{c}\hfill g_0^c(K_0^c(X_0^c,Y_0^c),Z_0^c)=g_0^c(K_0^c(X_0^c,Z_0^c),Y_0^c),K_0^c(X_0^c,Y_0^c)=K_0^c(Y_0^c,X_0^c),\end{array}$$

(31)

$$\begin{array}{c}\hfill {\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c={\ddot{\nabla }}_{X_0^c}^c{\ddot{\nabla }}_{Y_0^c}^c{Z}_0^c-{\ddot{\nabla }}_{Y_0^c}^c{\ddot{\nabla }}_{X_0^c}^c{Z}_0^c-{\ddot{\nabla }}_{[X_0^c,Y_0^c]}^c,\end{array}$$

(32)

$$\begin{array}{c}\hfill {\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c=-{\ddot{R}}^c(Y_0^c,X_0^c)Z_0^c,{\ddot{R}}^{\ast c}(X_0^c,Y_0^c)Z_0^c=-{\ddot{R}}^{\ast c}(Y_0^c,X_0^c)Z_0^c,\end{array}$$

(33)

$$\begin{array}{c}\hfill g_0^c({\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c,W_0^c)=-g_0^c({\ddot{R}}^{\ast c}(X_0^c,Y_0^c)W_0^c,Z_0^c),\end{array}$$

(34)

$${\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c+{\ddot{R}}^c(Y_0^c,Z_0^c)X_0^c+{\ddot{R}}^c(Z_0^c,X_0^c)Y_0^c=0.$$

(35)

Similarly, obtaining complete lifts on Equations (16)–(23) by mathematical operators, we obtain

$$\begin{array}{cc}\hfill {\left({\mathrm{\Phi }}_{0}A\right)}^c& =0,{\omega }_0^c\left(A^c\right)=1,g_0^c(A^c,X_0^c)={\omega }_0^c\left(X_0^c\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\left({\mathrm{\Phi }}_0^2{X}_0\right)}^c& =-X_0^c+{\omega }_0^c\left(X_0^c\right)A^v+{\omega }_0\left(X_0^c\right)A^c,\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,{\left({\mathrm{\Phi }}_0{Y}_0\right)}^c\right)& =& g_0^c(X_0^c,Y_0^c)-{\omega }_0^c\left(X_0^c\right){\omega }_0^v\left(Y_0^c\right)\hfill \\ & & -{\omega }_0^v\left(X_0^c\right){\omega }_0^c\left(Y_0^c\right),\hfill \end{array}$$

(38)

$$d{\omega }_0^c=0,{\tilde{\nabla }}_{X_0^c}^c{A}^c={\left({\mathrm{\Phi }}_0{X}_0\right)}^c,$$

(39)

$$\begin{array}{ccc}\hfill \left({\tilde{\nabla }}_{X_0^c}^c{\mathrm{\Phi }}_0^c\right)Y_0^c& =& {\omega }_0^c\left(Y_0^c\right)X_0^v+{\omega }_0^v\left(Y_0^c\right)X_0^c-g_0^c(X_0^c,Y_0^c)A^v\hfill \\ & & -g_0^v(X_0^c,Y_0^c)A^c,\hfill \end{array}$$

(40)

$$K_0^c\left(X_0^c,{\left({\mathrm{\Phi }}_0{Y}_0\right)}^c\right)+{\mathrm{\Phi }}_0^c{K}_0^c(X_0^c,Y_0^c)=0,$$

(41)

$$\begin{array}{ccc}\hfill {\ddot{\nabla }}_{X_0^c}^c{\mathrm{\Phi }}_0^c{Y}_0^c-{\mathrm{\Phi }}_0^c{\ddot{\nabla }}_{X_0^c}^{\ast c}{Y}_0^c& =& g_0^c(Y_0^c,A^c)X_0^v+g_0^c(Y_0^v,A^c)X_0^c\hfill \\ & & -g_0^c(Y_0^c,X_0^c)A^v-g_0^c(Y_0^v,X_0^c)A^c,\hfill \end{array}$$

(42)

$${\ddot{\nabla }}_{X_0^c}^c{A}^c={\mathrm{\Phi }}_0^c{X}_0^c+g_0^c(A^c,{\ddot{\nabla }}_{X_0^c}^c{A}^c)A^v+g_0^c(A^v,{\ddot{\nabla }}_{X_0^c}^c{A}^c)A^c.$$

(43)

4. Semi-Symmetric Metric Connection in the Tangent Bundle

Let M be an n-dimensional Riemannian manifold; the linear connection $\breve{\nabla }$ on M is given by [3]

$${\breve{\nabla }}_{X_0}{Y}_0={\tilde{\nabla }}_{X_0}{Y}_0+{\omega }_0\left(Y_0\right)X_0-g_0(X_0,Y_0)A-K_0(X_0,Y_0),$$

(44)

for all vector fields $X_0$ and $Y_0$$\in \chi \left(M^n\right)$, and ${\omega }_0$ is a one-form associated with vector field A and defined by

$${\omega }_0\left(X_0\right)=g_0(X_0,A).$$

(45)

By using (10) in (44), we obtain

$${\breve{\nabla }}_{X_0}{Y}_0={\tilde{\nabla }}_{X_0}^{\ast }{Y}_0+{\omega }_0\left(Y_0\right)X_0-g_0(X_0,Y_0)A+K_0(X_0,Y_0),$$

(46)

The torsion tensor is given by

$$\breve{T}(X_0,Y_0)={\omega }_0\left(Y_0\right)X_0-{\omega }_0\left(X_0\right)Y_0.$$

(47)

A linear connection satisfying (47) is called a semi-symmetric connection (SSC). For any vector fields $X_0,Y_0,Z_0$ on M, we have

$$\left({\breve{\nabla }}_{X_0}{g}_0\right)(Y_0,Z_0)=X_0{g}_0(Y_0,Z_0)-g_0({\breve{\nabla }}_{X_0}{Y}_0,Z_0)-g_0(Y_0,{\breve{\nabla }}_{X_0}{Z}_0).$$

(48)

Using (8), (11), and (44), we obtain

$$\left({\breve{\nabla }}_{X_0}{g}_0\right)(Y_0,Z_0)=0.$$

(49)

Hence, a linear connection $\breve{\nabla }$ defined by (44) satisfies (47) and (49) and is called an SSMC.

Let the tangent bundle be denoted by $T_{0}M$, and we obtain the complete lifts of Equations (44)–(49) by mathematical operators; we obtain

$$\begin{array}{ccc}\hfill {\breve{\nabla }}_{X_0^c}^c{Y}_0^c& =& {\tilde{\nabla }}_{X_0^c}^c{Y}_0^c+{\omega }_0^c\left(Y_0^c\right)X_0^v+{\omega }_0^v\left(Y_0^c\right)X_0^c-g_0^c(X_0^c,Y_0^c)A^v\hfill \\ & & -g_0^c(X_0^v,Y_0^c)A^c-K_0^c(X_0^c,Y_0^c),\hfill \end{array}$$

(50)

$${\omega }_0^c\left(X_0^c\right)=g_0^c(X_0^c,A^c),$$

(51)

$$\begin{array}{ccc}\hfill {\breve{\nabla }}_{X_0^c}^c{Y}_0^c& =& {\tilde{\nabla }}_{X_0^c}^{\ast c}{Y}_0^c+{\omega }_0^c\left(Y_0^c\right)X_0^v+{\omega }_0^v\left(Y_0^c\right)X_0^c-g_0^c(X_0^c,Y_0^c)A^v\hfill \\ & & -g_0^c(X_0^v,Y_0^c)A^c+K_0^c(X_0^c,Y_0^c),\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill {\breve{T}}^c(X_0^c,Y_0^c)& =& {\omega }_0^c\left(Y_0^c\right)X_0^v+{\omega }_0^v\left(Y_0^c\right)X_0^c-{\omega }_0^c\left(X_0^c\right)Y_0^v\hfill \\ & & -{\omega }_0^v\left(X_0^c\right)Y_0^c,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill \left({\breve{\nabla }}_{X_0^c}^c{g}^c\right)(Y_0^c,Z_0^c)& =X_0^c{g}^c(Y_0^v,Z_0^c)+X_0^v{g}^c(Y_0^c,Z_0^c)\hfill \\ \hfill & -g^c({\breve{\nabla }}_{X_0^c}^c{Y}_0^c,Z_0^c)-g^c(Y_0^c,{\breve{\nabla }}_{X_0^c}^c{Z}_0^c),\hfill \end{array}$$

(54)

$$\left({\breve{\nabla }}_{X_0^c}^c{g}_0^c\right)(Y_0^c,Z_0^c)=0.$$

(55)

Theorem 1.

Let M be a Sasakian statistical manifold admitting an SSMC in the tangent bundle $T_{0}M$, which satisfies (53) and (55), then the SSMC of the Sasakian statistical manifold in the tangent bundle is given by (50) and satisfies (52).

Proof. 

Let $({\breve{\nabla }}^c$ be a metric connection in the tangent bundle satisfying (53) on a Sasakian statistical manifold M defined by

$${\breve{\nabla }}_{X_0^c}^c{Y}_0^c={\ddot{\nabla }}_{X_0^c}^c{Y}_0^c+H^c(X_0^c,Y_0^c),$$

(56)

where ${\ddot{\nabla }}^c$ is a torsion-free connection and $H^c$ is the tensor field of type $(1,2)$ in the tangent bundle. Using (27) and (56), we obtain

$$\begin{array}{cc}\hfill 0& =\left({\breve{\nabla }}_{X_0^c}^c{g}^c\right)(Y_0^c,Z_0^c)\hfill \\ \hfill & =X_0^c{g}^c(Y_0^c,Z_0^c)-g^c({\breve{\nabla }}_{X_0^c}^c{Y}_0^c,Z_0^c)-g^c(Y_0^c,{\breve{\nabla }}_{X_0^c}^c{Z}_0^c)\hfill \\ \hfill & =X_0^c{g}^c(Y_0^c,Z_0^c)-g^c\left({\breve{\nabla }}_{X_0^c}^c{Y}_0^c+H^c(X_0^c,Y_0^c),Z_0^c\right)\hfill \\ \hfill & -g^c\left(Y_0^c,{\breve{\nabla }}_{X_0^c}^c{Z}_0^c+H^c(X_0^c,Z_0^c)\right)\hfill \\ \hfill & =-2g^c\left(K_0^c(X_0^c,Z_0^c),Y_0^c\right)-g^c\left(H^c(X_0^c,Y_0^c),Z_0^c\right)\hfill \\ \hfill & -g^c\left(H^c(X^c,Z^c),Y_0^c\right).\hfill \end{array}$$

(57)

So,

$$g^c\left(H^c(X^c,Y^c),Z^c\right)+g^c\left(H^c(X_0^c,Z_0^c),Y_0^c\right)=-2g^c\left(K_0^c(X_0^c,Z_0^c),Y_0^c\right).$$

(58)

Now, using (56), we obtain

$${\breve{T}}^c(X_0^c,Y_0^c)=H^c(X_0^c,Y_0^c)-H^c(Y_0^c,X_0^c).$$

(59)

By using (53), we have

$$\begin{array}{cc}\hfill & g^c\left({\breve{T}}^c(X_0^c,Y_0^c),Z_0^c\right)+g^c\left({\breve{T}}^c(Z_0^c,X_0^c),Y_0^c\right)+g^c\left({\breve{T}}^c(Z_0^c,Y_0^c),X_0^c\right)\hfill \\ \hfill & =g^c\left(H^c(X_0^c,Y_0^c)-H^c(Y_0^c,X_0^c),Z_0^c\right)+g^c\left(H^c(Z_0^c,X_0^c)-H^c(X_0^c,Z_0^c),Y_0^c\right)\hfill \\ \hfill & +g^c\left(H^c(Z_0^c,Y_0^c)-H^c(Y_0^c,Z_0^c),X_0^c\right)\hfill \\ \hfill & =2\left(g^c(H^c(X_0^c,Y_0^c),Z_0^c)+g^c(K_0^c(X_0^c,Z_0^c),Y_0^c)\right).\hfill \end{array}$$

(60)

Using (53) in (60), we obtain

$$\begin{array}{cc}\hfill g^c\left(H^c(X_0^c,Y_0^c),Z_0^c\right)& =\frac{1}{2}\left[g^c\right({\omega }^c\left(Y_0^c\right)X_0^v+{\omega }^v\left(Y_0^c\right)X_0^c\hfill \\ \hfill & -{\omega }^c\left(X_0^c\right)Y_0^v-{\omega }^v\left(X_0^c\right)Y_0^c,Z_0^c)\hfill \\ \hfill & +g^c({\omega }^c\left(X_0^c\right)Z_0^v+{\omega }^v\left(X_0^c\right)Z_0^c\hfill \\ \hfill & -{\omega }^c\left(Z_0^c\right)X_0^v-{\omega }^v\left(Z_0^c\right)X_0^c,Y_0^c)\hfill \\ \hfill & +g^c({\omega }^c\left(Y_0^c\right)Z_0^v+{\omega }^v\left(Y_0^c\right)Z_0^c\hfill \\ \hfill & -{\omega }^c\left(Z_0^c\right)Y_0^v-{\omega }^v\left(Z_0^c\right)Y_0^c,X_0^c\left)\right]\hfill \\ \hfill & -g^c\left(K_0^c(X_0^c,Z_0^c),Y_0^c\right).\hfill \end{array}$$

(61)

Hence, we obtain

$$H^c(X_0^c,Y_0^c)={\omega }^c\left(Y_0^c\right)X_0^v+{\omega }^v\left(Y_0^c\right)X_0^c-g^c(X_0^c,Y_0^c)A^v-g^c(X_0^v,Y_0^c)A^c-K_0^c(X_0^c,Y_0^c).$$

(62)

Further, by using Equations (29), (30), and (50), we obtain (52). □

5. Curvature Tensor of Semi-Symmetric Metric Connection on Statistical Manifolds in the Tangent Bundle

In an n-dimensional Riemannian manifold M and the statistical structure $({\ddot{\nabla }}^c,g_0^c)$ on M in the tangent bundle $T_{0}M$, we have the relation between the complete lifts of the SSMC ${\breve{\nabla }}^c$ and torsion-free connection ${\ddot{\nabla }}^c$ in the tangent bundle from (29) and (50) by

$$\begin{array}{cc}\hfill {\breve{\nabla }}_{X_0^c}^c{Y}_0^c& ={\ddot{\nabla }}_{X_0^c}^c{Y}_0^c+{\omega }_0^c\left(Y_0^c\right)X_0^v+{\omega }_0^v\left(Y_0^c\right)X_0^c\hfill \\ \hfill & -g_0^c(X_0^c,Y_0^c)A^v-g_0^c(X_0^v,Y_0^c)A^c-K_0^c(X_0^c,Y_0^c).\hfill \end{array}$$

(63)

The complete lifts of the Riemannian curvature tensor ${\breve{R}}^c$ of M associated with the SSMC ${\breve{\nabla }}^c$ in the tangent bundle are given by

$${\breve{R}}^c(X_0^c,Y_0^c)Z_0^c={\breve{\nabla }}_{X_0^c}^c{\breve{\nabla }}_{Y_0^c}^c{Z}_0^c-{\breve{\nabla }}_{Y_0^c}^c{\breve{\nabla }}_{X_0^c}^c{Z}_0^c-{\breve{\nabla }}_{[X_0^c,Y_0^c]}^c{Z}_0^c,$$

(64)

Using Equation (63) in (64), we obtain

$$\begin{array}{ccc}\hfill {\breve{R}}^c(X_0^c,Y_0^c)Z_0^c& & ={\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c+({\omega }_0^c\left(X_0^c\right)A^v+{\omega }_0^v\left(X_0^c\right)A^c\hfill \\ & & -{\omega }_0^c\left(A_0^c\right)X_0^v-{\omega }_0^v\left(A_0^c\right)X_0^c-{\ddot{\nabla }}_{X_0^c}^c{A}^c+K_0^c(X_0^c,A^c))\hfill \\ & & g_0^c(Y_0^c,Z_0^c)-({\omega }_0^c\left(Y_0^c\right)A^v+{\omega }_0^v\left(Y_0^c\right)A^c-{\omega }_0^c\left(A_0^c\right)Y_0^v\hfill \\ & & -{\omega }_0^v\left(A_0^c\right)Y_0^c-{\ddot{\nabla }}_{Y_0^c}^c{A}^c+K_0^c(Y_0^c,A^c))g_0^c(X_0^c,Z_0^c)\hfill \\ & & -g_0^c({\omega }_0^c\left(X_0^c\right)A^v+{\omega }_0^v\left(X_0^c\right)A^c-{\ddot{\nabla }}_{X_0^c}^c{A}^c+K_0^c(X_0^c,A^c)\hfill \\ & & ,Z_0^c)Y_0^c+g_0^c({\omega }_0^c\left(Y_0^c\right)A^v+{\omega }_0^v\left(Y_0^c\right)A^c-{\ddot{\nabla }}_{Y_0^c}^c{A}^c\hfill \\ & & +K_0^c(Y_0^c,A^c),Z_0^c)X_0^c-\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(Y_0^c,Z_0^c)\hfill \\ & & +\left({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c\right)(X_0^c,Z_0^c)+K_0^c\left(X_0^c,K_0^c(Y_0^c,Z_0^c)\right)\hfill \\ & & -K_0^c\left(Y_0^c,K_0^c(X_0^c,Z_0^c)\right),\hfill \end{array}$$

(65)

where ${\ddot{R}}^c$ is the complete lifts of the curvature tensor of M associated with torsion-free connection ${\ddot{\nabla }}^c$ in the tangent bundle $T_{0}M$ and is defined as

$${\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c={\ddot{\nabla }}_{X_0^c}^c{\ddot{\nabla }}_{Y_0^c}^c{Z}_0^c-{\ddot{\nabla }}_{Y_0^c}^c{\ddot{\nabla }}_{X_0^c}^c{Z}_0^c-{\ddot{\nabla }}_{[X_0^c,Y_0^c]}^c{Z}_0^c,$$

(66)

Similarly, we obtain the complete lifts of the relation between the SSMC ${\breve{\nabla }}^c$ and dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle from Equations (30) and (63) as

$$\begin{array}{cc}\hfill {\breve{\nabla }}_{X_0^c}^c{Y}_0^c& ={\ddot{\nabla }}_{X_0^c}^{\ast c}{Y}_0^c+{\omega }_0^c\left(Y_0^c\right)X_0^v+{\omega }_0^v\left(Y_0^c\right)X_0^c-g_0^c(X_0^c,Y_0^c)A^v\hfill \\ \hfill & -g_0^c(X_0^v,Y_0^c)A^c+K_0^c(X_0^c,Y_0^c).\hfill \end{array}$$

(67)

The relation between the complete lifts of the Riemannian curvature tensor ${\breve{R}}^c$ associated with the SSMC ${\breve{\nabla }}^c$ and Riemannian curvature tensor ${\breve{R}}^{\ast c}$ associated with the dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ is obtain by using Equation (67) in (64) by

$$\begin{array}{cc}\hfill {\breve{R}}^c(X_0^c,Y_0^c)& Z_0^c={\ddot{R}}^{\ast c}(X_0^c,Y_0^c)Z_0^c+({\omega }_0^c\left(X_0^c\right)A^v+{\omega }_0^v\left(X_0^c\right)A^c\hfill \\ \hfill & -{\omega }_0^c\left(A_0^c\right)X_0^v-{\omega }_0^v\left(A_0^c\right)X_0^c-{\ddot{\nabla }}_{X_0^c}^{\ast c}{A}^c-K_0^c(X_0^c,A^c))\hfill \\ \hfill & g_0^c(Y_0^c,Z_0^c)-({\omega }_0^c\left(Y_0^c\right)A^v+{\omega }_0^v\left(Y_0^c\right)A^c-{\omega }_0^c\left(A_0^c\right)Y_0^v\hfill \\ \hfill & -{\omega }_0^v\left(A_0^c\right)Y_0^c-{\ddot{\nabla }}_{Y_0^c}^{\ast c}{A}^c-K_0^c(Y_0^c,A^c))g_0^c(X_0^c,Z_0^c)\hfill \\ \hfill & -g_0^c({\omega }_0^c\left(X_0^c\right)A^v+{\omega }_0^v\left(X_0^c\right)A^c-{\ddot{\nabla }}_{X_0^c}^{\ast c}{A}^c-K_0^c(X_0^c,A^c)\hfill \\ \hfill & ,Z_0^c)Y_0^c+g_0^c({\omega }_0^c\left(Y_0^c\right)A^v+{\omega }_0^v\left(Y_0^c\right)A^c-{\ddot{\nabla }}_{Y_0^c}^{\ast c}{A}^c\hfill \\ \hfill & -K_0^c(Y_0^c,A^c),Z_0^c)X_0^c+\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(Y_0^c,Z_0^c)\hfill \\ \hfill & -\left({\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c\right)(X_0^c,Z_0^c)+K_0^c\left(X_0^c,K_0^c(Y_0^c,Z_0^c)\right)\hfill \\ \hfill & -K_0^c\left(Y_0^c,K_0^c(X_0^c,Z_0^c)\right),\hfill \end{array}$$

(68)

where ${\ddot{R}}^{\ast c}$ is the complete lifts of the Riemannian curvature tensor of M associated with the lift of the dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ and is defined as

$${\ddot{R}}^{\ast c}(X_0^c,Y_0^c)Z_0^c={\ddot{\nabla }}_{X_0^c}^{\ast c}{\ddot{\nabla }}_{Y_0^c}^{\ast c}{Z}_0^c-{\ddot{\nabla }}_{Y_0^c}^{\ast c}{\ddot{\nabla }}_{X_0^c}^{\ast c}{Z}_0^c-{\ddot{\nabla }}_{[X_0^c,Y_0^c]}^{\ast c}{Z}_0^c.$$

(69)

Theorem 2.

In a Riemannian manifold M, let the complete lifts of the statistical structure be $({\ddot{\nabla }}^c,g_0^c)$ in the tangent bundle $T_{0}M$. Then:

1.

The relation between the lifts of the Riemannian curvature tensor ${\breve{R}}^c$ of SSMC ${\breve{\nabla }}^c$ and the Riemannian curvature tensor ${\ddot{R}}^c$ of torsion-free connection ${\ddot{\nabla }}^c$ in the tangent bundle $T_{0}M$ is given by Equation (65).

2.

The relation between the lifts of the Riemannian curvature tensor ${\breve{R}}^c$ of SSMC ${\breve{\nabla }}^c$ and the Riemannian curvature tensor ${\ddot{R}}^{\ast c}$ of dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ is given by Equation (68).

Proposition 1.

For a statistical manifold in the tangent bundle, the following relations hold:

(i)

$-g_0^c({\ddot{\nabla }}_{X_0^c}^c{A}_0^c,W_0^c)+g_0^c(K_0^c(X_0^c,A_0^c),W_0^c)=-g_0^c({\ddot{\nabla }}_{X_0^c}^{\ast c}{A}_0^c,W_0^c)+g_0^c(K_0^c(X_0^c,A_0^c),W_0^c)$.

(ii)

$g_0^c(\left({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c\right)(X_0^c,Z_0^c),W_0^c)-g_0^c(\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(Y_0^c,Z_0^c),W_0^c)=g_0^c(\left({\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c\right)(X_0^c,W_0^c),Z_0^c)-g_0^c(\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(Y_0^c,W_0^c),Z_0^c)$.

(iii)

$g_0^c(K_0^c(X_0^c,K_0^c(Y_0^c,Z_0^c)),W_0^c)=g_0^c(K_0^c(Y_0^c,K_0^c(X_0^c,W_0^c)),Z_0^c)$.

Proof. 

Using (29) and (30), we obtain (i). For (ii), we use (31) and prove that

$$\begin{array}{cc}\hfill g_0^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^c{X}_0^c,Z_0^c),W_0^c)-& g_0^c(K_0^c({\ddot{\nabla }}_{X_0^c}^c{Y}_0^c,Z_0^c),W_0^c)=g_0^c(K_0^c([Y_0^c,X_0^c],Z_0^c),W_0^c)\hfill \\ \hfill & =g_0^c(K_0^c([Y_0^c,X_0^c],W_0^c),Z_0^c)\hfill \\ \hfill & =g_0^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^{\ast c}{X}_0^c,W_0^c),Z_0^c)-g_0^c(K_0^c({\ddot{\nabla }}_{X_0^c}^{\ast c}{Y}_0^c,W_0^c),Z_0^c).\hfill \end{array}$$

(70)

Now, using (27) and (31), we obtain

$$\begin{array}{cc}\hfill & g_0^c(\left({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c\right)(X_0^c,Z_0^c),W_0^c)\hfill \\ \hfill & =g_0^c({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c(X_0^c,Z_0^c),W_0^c)-g_0^c(K_0^c(X_0^c,{\ddot{\nabla }}_{Y_0^c}^c{Z}_0^c),W_0^c)-g_0^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^c{X}_0^c,Z_0^c),W_0^c)\hfill \\ \hfill & =Y_0^c{g}_0^c(W_0^v,K_0^c(X_0^c,Z_0^c))+Y_0^v{g}_0^c(W_0^c,K_0^c(X_0^c,Z_0^c))-g_0^c(K_0^c(X_0^c,Z_0^c),{\ddot{\nabla }}_{Y_0^c}^{\ast c}{W}_0^c)\hfill \\ \hfill & -g_0^c(K_0^c(X_0^c,W_0^c),{\ddot{\nabla }}_{Y_0^c}^c{Z}_0^c)-g_0^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^c{X}_0^c,Z_0^c),W_0^c)\hfill \\ \hfill & =Y_0^c{g}_0^c(W_0^v,K_0^c(X_0^c,Z_0^c))+Y_0^v{g}_0^c(W_0^c,K_0^c(X_0^c,Z_0^c))-g_0^c(K_0^c(X_0^c,{\ddot{\nabla }}_{Y_0^c}^{\ast c}{W}_0^c),Z_0^c)\hfill \\ \hfill & -Y_0^c{g}_0^c(Z_0^v,K_0^c(X_0^c,W_0^c))-Y_0^v{g}_0^c(Z_0^c,K_0^c(X_0^c,W_0^c))+g_0^c(Z_0^c,{\ddot{\nabla }}^{\ast c}{K}_0^c(X_0^c,W_0^c))\hfill \\ \hfill & -g^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^c{X}_0^c,Z_0^c),W_0^c)\hfill \\ \hfill & =g^c(Z_0^c,{\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c(X_0^c,W_0^c))-g^c(K_0^c(X_0^c,{\ddot{\nabla }}_{Y_0^c}^{\ast c}{W}_0^c),Z_0^c)-g_0^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^c{X}_0^c,Z_0^c),W_0^c).\hfill \end{array}$$

(71)

From (70) and (71), we obtain

$$\begin{array}{cc}\hfill & g_0^c(\left({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c\right)(X_0^c,Z_0^c),W_0^c)-g_0^c(\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(Y_0^c,Z_0^c),W_0^c)\hfill \\ \hfill & =g_0^c(Z_0^c,{\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c(X_0^c,W_0^c))-g_0^c(K_0^c(X_0^c,{\ddot{\nabla }}_{Y_0^c}^{\ast c}{W}_0^c),Z_0^c)-g_0^c(K_0^c({\ddot{\nabla }}_{Y_0^c}^{\ast c}{X}_0^c,W_0^c),Z_0^c)\hfill \\ \hfill & -g_0^c(Z_0^c,{\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c(Y_0^c,W_0^c))+g_0^c(K_0^c(Y_0^c,{\ddot{\nabla }}_{X_0^c}^{\ast c}{W}_0^c),Z_0^c)+g_0^c(K_0^c({\ddot{\nabla }}_{X_0^c}^{\ast c}{Y}_0^c,W_0^c),Z_0^c)\hfill \\ \hfill & =g_0^c(\left({\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c\right)(X_0^c,W_0^c),Z_0^c)-g_0^c(\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(Y_0^c,W_0^c),Z_0^c).\hfill \end{array}$$

(72)

For (iii), from the symmetry property of ${\ddot{K}}^c$, we deduce

$$\begin{array}{cc}\hfill g_0^c(K_0^c(X_0^c,K_0^c(Y_0^c,Z_0^c)),W_0^c)& =g_0^c(K_0^c(X_0^c,W_0^c),K_0^c(Y_0^c,Z_0^c))\hfill \\ \hfill & =g_0^c(K_0^c(Y_0^c,K_0^c(X_0^c,W_0^c)),Z_0^c).\hfill \end{array}$$

(73)

□

6. Curvature Tensor of Semi-Symmetric Metric Connection on Sasakian Statistical Manifolds in The Tangent Bundle

Let $(M,{\ddot{\nabla }}^c,g_0^c,{\mathrm{\Phi }}_0^c,{\omega }_0^c,A^c)$ be the complete lifts of $(2n+1)$-dimensional Sasakian statistical manifolds in the tangent bundle $T_{0}M$. Then, the complete lifts of the curvature tensors ${\breve{R}}^c$ associated with SSMC ${\breve{\nabla }}^c$ in the tangent bundle are given by

$${\breve{R}}^c(X_0^c,Y_0^c)Z_0^c={\breve{\nabla }}_{X_0^c}^c{\breve{\nabla }}_{Y_0^c}^c{Z}_0^c-{\breve{\nabla }}_{Y_0^c}^c{\breve{\nabla }}_{X_0^c}^c{Z}_0^c-{\breve{\nabla }}_{[X_0^c,Y_0^c]}^c{Z}_0^c.$$

(74)

Using Equations (36)–(43) and (63) in Equation (74), we can obtain the relation of the complete lifts of the curvature tensor ${\breve{R}}^c$ of the SSMC ${\breve{\nabla }}^c$ and the curvature tensor ${\ddot{R}}^c$ of the torsion-free connection ${\ddot{\nabla }}^c$ in the tangent bundle $T_{0}M$ as

$$\begin{array}{cc}\hfill {\breve{R}}^c(X_0^c,Y_0^c& )Z_0^c={\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c+\left({\left({\mathrm{\Phi }}_0^2{X}_0\right)}^c-{\left({\mathrm{\Phi }}_0{X}_0\right)}^c\right)g_0^c(Y_0^c,Z_0^c)\hfill \\ \hfill & -\left({\left({\mathrm{\Phi }}_0^2{Y}_0\right)}^c-{\left({\mathrm{\Phi }}_0{Y}_0\right)}^c\right)g_0^c(X_0^c,Z_0^c)+g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,Z_0^c\right)Y_0^v\hfill \\ \hfill & +g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^v,Z_0^c\right)Y_0^c-g_0^c\left({\left({\mathrm{\Phi }}_0{Y}_0\right)}^c,Z_0^c\right)X_0^v\hfill \\ \hfill & -g_0^c\left({\left({\mathrm{\Phi }}_0{Y}_0\right)}^v,Z_0^c\right)X_0^c-{\omega }_0^c\left(X_0^c\right){\omega }_0^c\left(Z_0^c\right)Y_0^v\hfill \\ \hfill & -{\omega }_0^c\left(X_0^c\right){\omega }_0^v\left(Z_0^c\right)Y_0^c-{\omega }_0^v\left(X_0^c\right){\omega }_0^c\left(Z_0^c\right)Y_0^c\hfill \\ \hfill & +{\omega }_0^c\left(Y_0^c\right){\omega }_0^c\left(Z_0^c\right)X_0^v+{\omega }_0^c\left(Y_0^c\right){\omega }_0^v\left(Z_0^c\right)X_0^c\hfill \\ \hfill & +{\omega }_0^v\left(Y_0^c\right){\omega }_0^c\left(Z_0^c\right)X_0^c-\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(Y_0^c,Z_0^c)\hfill \\ \hfill & +\left({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c\right)(X_0^c,Z_0^c)+K_0^c\left(X_0^c,K_0^c(Y_0^c,Z_0^c)\right)\hfill \\ \hfill & -K_0^c\left(Y_0^c,K_0^c(X_0^c,Z_0^c)\right).\hfill \end{array}$$

(75)

Similarly, we obtain the relation of the complete lifts of the curvature tensor ${\breve{R}}^c$ of the SSMC ${\breve{\nabla }}^c$ and the curvature tensor ${\ddot{R}}^{\ast c}$ of the dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ by using Equations (36)–(43) and (67) in (74) as

$$\begin{array}{cc}\hfill {\breve{R}}^c(X_0^c,Y_0^c& )Z_0^c={\ddot{R}}^{\ast c}(X_0^c,Y_0^c)Z_0^c+\left({\left({\mathrm{\Phi }}_0^2{X}_0\right)}^c-{\left({\mathrm{\Phi }}_0{X}_0\right)}^c\right)g_0^c(Y_0^c,Z_0^c)\hfill \\ \hfill & -\left({\left({\mathrm{\Phi }}_0^2{Y}_0\right)}^c-{\left({\mathrm{\Phi }}_0{Y}_0\right)}^c\right)g_0^c(X_0^c,Z_0^c)+g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,Z_0^c\right)Y_0^v\hfill \\ \hfill & +g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^v,Z_0^c\right)Y_0^c-g_0^c\left({\left({\mathrm{\Phi }}_0{Y}_0\right)}^c,Z_0^c\right)X_0^v\hfill \\ \hfill & -g_0^c\left({\left({\mathrm{\Phi }}_0{Y}_0\right)}^v,Z_0^c\right)X_0^c-{\omega }_0^c\left(X_0^c\right){\omega }_0^c\left(Z_0^c\right)Y_0^v\hfill \\ \hfill & -{\omega }_0^c\left(X_0^c\right){\omega }_0^v\left(Z_0^c\right)Y_0^c-{\omega }_0^v\left(X_0^c\right){\omega }_0^c\left(Z_0^c\right)Y_0^c\hfill \\ \hfill & +{\omega }_0^c\left(Y_0^c\right){\omega }_0^c\left(Z_0^c\right)X_0^v+{\omega }_0^c\left(Y_0^c\right){\omega }_0^v\left(Z_0^c\right)X_0^c\hfill \\ \hfill & +{\omega }_0^v\left(Y_0^c\right){\omega }_0^c\left(Z_0^c\right)X_0^c+\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(Y_0^c,Z_0^c)\hfill \\ \hfill & -\left({\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c\right)(X_0^c,Z_0^c)+K_0^c\left(X_0^c,K_0^c(Y_0^c,Z_0^c)\right)\hfill \\ \hfill & -K_0^c\left(Y_0^c,K_0^c(X_0^c,Z_0^c)\right).\hfill \end{array}$$

(76)

Proposition 2.

Let $(M,{\ddot{\nabla }}^c,g_0^c,{\mathrm{\Phi }}_0^c,{\omega }_0^c,A^c)$ be the complete lifts of a $(2n+1)$-dimensional Sasakian statistical manifold in the tangent bundle $T_{0}M$. Then, we have:

1.

${\breve{R}}^c(X_0^c,Y_0^c)A^c={\ddot{R}}^c(X_0^c,Y_0^c)A^c+{\omega }_0^v\left(X_0^c\right){\left({\mathrm{\Phi }}_0{Y}_0\right)}^c+{\omega }_0^c\left(X_0^c\right){\left({\mathrm{\Phi }}_0{Y}_0\right)}^v-{\omega }_0^v\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^c-{\omega }_0^c\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^v-\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(Y_0^c,A^c)+\left({\ddot{\nabla }}_{Y_0^c}^c{K}_0^c\right)(X_0^c,A^c)$.

2.

${\breve{R}}^c(X_0^c,Y_0^c)A^c={\ddot{R}}^{\ast c}(X_0^c,Y_0^c)A^c+{\omega }_0^v\left(X_0^c\right){\left({\mathrm{\Phi }}_0{Y}_0\right)}^c+{\omega }_0^c\left(X_0^c\right){\left({\mathrm{\Phi }}_0{Y}_0\right)}^v-{\omega }_0^v\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^c-{\omega }_0^c\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^v+\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(Y_0^c,A^c)-\left({\ddot{\nabla }}_{Y_0^c}^{\ast c}{K}_0^c\right)(X_0^c,A^c)$.

3.

${\breve{R}}^c(A^c,X_0^c)Y_0^c={\ddot{R}}^c(A^c,X_0^c)Y_0^c+{\omega }_0^c\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^v+{\omega }_0^v\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^c-g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,Y_0^c\right)A^v-g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^v,Y_0^c\right)A^c-\left({\ddot{\nabla }}_{A^c}^c{K}_0^c\right)(X_0^c,Y_0^c)+\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(A_0^c,Y_0^c)+K_0^c\left(A^c,K_0^c(X_0^c,Y_0^c)\right)-K_0^c\left(X_0^c,K_0^c(A^c,Y_0^c)\right)$.

4.

${\breve{R}}^c(A^c,X_0^c)Y_0^c={\ddot{R}}^{\ast c}(A^c,X_0^c)Y_0^c+{\omega }_0^c\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^v+{\omega }_0^v\left(Y_0^c\right){\left({\mathrm{\Phi }}_0{X}_0\right)}^c-g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,Y_0^c\right)A^v-g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^v,Y_0^c\right)A^c+\left({\ddot{\nabla }}_{A^c}^{\ast c}{K}_0^c\right)(X_0^c,Y_0^c)-\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(A_0^c,Y_0^c)+K_0^c\left(A^c,K_0^c(X_0^c,Y_0^c)\right)-K_0^c\left(X_0^c,K_0^c(A^c,Y_0^c)\right)$.

Proof. 

In a Sasakian manifold, we have from [6]

$${\ddot{\nabla }}_{X_0}A={\mathrm{\Phi }}_0{X}_0+{\omega }_0\left({\ddot{\nabla }}_{X_0}A\right)A.$$

(77)

Obtaining the complete lifts of Equation (77), we have

$${\ddot{\nabla }}_{X_0^c}^c{A}^c={\left({\mathrm{\Phi }}_0{X}_0\right)}^c+{\omega }_0^c\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right)A^v+{\omega }_0^v\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right)A^c.$$

(78)

From Equation (29), we have

$$K_0^c(X_0^c,A^c)={\omega }_0^c\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right)A^v+{\omega }_0^v\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right)A^c.$$

(79)

So,

$$\begin{array}{cc}\hfill K_0^c\left(X_0^c,K_0^c(Y_0^c,A^c)\right)& ={\omega }_0^c\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right){\omega }_0^c\left({\ddot{\nabla }}_{Y_0^c}^c{A}^c\right)A^v\hfill \\ \hfill & +{\omega }_0^c\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right){\omega }_0^v\left({\ddot{\nabla }}_{Y_0^c}^c{A}^c\right)A^c\hfill \\ \hfill & +{\omega }_0^v\left({\ddot{\nabla }}_{X_0^c}^c{A}^c\right){\omega }_0^c\left({\ddot{\nabla }}_{Y_0^c}^c{A}^c\right)A^c.\hfill \end{array}$$

(80)

Hence, we obtain

$$K_0^c\left(X_0^c,K_0^c(Y_0^c,A^c)\right)=K_0^c\left(Y_0^c,K_0^c(X_0^c,A^c)\right).$$

(81)

Using Equations (36)–(38) and (81) in Equations (75) and (76), we obtain (1)–(4). □

6.1. Ricci Tensor Associated with Semi-Symmetric Metric Connection of Sasakian Statistical Manifolds in the Tangent Bundle

The complete lifts of the Ricci tensor ${\breve{S}}^c$ associated with the connection ${\breve{\nabla }}^c$ in the tangent bundle $T_{0}M$ are given as

$${\breve{S}}^c(X_0^c,Y_0^c)=\mathrm{Trace}\mathrm{of}\mathrm{the}\mathrm{map}:Z_0^c\to {\ddot{R}}^c(X_0^c,Y_0^c)Z_0^c.$$

(82)

Then, from (75) and (76), we have

$$\begin{array}{cc}\hfill {\breve{S}}^c(X_0^c,Y_0^c)& ={\ddot{S}}^c(X_0^c,Y_0^c)-(2n+1)g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,{\left({\mathrm{\Phi }}_0{Y}_0\right)}^c+Y_0^c\right)\hfill \\ \hfill & -\sum _{i=1}^{2n+1}{g}_0^c(\left({\ddot{\nabla }}_{X_0^c}^c{K}_0^c\right)(e_i^c,e_i^c)-\left({\ddot{\nabla }}_{e_i^c}^c{K}_0^c\right)(X_0^c,e_i^c)\hfill \\ \hfill & -K_0^c\left(X_0^c,K_0^c(e_i^c,e_i^c)\right)+K_0^c\left(e_i^c,K_0^c(X_0^c,e_i^c)\right),Y_0^c),\hfill \end{array}$$

(83)

and

$$\begin{array}{cc}\hfill {\breve{S}}^c(X_0^c,Y_0^c)& ={\ddot{S}}^{\ast c}(X_0^c,Y_0^c)-(2n+1)g_0^c\left({\left({\mathrm{\Phi }}_0{X}_0\right)}^c,{\left({\mathrm{\Phi }}_0{Y}_0\right)}^c+Y_0^c\right)\hfill \\ \hfill & -\sum _{i=1}^{2n+1}{g}_0^c(\left({\ddot{\nabla }}_{X_0^c}^{\ast c}{K}_0^c\right)(e_i^c,e_i^c)-\left({\ddot{\nabla }}_{e_i^c}^{\ast c}{K}_0^c\right)(X_0^c,e_i^c)\hfill \\ \hfill & +K_0^c\left(X_0^c,K_0^c(e_i^c,e_i^c)\right)-K_0^c\left(e_i^c,K_0^c(X_0^c,e_i^c)\right),Y_0^c).\hfill \end{array}$$

(84)

Theorem 3.

Let $(M,{\ddot{\nabla }}^c,g_0^c,{\mathrm{\Phi }}_0^c,{\omega }_0^c,A^c)$ be the complete lifts of a $(2n+1)$-dimensional Sasakian statistical manifold in the tangent bundle $T_{0}M$. Then:

1.

The relation between the Ricci tensor ${\breve{S}}^c$ of the SSMC ${\breve{\nabla }}^c$ and the Ricci tensor ${\ddot{S}}^c$ of the torsion-free connection ${\ddot{\nabla }}^c$ in the tangent bundle $T_{0}M$ is given by Equation (83).

2.

The relation between the Ricci tensor ${\breve{S}}^c$ of the SSMC ${\breve{\nabla }}^c$ and the Ricci tensor ${\ddot{S}}^{\ast c}$ of the dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ is given by Equation (84).

6.2. Scalar Curvature Associated with Semi-Symmetric Metric Connection of Sasakian Statistical Manifolds in the Tangent Bundle

From Equations (83) and (84), it follows that

$$\begin{array}{cc}\hfill {\breve{r}}^c& ={\ddot{r}}^c+2n-4n^2-\sum _{i,j=1}^{2n+1}{g}_0^c(\left({\ddot{\nabla }}_{e_j^c}^c{K}_0^c\right)(e_i^c,e_i^c)-\left({\ddot{\nabla }}_{e_i^c}^c{K}_0^c\right)(e_j^c,e_i^c)\hfill \\ \hfill & -K_0^c\left(e_j^c,K_0^c(e_i^c,e_i^c)\right)+K_0^c\left(e_i^c,K_0^c(e_j^c,e_i^c)\right),e_j^c),\hfill \end{array}$$

(85)

and

$$\begin{array}{cc}\hfill {\breve{r}}^c& ={\ddot{r}}^{\ast c}+2n-4n^2+\sum _{i,j=1}^{2n+1}{g}_0^c(\left({\ddot{\nabla }}_{e_j^c}^{\ast c}{K}_0^c\right)(e_i^c,e_i^c)-\left({\ddot{\nabla }}_{e_i^c}^{\ast c}{K}_0^c\right)(e_j^c,e_i^c)\hfill \\ \hfill & +K_0^c\left(e_j^c,K_0^c(e_i^c,e_i^c)\right)-K_0^c\left(e_i^c,K_0^c(e_j^c,e_i^c)\right),e_j^c).\hfill \end{array}$$

(86)

where ${\breve{r}}^c$, ${\ddot{r}}^c$, and ${\ddot{r}}^{\ast c}$ are the complete lifts of the scalar curvatures associated with the SSMC ${\breve{\nabla }}^c$, torsion-free connection ${\ddot{\nabla }}^c$, and dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$, respectively.

Theorem 4.

Let $(M,{\ddot{\nabla }}^c,g_0^c,{\mathrm{\Phi }}_0^c,{\omega }_0^c,A^c)$ be the complete lifts of a $(2n+1)$-dimensional Sasakian statistical manifold in the tangent bundle $T_{0}M$. Then:

1.

The relation between the scalar curvature ${\breve{r}}^c$ of the SSMC ${\breve{\nabla }}^c$ and the scalar curvature ${\ddot{r}}^c$ of the torsion-free connection ${\ddot{\nabla }}^c$ in the tangent bundle $T_{0}M$ is given by Equation (85).

2.

The relation between the scalar curvature ${\breve{r}}^c$ of the SSMC ${\breve{\nabla }}^c$ and the scalar curvature ${\ddot{r}}^{\ast c}$ of the dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ is given by Equation (86).

Theorem 5.

In a $(2n+1)$-dimensional Sasakian statistical manifold in the tangent bundle $T_{0}M$, the complete lift of the Ricci tensor of the Sasakian statistical manifold associated with the SSMC in the tangent bundle $T_{0}M$ is said to be Ricci flat if the complete lift of the scalar curvature with respect to torsion-free connection ${\ddot{\nabla }}^c$ and dual connection ${\ddot{\nabla }}^{\ast c}$ in the tangent bundle $T_{0}M$ satisfies

$$\begin{array}{cc}\hfill {\ddot{r}}^c& =-2n+4n^2+\sum _{i,j=1}^{2n+1}{g}^c(\left({\ddot{\nabla }}_{e_j^c}^c{K}_0^c\right)(e_i^c,e_i^c)-\left({\ddot{\nabla }}_{e_i^c}^c{K}_0^c\right)(e_j^c,e_i^c)\hfill \\ \hfill & -K_0^c\left(e_j^c,K_0^c(e_i^c,e_i^c)\right)+K_0^c\left(e_i^c,K_0^c(e_j^c,e_i^c)\right),e_j^c),\hfill \end{array}$$

(87)

and

$$\begin{array}{cc}\hfill {\ddot{r}}^{\ast c}& =-2n+4n^2-\sum _{i,j=1}^{2n+1}{g}^c(\left({\ddot{\nabla }}_{e_j^c}^{\ast c}{K}_0^c\right)(e_i^c,e_i^c)-\left({\ddot{\nabla }}_{e_i^c}^{\ast c}{K}_0^c\right)(e_j^c,e_i^c)\hfill \\ \hfill & +K_0^c\left(e_j^c,K_0^c(e_i^c,e_i^c)\right)-K_0^c\left(e_i^c,K_0^c(e_j^c,e_i^c)\right),e_j^c).\hfill \end{array}$$

(88)

Proof. 

Treating ${\breve{S}}^c(X_0^c,Y_0^c)=0$ in Equations (83) and (84), then contracting it with respect to $X_0^c$ and $Y_0^c$ in the tangent bundle $T_{0}M$, we obtain the results. □

7. Example

In this section, we shall show an example of the lifts of a Sasakian statistical manifold in the tangent bundle $T_{0}M$. Let us consider M to be a three-dimensional manifold, which is defined as

$$M=\left\{(x_1,x_2,x_3)\in {\mathbb{R}}^3;x_3\ne 0\right\},$$

(89)

where $\mathbb{R}$ is the set of real numbers. Let $x_1,x_2,x_3$ be given by

$$e_1=\frac{\partial }{\partial x_1},e_2=-x_1\left(\frac{\partial }{\partial x_1}-\frac{\partial }{\partial x_3}\right)+\frac{\partial }{\partial x_2},e_3=\frac{1}{2}\frac{\partial }{\partial x_3},$$

where $\{e_1,e_2,e_3\}$ are the linearly independent global frame on M. Let $T_{0}M$ be the tangent bundle, and let the one-form ${\omega }_0$ be given by

$${\omega }_0\left(X_0\right)=g_0(X_0,e_3).$$

The Riemannian metric $g_0$ is defined by

$$g_0(e_i,e_j)=\left\{\begin{array}{cc}1,\hfill & i=j=1,2,3\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let ${\mathrm{\Phi }}_0$ be the tensor field defined by

$${\mathrm{\Phi }}_0{e}_i=\left\{\begin{array}{cc}-e_2,\hfill & i=1\hfill \\ e_1,\hfill & i=2\hfill \\ 0,\hfill & i=3.\hfill \end{array}\right.$$

Using the linearity of ${\mathrm{\Phi }}_0$ and $g_0$, we acquire ${\omega }_0\left(e_3\right)=1$, ${\mathrm{\Phi }}_0^2{X}_0=-X_0+{\omega }_0\left(X_0\right)e_3$ and $g_0({\mathrm{\Phi }}_0{X}_0,{\mathrm{\Phi }}_0{Y}_0)=g_0(X_0,Y_0)-{\omega }_0\left(X_0\right){\omega }_0\left(Y_0\right).$ Thus, for $e_3=A$, the structure $({\mathrm{\Phi }}_0,A,{\omega }_0,g_0))$ is an almost-contact metric structure on M. In addition, M satisfies

$$\left({\tilde{\nabla }}_{X_0}{\mathrm{\Phi }}_0\right)Y_0={\omega }_0\left(Y_0\right)X_0-g_0(X_0,Y_0)e_3.$$

Thus, for $e_3=A$, M is a Sasakian manifold.

Also, M satisfies

$$\begin{array}{c}\hfill K_0(X_0,{\tilde{\nabla }}_{Y_0}{e}_3)+{\mathrm{\Phi }}_0{K}_0(X_0,Y_0)=0,\end{array}$$

(90)

$$\begin{array}{c}\hfill {\ddot{\nabla }}_{X_0}\left({\tilde{\nabla }}_{Y_0}{e}_3\right)-{\mathrm{\Phi }}_0{\ddot{\nabla }}_{X_0}^{\ast }{Y}_0=g_0(Y_0,e_3)X_0-g_0(Y_0,X_0)e_3,\end{array}$$

(91)

$$\begin{array}{c}\hfill {\ddot{\nabla }}_{X_0}{e}_3={\tilde{\nabla }}_{X_0}{e}_3+g_0({\ddot{\nabla }}_{X_0}{e}_3,e_3)e_3.\end{array}$$

(92)

Then, M is called a Sasakian statistical manifold.

In tangent bundle $T_{0}M$, let the complete and vertical lifts of $e_1,e_2,e_3$ be $e_1^c,e_2^c,e_3^c$ and $e_1^v,e_2^v,e_3^v$ on M, and let $g_0^c$ be the complete lift of the Riemannian metric $g_0$ on $T_{0}M$ such that

$$\begin{array}{c}\hfill g^c(X_0^v,e_3^c)={\left(g^c(X_0,e_3)\right)}^v={\left({\omega }_0\left(X_0\right)\right)}^v\end{array}$$

(93)

$$\begin{array}{c}\hfill g^c(X_0^c,e_3^c)={\left(g^c(X_0,e_3)\right)}^c={\left({\omega }_0\left(X_0\right)\right)}^c\end{array}$$

(94)

$$\begin{array}{c}\hfill g^c(e_3^c,e_3^c)=1,g^v(X_0^v,e_3^c)=0,g^v(e_3^v,e_3^c)=0,\end{array}$$

(95)

and so on. Let ${\mathrm{\Phi }}_0^c$ and ${\mathrm{\Phi }}_0^v$ be the complete and vertical lifts of the $(1,1)$ tensor field ${\mathrm{\Phi }}_0$ defined by

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^v\left(e_3^v\right)={\mathrm{\Phi }}_0^c\left(e_3^c\right)=0,\end{array}$$

(96)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^v\left(e_1^v\right)=e_1^v,{\mathrm{\Phi }}_0^c\left(e_1^c\right)=e_1^c,\end{array}$$

(97)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^v\left(e_2^v\right)=e_2^v,{\mathrm{\Phi }}_0^c\left(e_2^c\right)=e_2^c,\end{array}$$

(98)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^v\left(e_3^v\right)=e_3^v,{\mathrm{\Phi }}_0^c\left(e_3^c\right)=e_3^c.\end{array}$$

(99)

Using the linearity of ${\mathrm{\Phi }}_0$ and $g_0$, we infer that

$$\begin{array}{c}\hfill {\left({\mathrm{\Phi }}_0^2{Y}_0\right)}^c=-Y_0^c+{\omega }_0^c\left(Y_0\right)e_3^v+{\omega }_0^v\left(Y_0\right)e_3^c,\end{array}$$

(100)

$$\begin{array}{c}\hfill g^c\left({\left({\mathrm{\Phi }}_0{e}_3\right)}^c,{\left({\mathrm{\Phi }}_0{e}_2\right)}^c\right)=g_0^c(e_3^c,e_2^c)-{\omega }_0^c\left(e_3^c\right){\omega }_0^v\left(e_2^c\right)-{\omega }_0^v\left(e_3^c\right){\omega }_0^c\left(e_2^c\right).\end{array}$$

(101)

$$\left({\tilde{\nabla }}_{e_3^c}^c{\mathrm{\Phi }}_0^c\right)e_2^c={\omega }_0^c\left(e_2^c\right)e_3^v+{\omega }_0^v\left(e_2^c\right)e_3^c-g_0^c(e_3^c,e_2^c)A^v-g_0^c(e_3^v,e_2^c)A^c.$$

(102)

Also,

$$\begin{array}{c}\hfill K_0^c(e_3^c,{\tilde{\nabla }}_{e_2^c}^c{A}^c)+{\mathrm{\Phi }}_0^c{K}_0^c(e_3^c,e_2^c)=0,\end{array}$$

(103)

$$\begin{array}{cc}\hfill {\ddot{\nabla }}_{e_3^c}^c\left({\tilde{\nabla }}_{e_2^c}^c{A}^c\right)-{\mathrm{\Phi }}_0^c{\ddot{\nabla }}_{e_3^c}^{\ast c}{e}_2^c& =g_0^c(e_2^c,A^c)e_3^v+g_0^c(e_2^v,A^c)e_3^c-g_0^c(e_2^c,e_3^c)A^c\hfill \\ \hfill & -g_0^c(e_2^v,e_3^c)A^c,\hfill \end{array}$$

(104)

$${\ddot{\nabla }}_{e_3^c}^c{A}^c={\tilde{\nabla }}_{e_3^c}^c{A}^c+g_0^c(A^c,{\ddot{\nabla }}_{e_3^c}^c{A}^c)A^v+g_0^c(A^v,{\ddot{\nabla }}_{e_3^c}^c{A}^c)A^v.$$

(105)

Then, for $e_3=A$ in Equations (93)–(100), the structure $({\mathrm{\Phi }}_0^c,A^c,{\omega }_0^c,g_0^c)$ is an almost-contact metric structure on $T_{0}M$ and satisfies the relation (102). Then, $({\mathrm{\Phi }}_0^c,A^c,{\omega }_0^c,g_0^c,T_0^M)$ is a Sasakian manifold. Also, this structure satisfies the relation (103)–(105) and is called a Sasakian statistical manifold.