Infinitesimal Affine Transformations and Mutual Curvatures on Statistical Manifolds and Their Tangent Bundles

Abstract

The purpose of this paper is to find some conditions under which the tangent bundle $TM$ has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle $TM$ and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle $TM$ (the complete, horizontal, and Sasaki connections) and we study the vanishing of them.

1. Introduction

Concerning the necessity of the study of tangent bundles, it can be stated that the concept of the tangent bundle is used widely in classical mechanics and especially in Lagrangian formalism. The tangent bundle can also describe the motion of objects in all classical mechanics scenarios, with a coordinate system such that the first n tuples of it represent the position of an object and the second n tuples of it represent the velocity of the object. This coordinate system is an effective tool in the study of geometric structures of the $TM$. So, we can regard the $TM$ as the state space in classical mechanics. In fact, the tangent bundle of a differential manifold M assembles all of the tangent vectors in M. On the other hand, vector fields on tangent bundles belong to basic concepts of pure and applied differential geometry, global analysis, and mathematical physics. Semisprays, sprays, and geodesic sprays are important classes of vector fields on tangent bundles. For instance, the semispray theory has been used in the calculus of variations on manifolds to characterize extremal curves of a variational functional as integral curves of the Hamilton or Euler–Lagrange vector fields. Sprays and semisprays also provide a natural framework for the extension of classical results of analytical mechanics to contemporary mechanical problems and stimulate a broad research field in the global theory of nonconservative systems, symmetries, and constraint theory (see [1] for more details).

The geometry of tangent bundles with Riemannian lift metrics has been studied very extensively in recent years (see [2,3,4,5,6,7,8], for instance). In this paper, we consider one of the famous Riemannian lift metrics, the Sasaki metric. Then, using this metric and its Levi-Civita connection, we investigate some of the geometric structures on the $TM$. The focus of this paper is concerned with the equations of a dualistic structure on the $TM$. Then, we investigate the mutual curvature of the $TM$ and its relation with the curvature of M and the mutual curvature of M.

Information geometry is the combination and interaction of differential geometry and statistics [9]. In addition, it is an important and useful bridge between applicable and pure sciences (see [9,10], for instance). In this area, we use and extend the methods of differential geometry in probability theory. The mathematical point of view of information geometry started with C. R. Rao. He showed that a statistical model should be a differentiable Riemannian manifold, via the Fisher information matrix. This means that we can define a Riemannian metric in the space of probability distributions. In fact, information geometry is the study of natural geometric structures using families of probability distributions. Two of the main objects in this area are statistical connections and statistical manifolds. They have applications in fields such as computer science and physics. In fact, a statistical manifold is a manifold whose points are probability distributions (see [9,10,11,12]).

A statistical structure on a manifold M is a pair $(g,\nabla )$ such that g is a Riemannian (semi-Riemannian) metric and ∇ is a torsion-free linear connection such that $\nabla g$ is totally symmetric. A Riemannian (semi-Riemannian) manifold $(M,g)$ together with the Levi-Civita connection ∇ of g is a typical example of a statistical manifold. In other words, statistical manifolds can be regarded as generalizations of Riemannian (semi-Riemannian) manifolds. Statistical manifolds provide geometric models of probability distributions. The geometries of statistical manifolds have been applied to various fields of information science, information theory, neural networks, machine learning, image processing, statistical mechanics, etc. (see [9,10], for instance).

The organization of this paper is as follows: In the first part we introduce the concept of the infinitesimal affine transformation of the Riemannian manifold $(M,g)$ with respect to the affine connection ∇. Then, using an explicit example we find conditions such that a vector field X be an infinitesimal affine transformation of the 2-dimensional statistical manifold $M_2$. In fact, we solve a system of partial differential equations and this solution gives us the general form of any infinitesimal affine transformation of $M_2$. To continue, we also find conditions under which the vector fields $X^V$ and $X^C$ are an infinitesimal affine transformation of the $TM$ with respect to the $(\alpha )$-connection ${\nabla }^{(\alpha )}$.

In the second part, we study the geometry of the dualistic structure of M and the mutual curvature on the statistical manifolds. Then, we prove that under which conditions R and $R^{*}$ are parallel with respect to the $(\alpha )$-connection ${\nabla }^{(\alpha )}$, where R and $R^{*}$ are curvature tensors of the dualistic structure $(\nabla ,{\nabla }^{*})$ on M. Then, we extend this problem to the mutual curvature $R_{\nabla ,{\nabla }^{*}}$ as mentioned in [13]. Moreover, we find conditions such that the $(\alpha )$-curvature $R^{(\alpha )}$ is parallel with respect to the $(\beta )$-connection ${\nabla }^{(\beta )}$. In the next part, we provide conditions such that the $TM$ equipped with them has a dualistic structure, and then, we find equations in which the $TM$ equipped with them is a conjugate symmetric space. We also study the mutual curvature ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ on the $TM$ and its relation with the curvature of M, where ∇ is an affine connection on M and $\stackrel{C}{\nabla }$ and $\stackrel{H}{\nabla }$ are the complete lift and horizontal lift connections on the $TM$, respectively. At the end, we investigate the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$, where $\{\stackrel{1}{\nabla },\stackrel{2}{\nabla }\}$ is a pair of the Levi-Civita connections of two non-isometric Riemannian metrics $g_1$ and $g_2$, and $\stackrel{1}{{\nabla }^S}$ and $\stackrel{2}{{\nabla }^S}$ are the Levi-Civita connections of the Sasaki lift metrics $g_1^S$ and $g_2^S$ on the $TM$, respectively. Moreover, we prove that the mutual curvature ${\tilde{R}}_{\stackrel{S}{\nabla },\stackrel{C}{\nabla }}$ vanishes if and only if M is a flat space with respect to the ∇, where ∇ is the Levi-Civita connection of the metric g and ${\nabla }^S$ is the Levi-Civita connection of the Sasaki lift metric $g^S$. Moreover, in this case we prove that the mutual curvature ${\tilde{R}}_{\stackrel{S}{\nabla },\stackrel{C}{\nabla }}$ reduces to the Riemannian curvature tensor of the Levi-Civita connection ${\nabla }^S$. At the end of this paper, we give an explicit example of the mutual curvature.

2. Preliminaries

In this section, we introduce some basic facts that we use throughout the paper.

Let $(x^i)$, $i=1,\cdots ,n$, be a coordinate system on M, $(x^i,y^i)$ be the induced coordinate system on the $TM$, and $\{\frac{\partial }{\partial x^i}{\mid }_{(x,y)},\frac{\partial }{\partial y^i}{\mid }_{(x,y)}\}$ be the natural basis of $T_{(x,y)}TM$. Then, the various lifts of a vector field $X=X^i{\partial }_i$ on M (complete lift, horizontal lift, and vertical lift, respectively) are defined as follows:

$$\begin{array}{c}\hfill X^C=X^i\frac{\partial }{\partial x^i}+y^a({\partial }_a{X}^i)\frac{\partial }{\partial y^i},X^H=X^i\frac{\partial }{\partial x^i}-y^a{\mathrm{\Gamma }}_{ai}^k{X}^i\frac{\partial }{\partial y^k},X^V=X^i\frac{\partial }{\partial y^i}.\end{array}$$

It is known that $T_{(x,y)}TM$ can be decomposed to $H_{(x,y)}TM\oplus V_{(x,y)}TM$, where $H_{(x,y)}TM$ is spanned by $\{\frac{\delta }{\delta x^i}{\mid }_{(x,y)}:={(\frac{\partial }{\partial x^i})}^h=\frac{\partial }{\partial x^i}{\mid }_{(x,y)}-y^k{\mathrm{\Gamma }}_{ki}^j(x)\frac{\partial }{\partial y^j}{\mid }_{(x,y)}\}$ and $V_{(x,y)}$ is spanned by $\{\frac{\partial }{\partial y^i}{\mid }_{(x,y)}:={(\frac{\partial }{\partial x^i})}^v\}$. For simplicity, we write ${\partial }_i$, ${\delta }_i$, and ${\partial }_{\overline{i}}$ instead of $\frac{\partial }{\partial x^i}$, $\frac{\delta }{\delta x^i}$, and $\frac{\partial }{\partial y^i}$.

Let $(M,g)$ be a Riemannian manifold. Similar to the lifts of vector fields, we can construct the Sasaki lift metric $g^S$ on the $TM$ as follows:

$$\begin{array}{c}\hfill g_{(x,y)}^S(X^H,Y^H)=g_x(X,Y),g_{(x,y)}^S(X^V,Y^H)=0,g_{(x,y)}^S(X^V,Y^V)=g_x(X,Y).\end{array}$$

(1)

The Levi-Civita connection of the Sasaki metric $g^S$ is as follows:

$$\begin{array}{c}\hfill {({\nabla }_{X^H}^S{Y}^H)}_{(x,y)}={({\nabla }_{X}Y)}_{(x,y)}^H-\frac{1}{2}{(R_x(X,Y)y)}^V,{({\nabla }_{X^V}^S{Y}^H)}_{(x,y)}=\frac{1}{2}{(R_x(y,X)Y)}^H,\end{array}$$

(2)

$$\begin{array}{c}\hfill {({\nabla }_{X^H}^S{Y}^V)}_{(x,y)}={({\nabla }_{X}Y)}_{(x,y)}^V+\frac{1}{2}{(R_x(y,Y)X)}^H,{({\nabla }_{X^V}^S{Y}^V)}_{(x,y)}=0,\end{array}$$

(3)

for all vector fields $X,Y$ on M, and $(x,y)\in TM$.

If ∇ is a linear connection, then the horizontal lift connection $\stackrel{H}{\nabla }$ and complete lift connection $\stackrel{C}{\nabla }$ of ∇ are, respectively, defined by [14]:

$$\begin{array}{c}\hfill {\stackrel{H}{\nabla }}_{X^H}{Y}^H={({\nabla }_{X}Y)}^H,{\stackrel{H}{\nabla }}_{X^H}{Y}^V={({\nabla }_{X}Y)}^V,{\stackrel{H}{\nabla }}_{X^V}{Y}^H={\stackrel{H}{\nabla }}_{X^V}{Y}^V=0,\\ \hfill {\stackrel{C}{\nabla }}_{X^H}{Y}^H={({\nabla }_{X}Y)}^H+{(R(y,X)Y)}^V,{\stackrel{C}{\nabla }}_{X^V}{Y}^H={\stackrel{C}{\nabla }}_{X^V}{Y}^V=0,\\ \hfill {\stackrel{C}{\nabla }}_{X^H}{Y}^V={({\nabla }_{X}Y)}^V,{\stackrel{C}{\nabla }}_{X^C}{Y}^C={({\nabla }_{X}Y)}^C,{\stackrel{C}{\nabla }}_{X^C}{Y}^V={\stackrel{C}{\nabla }}_{X^V}{Y}^C={({\nabla }_{X}Y)}^V.\end{array}$$

(4)

According to [14], the Lie brackets of the horizontal lift and vertical lift of vector fields are as follows:

$$\begin{array}{c}\hfill [X^H,Y^H]={[X,Y]}^H-{(R(X,Y)y)}^V,[X^H,Y^V]={({\nabla }_{X}Y)}^V-T{(X,Y)}^V,[X^V,Y^V]=0.\end{array}$$

(5)

Let $(M,g)$ be an n-dimensional Riemannian manifold and ∇ be an affine connection on M. A Codazzi couple on M is a pair $(g,\nabla )$ such that the cubic tensor field $C=\nabla g$ is totally symmetric, namely, the Codazzi equations hold:

$$\begin{array}{c}\hfill ({\nabla }_{X}g)(Y,Z)=({\nabla }_{Y}g)(Z,X)=({\nabla }_{Z}g)(X,Y),\forall X,Y,Z\in \chi (M).\end{array}$$

In this case, the triplet $(M,g,\nabla )$ is called a Codazzi manifold and ∇ is called a Codazzi connection. Furthermore, if ∇ is torsion free, then $(M,g,\nabla )$ is a statistical manifold, $(g,\nabla )$ is a statistical couple, and ∇ is a statistical connection. In local coordinates, C has the following form:

$$\begin{array}{c}\hfill C({\partial }_i,{\partial }_j,{\partial }_k)={\partial }_{i}g({\partial }_j,{\partial }_k)-g({\nabla }_{{\partial }_i}{\partial }_j,{\partial }_k)-g({\partial }_j,{\nabla }_{{\partial }_i}{\partial }_k),\end{array}$$

and so $C_{ijk}={\partial }_i{g}_{jk}-{\mathrm{\Gamma }}_{ij}^r{g}_{rk}-{\mathrm{\Gamma }}_{ik}^r{g}_{jr}$, $C_{ijk}=C_{jki}=C_{kij}$, where ${\mathrm{\Gamma }}_{ij}^r$ are the connection coefficients of ∇.

We also recall that if $(M,g)$ is a Riemannian (pseudo-Riemannian) manifold, two affine torsion-free connections ∇ and ${\nabla }^{*}$ on M are said to be dual connections with respect to g if the following equation is satisfied:

$$\begin{array}{c}\hfill Xg(Y,Z)=g({\nabla }_{X}Y,Z)+g(Y,{\nabla }_X^{*}Z),\forall X,Y,Z\in \chi (M),\end{array}$$

(6)

and in this case, we call $(g,\nabla ,{\nabla }^{*})$ a dualistic structure on M. Furthermore, if we denote by R and $R^{*}$ the curvature tensors of ∇ and ${\nabla }^{*}$, then we say that $(M,g)$ is a conjugate symmetric space if $R=R^{*}$. Moreover, if $(g,\nabla ,{\nabla }^{*})$ is a dualistic structure on M, then ${\{{\nabla }^{(\alpha )}\}}_{\alpha \in R}$ given by

$$\begin{array}{c}\hfill {\nabla }^{(\alpha )}=\frac{1+\alpha }{2}\nabla +\frac{1-\alpha }{2}{\nabla }^{*},\forall \alpha \in \mathbb{R},\end{array}$$

(7)

is a family of affine connections, which is called an $\alpha$-connection. It is known that if ∇ and ${\nabla }^{*}$ are statistical connections, then ${\nabla }^{(\alpha )}$ is a statistical connection for any $\alpha \in \mathbb{R}$ [15,16].

3. Infinitesimal Affine Transformations on Statistical Manifolds

Definition 1.

Let $(M,g)$ be a Riemannian manifold, X be a vector field, and ∇ be an affine connection on M. Then, X is said to be an infinitesimal affine transformation of M with respect to ∇ if $L_X\nabla =0$, where $L_X\nabla$ is the Lie derivative of ∇ with respect to X given by

$$\begin{array}{c}\hfill (L_X\nabla )(Y,Z)=L_X({\nabla }_{Y}Z)-{\nabla }_Y(L_{X}Z)-{\nabla }_{[X,Y]}Z.\end{array}$$

(8)

Setting $X=X^i{\partial }_i$, $Y={\partial }_j$, and $Z={\partial }_k$, then the local expression of (8) is as follows:

$$\begin{array}{c}\hfill (L_{X^i{\partial }_i}\nabla )({\partial }_j,{\partial }_k)=\{X^i{\partial }_i({\mathrm{\Gamma }}_{jk}^t)-{\mathrm{\Gamma }}_{jk}^r{\partial }_r(X^t)+{\partial }_j{\partial }_k(X^t)+{\partial }_k(X^i){\mathrm{\Gamma }}_{ji}^t+{\partial }_j(X^i){\mathrm{\Gamma }}_{ik}^t\}{\partial }_t,\end{array}$$

(9)

Example 1.

Here, we consider a p-dimensional statistical manifold. The importance of the distribution family introduced below lies in the fact that its member is a non-Gaussian multivariate distribution while the marginal distribution is Gaussian, which implies that a set of marginal distributions does not uniquely determine the multivariate normal distribution [17]. A p-dimensional statistical manifold is defined by

$$M=\left\{f(\mathbf{x};\lambda )|f(\mathbf{x};\lambda )=2{\mathrm{\Pi }}_{i=1}^p\frac{\sqrt{{\lambda }_i}}{\sqrt{2\pi }}{e}^{-\frac{{\lambda }_i{x}_i^2}{2}},\mathbf{x}\in {\mathrm{\Omega }}_p,\lambda \in {\mathbb{R}}_{+}^p\right\},$$

where

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_p& =\{x=(x_1,\cdots ,x_p)\in {\mathbb{R}}^p|{\mathrm{\Pi }}_{i=1}^p{x}_i>0\},\hfill \\ \hfill {\mathbb{R}}_{+}^p& =\{x=(x_1,\cdots ,x_p)\in {\mathbb{R}}^p|x_i>0,i=1,\cdots ,p\}.\hfill \end{array}$$

The distribution in M can be rewritten as

$$f(\mathbf{x};\lambda )=e^{\frac{1}{2}}\sum _{i=1}^{p}log(-{\theta }_i)+\sum _{i=1}^p{\theta }_i{x}_i^2+\frac{p}{2}log2-log\sqrt{2\pi },$$

where ${\theta }_i=-\frac{1}{2}{\lambda }_i$. This is one member of the exponential family with the natural coordinates $({\theta }_1,\cdots ,{\theta }_p)$ and the potential function $\psi (\theta )=-\frac{1}{2}{\sum }_{i=1}^{p}log(-{\theta }_i)$. It is known that for the exponential family, the Fisher information is just the second derivative of the potential function

$$g_{ij}=\frac{{\partial }^2\psi }{\partial {\theta }_i\partial {\theta }_j}=-\frac{1}{2}\frac{1}{{\theta }_i{\theta }_j}{\delta }_{ij},$$

(10)

and the α-connection is the third derivative of the potential function

$${\mathrm{\Gamma }}_{ijk}^{(\alpha )}=\frac{1-\alpha }{2}\frac{{\partial }^3\psi }{\partial {\theta }_i\partial {\theta }_j\partial {\theta }_k}=-\frac{1-\alpha }{2}\frac{1}{{\theta }_i{\theta }_j{\theta }_k}{\delta }_{ijk},$$

(11)

where ${\delta }_{ii}=1$ for $i=1,\cdots ,p$, ${\delta }_{ij}=0$ for $i\ne j$, and ${\delta }_{iii}=1$ for $i=1,\cdots ,p$, ${\delta }_{ijk}=0$ for unequal i, j, k (see [18] for more details). For $p=2$, the matrix expression of the metric g given by (10) and its inverse matrix are as follows:

$$\begin{array}{c}\hfill g=\left[\begin{array}{cc}-\frac{1}{2{\theta }_1^2}& 0\\ 0& -\frac{1}{2{\theta }_2^2}\end{array}\right],g^{-1}=\left[\begin{array}{cc}-2{\theta }_1^2& 0\\ 0& -2{\theta }_2^2\end{array}\right].\end{array}$$

(12)

Combining (11) and (10) we get

$${\mathrm{\Gamma }}_{11}^{(\alpha )1}=\frac{1-\alpha }{{\theta }_1},{\mathrm{\Gamma }}_{22}^{(\alpha )2}=\frac{1-\alpha }{{\theta }_2},{\mathrm{\Gamma }}_{ij}^{(\alpha )k}=0,\mathit{for}\mathit{unequal}i,j,k.$$

(13)

Now, let $X=X^1{\partial }_1+X^2{\partial }_2$ be an infinitesimal affine transformation of M with respect to ∇, where ${\partial }_1=\frac{\partial }{\partial {\theta }_1}$ and ${\partial }_2=\frac{\partial }{\partial {\theta }_2}$. Then, from (9) we obtain

$$\begin{array}{c}\hfill \{X^i{\partial }_i({\mathrm{\Gamma }}_{jk}^t)-{\mathrm{\Gamma }}_{jk}^r{\partial }_r(X^t)+{\partial }_j{\partial }_k(X^t)+{\partial }_k(X^i){\mathrm{\Gamma }}_{ji}^t+{\partial }_j(X^i){\mathrm{\Gamma }}_{ik}^t\}=0.\end{array}$$

(14)

If we take $j,k,t\in \{1,2\}$, then we obtain the following system of equations

$$\begin{array}{c}\hfill \frac{X^1(\alpha -1)}{{\theta }_1^2}+{\partial }_1^2(X^1)+{\partial }_1(X^1)(\frac{1-\alpha }{{\theta }_1})=0,\end{array}$$

(15)

$$\begin{array}{c}\hfill \frac{X^2(\alpha -1)}{{\theta }_2^2}+{\partial }_2^2(X^2)+{\partial }_2(X^2)(\frac{1-\alpha }{{\theta }_2})=0,\end{array}$$

(16)

$$\begin{array}{c}\hfill (\frac{\alpha -1}{{\theta }_1}){\partial }_1(X^2)+{\partial }_1^2(X^2)=0,\end{array}$$

(17)

$$\begin{array}{c}\hfill (\frac{\alpha -1}{{\theta }_2}){\partial }_2(X^1)+{\partial }_2^2(X^1)=0,\end{array}$$

(18)

$$\begin{array}{c}\hfill {\partial }_1{\partial }_2(X^1)+{\partial }_2(X^1)(\frac{1-\alpha }{{\theta }_1})=0,\end{array}$$

(19)

$$\begin{array}{c}\hfill {\partial }_1{\partial }_2(X^2)+{\partial }_1(X^2)(\frac{1-\alpha }{{\theta }_2})=0.\end{array}$$

(20)

From (19), we have ${\partial }_2({\partial }_1{X}^1+X^1(\frac{1-\alpha }{{\theta }_1}))=0$. So, $({\partial }_1{X}^1+X^1(\frac{1-\alpha }{{\theta }_1}))$ is a function with respect to ${\theta }_1$ only, i.e., ${\partial }_1{X}^1+X^1(\frac{1-\alpha }{{\theta }_1})=f({\theta }_1)$. Thus,

$$\begin{array}{c}\hfill X^1(\frac{1-\alpha }{{\theta }_1})=f({\theta }_1)-{\partial }_1{X}^1.\end{array}$$

(21)

From (15), we find

$$\begin{array}{c}\hfill {\partial }_1(X^1(\frac{1-\alpha }{{\theta }_1}))+{\partial }_1^2{X}^1=0.\end{array}$$

(22)

Substituting (21) in (22), we obtain ${\partial }_{1}f=0$, which gives $f({\theta }_1)=A$, where A is a constant. So, we have

$$\begin{array}{c}\hfill {\partial }_1{X}^1+X^1(\frac{1-\alpha }{{\theta }_1})=A,\end{array}$$

(23)

which is a linear differential equation with respect to ${\theta }_1$. Thus, we have the following solution:

$$\begin{array}{cc}\hfill X^1& =e^{-\int \frac{1-\alpha }{{\theta }_1}d{\theta }_1}[\int A{e}^{\int \frac{1-\alpha }{{\theta }_1}d{\theta }_1}d{\theta }_1+C({\theta }_2)]\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & ={\theta }_1^{\alpha -1}(A\frac{{\theta }_1^{2-\alpha }}{2-\alpha }+C({\theta }_2))=\frac{A{\theta }_1}{2-\alpha }+C({\theta }_2){\theta }_1^{\alpha -1}.\hfill \end{array}$$

(25)

It is easy to check that $X^1=\frac{A{\theta }_1}{2-\alpha }+C({\theta }_2){\theta }_1^{\alpha -1}$ satisfies (15) and (19). Setting $X^1$ in (18) we obtain

$$\begin{array}{c}\hfill {\theta }_1^{\alpha -1}(C^{″}({\theta }_2)+\frac{\alpha -1}{{\theta }_2}{C}^{\prime }({\theta }_2))=0.\end{array}$$

(26)

So, we have the following ordinary differential equation

$$\begin{array}{c}\hfill C^{″}({\theta }_2)+\frac{\alpha -1}{{\theta }_2}{C}^{\prime }({\theta }_2)=0.\end{array}$$

(27)

It is easily seen that the above ODE has the solution

$$\begin{array}{c}\hfill C({\theta }_2)=\frac{E}{\alpha }{\theta }_2^{\alpha }+F,\end{array}$$

(28)

where E and F are constants. Therefore, we have

$$\begin{array}{c}\hfill X^1=\frac{A}{2-\alpha }{\theta }_1+(\frac{E}{\alpha }{\theta }_2^{\alpha }+F){\theta }_1^{\alpha -1}.\end{array}$$

(29)

Similarly, we obtain

$$\begin{array}{c}\hfill X^2=\frac{B}{2\alpha }{\theta }_2+(\frac{G}{\alpha }{\theta }_1^{\alpha }+H){\theta }_2^{\alpha -1},\end{array}$$

(30)

where B, G, and H are constants.

Let ∇ be an affine connection on the Riemannian manifold M and let X be a vector field on M. Then, we have (see [14])

$$\begin{array}{c}\hfill L_{X^V}\stackrel{C}{\nabla }={(L_X\nabla )}^V,L_{X^C}\stackrel{C}{\nabla }={(L_X\nabla )}^C.\end{array}$$

(31)

Now, we assume that $(g,\nabla ,{\nabla }^{*})$ is a dualistic structure on the Riemannian manifold M and let X be an infinitesimal affine transformation of M with respect to ∇ and ${\nabla }^{*}$. Then, (7) gives us

$$\begin{array}{c}\hfill (L_X{\nabla }^{(\alpha )})(Y,Z)=\frac{1+\alpha }{2}(L_X\nabla )(Y,Z)+\frac{1-\alpha }{2}(L_X{\nabla }^{*})(Y,Z).\end{array}$$

(32)

The above equation means that X is an infinitesimal affine transformation of M with respect to the $\alpha$-connection ${\nabla }^{(\alpha )}$. Now, if we replace ∇ by ${\nabla }^{(\alpha )}$ in (31), we obtain

$$\begin{array}{c}\hfill L_{X^V}{({\nabla }^{(\alpha )})}^C={(L_X{\nabla }^{(\alpha )})}^V,L_{X^C}{({\nabla }^{(\alpha )})}^C={(L_X{\nabla }^{(\alpha )})}^C.\end{array}$$

(33)

From the hypothesis and (33), we deduce that $X^V$ and $X^C$ are infinitesimal affine transformations of the $TM$ with respect to ${({\nabla }^{(\alpha )})}^C$. So, we conclude the following theorem.

Theorem 1.

Let $(g,\nabla ,{\nabla }^{*})$ be a dualistic structure on the Riemannian manifold M. If X is an infinitesimal affine transformation of M with respect to ∇ and ${\nabla }^{*}$, then X is an infinitesimal affine transformation of M with respect to the α-connection ${\nabla }^{(\alpha )}$. Moreover, $X^V$ and $X^C$ are infinitesimal affine transformations of the $TM$ with respect to ${({\nabla }^{(\alpha )})}^C$.

4. Mutual Curvature on Statistical Manifolds

In this section, we introduce the concept of mutual curvature for the Riemannian manifold $(M,g)$, and to continue, we consider the dualistic structure $(\nabla ,{\nabla }^{*})$ on M. Then, we show under which conditions the mutual curvature $R_{\nabla ,{\nabla }^{*}}$ is parallel with respect to the $\alpha$-connection ${\nabla }^{(\alpha )}$. Finally, we find conditions under which the mutual curvature $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}$ is parallel with respect to the $\gamma$-connection ${\nabla }^{(\gamma )}$.

Definition 2.

([13]). Let $(M,g)$ be a Riemannian manifold and let $(\stackrel{1}{\nabla },\stackrel{2}{\nabla })$ be a pair of connections. Then, their mutual curvature is the $(1,3)$-tensor $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$, which is defined by the following formula:

$$\begin{array}{c}\hfill R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(X,Y)Z=\frac{1}{2}\{{\stackrel{1}{\nabla }}_X{\stackrel{2}{\nabla }}_{Y}Z-{\stackrel{1}{\nabla }}_Y{\stackrel{2}{\nabla }}_{X}Z-{\stackrel{1}{\nabla }}_{[X,Y]}Z+{\stackrel{2}{\nabla }}_X{\stackrel{1}{\nabla }}_{Y}Z-{\stackrel{2}{\nabla }}_Y{\stackrel{1}{\nabla }}_{X}Z-{\stackrel{2}{\nabla }}_{[X,Y]}Z\},\end{array}$$

(34)

for all $X,Y,Z\in \chi (M)$.

It should be noted that the definition of mutual (or relative) curvature was previously presented by the authors in [16,19] in two different ways. However, as D. Iosifidis showed in [13], none of these are tensors.

Theorem 2.

Let $(\nabla ,{\nabla }^{*})$ be a dualistic structure on the Riemannian manifold $(M,g)$. Then, the following relation holds:

$$\begin{array}{c}\hfill R_{\nabla ,{\nabla }^{*}}(X,Y)Z=R_{\nabla }(X,Y)Z+({\nabla }_{Y}K)(X,Z)-({\nabla }_{X}K)(Y,Z),\forall X,Y,Z\in \chi (M),\end{array}$$

(35)

where K is the difference tensor of ∇. Moreover, the difference tensor K is Codazzi-coupled (i.e., $({\nabla }_{X}K)(Y,Z)=({\nabla }_{Y}K)(X,Z)$) if and only if $R_{\nabla ,{\nabla }^{*}}$ and $R_{\nabla }$ coincide. Furthermore, the mutual curvature $R_{\nabla ,{\nabla }^{*}}$ reduces to $R_{\nabla }$ whenever ∇ is the Levi-Civita connection.

Proof. 

Using ${\nabla }^{*}=\nabla -2K$ and direct computations, we obtain

$$\begin{array}{cc}\hfill & R_{\nabla ,{\nabla }^{*}}(X,Y)Z\hfill \\ & =\frac{1}{2}\{{\nabla }_X{\nabla }_Y^{*}Z-{\nabla }_Y{\nabla }_X^{*}Z-{\nabla }_{[X,Y]}Z+{\nabla }_X^{*}{\nabla }_{Y}Z-{\nabla }_Y^{*}{\nabla }_{X}Z-{\nabla }_{[X,Y]}^{*}Z\}\hfill \\ \hfill & =\frac{1}{2}\{{\nabla }_X({\nabla }_{Y}Z-2K_{Y}Z)-{\nabla }_Y({\nabla }_{X}Z-2K_{X}Z)-{\nabla }_{[X,Y]}Z\hfill \\ \hfill & +({\nabla }_X{\nabla }_{Y}Z-2K_X{\nabla }_{Y}Z)-({\nabla }_Y{\nabla }_{X}Z-2K_Y{\nabla }_{X}Z)-({\nabla }_{[X,Y]}Z-2K_{[X,Y]}Z)\}\hfill \\ \hfill & =\frac{1}{2}\{2R_{\nabla }(X,Y)Z+2({\nabla }_Y(K_{X}Z)-K_X({\nabla }_{Y}Z)+K_Y({\nabla }_{X}Z)+K_{[X,Y]}Z-{\nabla }_X(K_{Y}Z))\}\hfill \\ \hfill & =R_{\nabla }(X,Y)Z+({\nabla }_{Y}K)(X,Z)-({\nabla }_{X}K)(Y,Z).\hfill \end{array}$$

□

If we put $X={\partial }_i$, $Y={\partial }_j$, and $Z={\partial }_k$ in (35) and we denote the connection coefficients of ∇ and ${\nabla }^{(0)}$ by ${\mathrm{\Gamma }}_{ij}^r$ and ${\mathrm{\Gamma }}_{ij}^{(0)r}$, respectively, then we obtain the local expression of (35) as follows:

$$\begin{array}{cc}\hfill & R_{\nabla ,{\nabla }^{*}}({\partial }_i,{\partial }_j){\partial }_k=R_{\nabla ijk}^m{\partial }_m\hfill \\ \hfill & +\{{\partial }_j({\mathrm{\Gamma }}_{ik}^m-{\mathrm{\Gamma }}_{ik}^{(0)m})+({\mathrm{\Gamma }}_{ik}^r-{\mathrm{\Gamma }}_{ik}^{(0)r}){\mathrm{\Gamma }}_{jr}^m-{\mathrm{\Gamma }}_{ji}^r({\mathrm{\Gamma }}_{rk}^m-{\mathrm{\Gamma }}_{rk}^{(0)m})-{\mathrm{\Gamma }}_{jk}^r({\mathrm{\Gamma }}_{ir}^m-{\mathrm{\Gamma }}_{ir}^{(0)m})\}{\partial }_m\hfill \\ \hfill & -\{{\partial }_j({\mathrm{\Gamma }}_{jk}^m-{\mathrm{\Gamma }}_{jk}^{(0)m})+({\mathrm{\Gamma }}_{jk}^r-{\mathrm{\Gamma }}_{jk}^{(0)r}){\mathrm{\Gamma }}_{jr}^m-{\mathrm{\Gamma }}_{ij}^r({\mathrm{\Gamma }}_{rk}^m-{\mathrm{\Gamma }}_{rk}^{(0)m})-{\mathrm{\Gamma }}_{ik}^r({\mathrm{\Gamma }}_{jr}^m-{\mathrm{\Gamma }}_{jr}^{(0)m})\}{\partial }_m.\hfill \end{array}$$

(36)

From (34) and by direct computations we obtain the following:

Lemma 1.

Let $(M,g)$ be a Riemannian manifold and let $(\stackrel{1}{\nabla },\stackrel{2}{\nabla })$ be a pair of connections such that ${\mathrm{\Gamma }}_{ij}^{1r}$ and ${\mathrm{\Gamma }}_{ij}^{2r}$ are the connection coefficients of $\stackrel{1}{\nabla }$ and $\stackrel{2}{\nabla }$, respectively. Then, the following assertions hold:

(1) 

$R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(X,Y)Z=R_{\stackrel{2}{\nabla },\stackrel{1}{\nabla }}(X,Y)Z$ and $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(X,Y)Z=-R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(Y,X)Z$ for all $X,Y,Z\in \chi (M)$.

(2) 

$R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(X,Y)Z+R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(Y,Z)X+R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}(Z,X)Y=0$, whenever $\stackrel{1}{\nabla },\stackrel{2}{\nabla }$ are torsion-free connections.

(3) 

The local expression of the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ is in the following form: $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}({\partial }_i,{\partial }_j){\partial }_k=\frac{1}{2}\{({\partial }_i({\mathrm{\Gamma }}_{jk}^{2t})+{\mathrm{\Gamma }}_{jk}^{2r}{\mathrm{\Gamma }}_{ir}^{1t}-{\partial }_j({\mathrm{\Gamma }}_{ik}^{2t})-{\mathrm{\Gamma }}_{ik}^{2r}{\mathrm{\Gamma }}_{jr}^{1t}+{\partial }_i({\mathrm{\Gamma }}_{jk}^{1t})+{\mathrm{\Gamma }}_{jk}^{1r}{\mathrm{\Gamma }}_{ir}^{2t}-{\partial }_j({\mathrm{\Gamma }}_{ik}^{1t})-{\mathrm{\Gamma }}_{ik}^{1r}{\mathrm{\Gamma }}_{jr}^{2t})\}{\partial }_t$.

Example 2.

Now, we compute the mutual curvature $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}$ in Example 1. Direct computations give us the following

$$\begin{array}{cc}\hfill R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}({\partial }_1,{\partial }_2){\partial }_1& =\frac{1}{2}\{{\nabla }_{{\partial }_1}^{(\alpha )}{\nabla }_{{\partial }_2}^{(\beta )}{\partial }_1-{\nabla }_{{\partial }_2}^{(\alpha )}{\nabla }_{{\partial }_1}^{(\beta )}{\partial }_1-{\nabla }_{[{\partial }_1,{\partial }_2]}^{(\alpha )}{\partial }_1\hfill \\ \hfill & +{\nabla }_{{\partial }_1}^{(\beta )}{\nabla }_{{\partial }_2}^{(\alpha )}{\partial }_1-{\nabla }_{{\partial }_2}^{(\beta )}{\nabla }_{{\partial }_1}^{(\alpha )}{\partial }_1-{\nabla }_{[{\partial }_1,{\partial }_2]}^{(\beta )}{\partial }_1\}.\hfill \end{array}$$

(37)

Using (13), we obtain

$$\begin{array}{c}\hfill {\nabla }_{{\partial }_2}^{(\beta )}{\partial }_1=0,{\nabla }_{{\partial }_1}^{(\beta )}{\partial }_1=\frac{1-\beta }{{\theta }_1}{\partial }_1,{\nabla }_{{\partial }_2}^{(\alpha )}(\frac{1-\beta }{{\theta }_1}){\partial }_1=0,\end{array}$$

(38)

$$\begin{array}{c}\hfill {\nabla }_{{\partial }_2}^{(\alpha )}{\partial }_1=0,{\nabla }_{{\partial }_1}^{(\alpha )}{\partial }_1=\frac{1-\alpha }{{\theta }_1}{\partial }_1,{\nabla }_{{\partial }_2}^{(\beta )}(\frac{1-\alpha }{{\theta }_1}){\partial }_1=0.\end{array}$$

(39)

Putting (38) and (39) into (37) gives us $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}({\partial }_1,{\partial }_2){\partial }_1=0$. Similar computations imply

$$\begin{array}{c}\hfill {\nabla }_{{\partial }_1}^{(\beta )}{\partial }_2=0,{\nabla }_{{\partial }_2}^{(\beta )}{\partial }_2=\frac{1-\beta }{{\theta }_2}{\partial }_2,{\nabla }_{{\partial }_1}^{(\alpha )}(\frac{1-\beta }{{\theta }_2}){\partial }_2=0,\end{array}$$

(40)

$$\begin{array}{c}\hfill {\nabla }_{{\partial }_1}^{(\alpha )}{\partial }_2=0,{\nabla }_{{\partial }_2}^{(\alpha )}{\partial }_2=\frac{1-\alpha }{{\theta }_2}{\partial }_2,{\nabla }_{{\partial }_1}^{(\beta )}(\frac{1-\alpha }{{\theta }_2}){\partial }_2=0.\end{array}$$

(41)

Substituting (40) and (41) into (34), we derive that $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}({\partial }_2,{\partial }_1){\partial }_2=0$. From the first item of Lemma 1, we deduce that the other components of the mutual curvature are zero.

Definition 3.

Let $(M,g)$ be a Riemannian manifold and let ∇ be an affine connection with the curvature tensor R. Then, we say that M is a locally symmetric space if $\nabla R=0$, i.e., ${\nabla }_{X}R=0$, for each $X\in \chi (M)$, this means that R is parallel with respect to ∇.

Now, we assume that $(g,\nabla ,{\nabla }^{*})$ is a dualistic structure on the Riemannian manifold M and let R and $R^{*}$ be curvature tensors of ∇ and ${\nabla }^{*}$, respectively. If R is parallel with respect to ∇ and ${\nabla }^{*}$, then by direct computations we have

$$\begin{array}{cc}\hfill & ({\nabla }_X^{(\alpha )}R)(Y,Z)W\hfill \\ & ={\nabla }_X^{(\alpha )}R(Y,Z)W-R({\nabla }_X^{(\alpha )}Y,Z)W-R(Y,{\nabla }_X^{(\alpha )}Z)W-R(Y,Z){\nabla }_X^{(\alpha )}W\hfill \\ \hfill & =\{\frac{1+\alpha }{2}{\nabla }_{X}R(Y,Z)W+\frac{1-\alpha }{2}{\nabla }_X^{*}R(Y,Z)W\}\hfill \\ \hfill & -R(\frac{1+\alpha }{2}{\nabla }_{X}Y+\frac{1-\alpha }{2}{\nabla }_X^{*}Y,Z)W\hfill \\ \hfill & -R(Y,\frac{1+\alpha }{2}{\nabla }_{X}Z+\frac{1-\alpha }{2}{\nabla }_X^{*}Z)W\hfill \\ \hfill & -R(Y,Z)(\frac{1+\alpha }{2}{\nabla }_{X}W+\frac{1-\alpha }{2}{\nabla }_X^{*}W)\hfill \\ \hfill & =\frac{1+\alpha }{2}\{{\nabla }_{X}R(Y,Z)W-R({\nabla }_{X}Y,Z)W-R(Y,{\nabla }_{X}Z)W-R(Y,Z){\nabla }_{X}W\}\hfill \\ \hfill & +\frac{1-\alpha }{2}\{{\nabla }_X^{*}R(Y,Z)W-R({\nabla }_X^{*}Y,Z)W-R(Y,{\nabla }_X^{*}Z)W-R(Y,Z){\nabla }_X^{*}W\}\hfill \\ \hfill & =\frac{1+\alpha }{2}({\nabla }_{X}R)(Y,Z)W+\frac{1-\alpha }{2}({\nabla }_X^{*}R)(Y,Z)W.\hfill \end{array}$$

(42)

Since R is parallel with respect to ∇ and ${\nabla }^{*}$, then we derive that R is parallel with respect to the $\alpha$-connection ${\nabla }^{(\alpha )}$. Moreover, if we replace R by $R^{*}$ in the above relations and assume that $R^{*}$ is parallel to ∇ and ${\nabla }^{*}$, then we obtain that $R^{*}$ is parallel to the $\alpha$-connection ${\nabla }^{(\alpha )}$. Now, we consider a pair $(\stackrel{1}{\nabla },\stackrel{2}{\nabla })$ of connections on M. If in (42), we replace the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ instead of R, then, the same as in (42), we obtain

$$\begin{array}{c}\hfill ({\nabla }_X^{(\alpha )}{R}_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }})(Y,Z)W=\frac{1+\alpha }{2}({\nabla }_X{R}_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }})(Y,Z)W+\frac{1-\alpha }{2}({\nabla }_X^{*}{R}_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }})(Y,Z)W.\end{array}$$

(43)

If the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ is parallel with respect to ∇ and ${\nabla }^{*}$, then from (43), we obtain that $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ is parallel with respect to ${\nabla }^{(\alpha )}$. According to the above discussion we obtain the following:

Proposition 1.

Let $(g,\nabla ,{\nabla }^{*})$ be a dualistic structure on the Riemannian manifold M, and R and $R^{*}$ be curvature tensors of ∇ and ${\nabla }^{*}$. Then, the following statements hold:

(1) 

If R (respectively, $R^{*}$) is parallel with respect to ∇ and ${\nabla }^{*}$, then R (respectively, $R^{*}$) is parallel with respect to the α-connection ${\nabla }^{(\alpha )}$.

(2) 

The mutual curvature $R_{\nabla ,{\nabla }^{*}}$ is parallel with respect to ${\nabla }^{(\alpha )}$ whenever $R_{\nabla ,{\nabla }^{*}}$ is parallel with respect to ∇ and ${\nabla }^{*}$.

Now, we consider the dualistic structure $(g,\nabla ,{\nabla }^{*})$ on the Riemannian manifold M and let ${\{{\nabla }^{(\alpha )}\}}_{\alpha \in R}$ be a family of $\alpha$-connections. Equations (7), (34), (43), and direct computations give us the following:

Lemma 2.

Let $(g,\nabla ,{\nabla }^{*})$ be a dualistic structure on the Riemannian manifold M, and let ${\{{\nabla }^{(\alpha )}\}}_{\alpha \in R}$ be a family of α-connections on M. Then, the following statements hold:

(1) 

$$\begin{array}{cc}\hfill R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}(X,Y)Z& =\frac{(1+\alpha )(1+\beta )}{4}{R}_{\nabla }(X,Y)Z+\frac{(1-\alpha \beta )}{2}{R}_{\nabla ,{\nabla }^{*}}(X,Y)Z\hfill \\ \hfill & +\frac{(1-\alpha )(1-\beta )}{4}{R}_{{\nabla }^{*}}(X,Y)Z,\forall X,Y,Z\in \chi (M).\hfill \end{array}$$

(2) 

If M is a flat space with respect to ∇ and ${\nabla }^{*}$ and the mutual curvature $R_{\nabla ,{\nabla }^{*}}$ vanishes, then the mutual curvature $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}$ vanishes.

(3) 

The mutual curvature $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}$ reduces to $R_{\nabla }$ (respectively, $R_{{\nabla }^{*}}$) whenever $\alpha =\beta =1$ (respectively, $\alpha =\beta =-1$).

(4) 

The mutual curvature $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}$ reduces to the mutual curvature $R_{\nabla ,{\nabla }^{*}}$ whenever $\alpha =-\beta =1$.

(5) 

The mutual curvature $R_{{\nabla }^{(\alpha )},{\nabla }^{(\beta )}}$ is parallel with respect to the γ-connection ${\nabla }^{(\gamma )}$ whenever $R_{\nabla }$, $R_{{\nabla }^{*}}$, and the mutual curvatures $R_{\nabla ,{\nabla }^{*}}$ are parallel with respect to ∇ and ${\nabla }^{*}$.

As a consequence of the above lemma, if we consider $\alpha =\beta$, then we obtain the $(\alpha )$-curvature $R^{(\alpha )}$ of the $(\alpha )$-connection ${\nabla }^{(\alpha )}$ as follows:

$$\begin{array}{c}\hfill R^{(\alpha )}(X,Y)Z=\frac{{(1+\alpha )}^2}{4}{R}_{\nabla }(X,Y)Z+\frac{(1-{\alpha }^2)}{2}{R}_{\nabla ,{\nabla }^{*}}(X,Y)Z+\frac{{(1-\alpha )}^2}{4}{R}_{{\nabla }^{*}}(X,Y)Z.\end{array}$$

(44)

To continue, we investigate under which conditions the $(\alpha )$-curvature $R^{(\alpha )}$ of the $(\alpha )$-connection ${\nabla }^{(\alpha )}$ is parallel with respect to the $(\beta )$-connection ${\nabla }^{(\beta )}$. Direct computations and (44) give us the following

$$\begin{array}{cc}\hfill ({\nabla }_X^{(\beta )}{R}^{(\alpha )})(Y,Z)W& =\frac{{(1+\alpha )}^2}{4}({\nabla }_X^{(\beta )}{R}_{\nabla })(Y,Z)W+\frac{(1-{\alpha }^2)}{2}({\nabla }_X^{(\beta )}{R}_{\nabla ,{\nabla }^{*}})(Y,Z)W\hfill \\ \hfill & +\frac{{(1-\alpha )}^2}{4}({\nabla }_X^{(\beta )}{R}_{{\nabla }^{*}})(Y,Z)W.\hfill \end{array}$$

(45)

Let $(\nabla ,{\nabla }^{*})$ be a dualistic structure on the Riemannian manifold $(M,g)$ with the curvature tensors $R_{\nabla }$ and $R_{{\nabla }^{*}}$, respectively, and let X be a vector field on M. Then, (44) implies the following;

$$\begin{array}{cc}\hfill (L_X{R}^{(\alpha )})(Y,Z,W)& =\frac{{(1+\alpha )}^2}{4}(L_X{R}_{\nabla })(Y,Z,W)+\frac{(1-{\alpha }^2)}{2}(L_X{R}_{\nabla ,{\nabla }^{*}})(Y,Z,W)\hfill \\ \hfill & +\frac{{(1-\alpha )}^2}{4}(L_X{R}_{{\nabla }^{*}})(Y,Z,W).\hfill \end{array}$$

(46)

Corollary 1.

Let $(g,\nabla ,{\nabla }^{*})$ be a dualistic structure on the Riemannian manifold M. Then, the following statements hold:

(1) 

The $(\alpha )$-curvature $R^{(\alpha )}$ is parallel with respect to the $(\beta )$-connection ${\nabla }^{(\beta )}$ whenever $R_{\nabla },R_{{\nabla }^{*}}$ and the mutual curvature $R_{\nabla ,{\nabla }^{*}}$ are parallel with respect to ∇ and ${\nabla }^{*}$.

(2) 

The Lie derivative of the $(\alpha )$-curvature $R^{(\alpha )}$ along X vanishes if the Lie derivatives of $R_{\nabla }$, $R_{{\nabla }^{*}}$, and $R_{\nabla ,{\nabla }^{*}}$ along X vanish.

5. Dualistic Structure on the Tangent Bundle

In this section, we consider an arbitrary Riemannian metric $\overline{g}$ and two affine torsion-free connections $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ on the $TM$. Then, we investigate under which conditions $(\overline{g},\tilde{\nabla },{\tilde{\nabla }}^{*})$ is a dualistic structure on the tangent bundle $TM$.

Let $\{{\delta }_i,{\partial }_{\overline{i}}\}$ be a basis for $T_v(TM)$, where $v\in TM$. According to this basis, we consider the Riemannian metric $\overline{g}={\alpha }_{ij}d{x}^{i}d{x}^j+2{\beta }_{ij}d{x}^i\delta y^j+{\gamma }_{ij}\delta y^i\delta y^j$ on the $TM$, where ${\alpha }_{ij},{\beta }_{ij},{\gamma }_{ij}\in C^{\infty }(TM)$. Since $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ are affine torsion-free connections on the $TM$ and $\{{\delta }_i,{\partial }_{\overline{i}}\}$ is a basis, so the following identities hold:

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\delta }_i}{\delta }_j={\mathrm{\Gamma }}_{ij}^r{\delta }_r+{\mathrm{\Gamma }}_{ij}^{\overline{r}}{\partial }_{\overline{r}},{\tilde{\nabla }}_{{\partial }_{\overline{i}}}{\partial }_{\overline{j}}={\mathrm{\Gamma }}_{\overline{i}\overline{j}}^r{\delta }_r+{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{r}}{\partial }_{\overline{r}},\end{array}$$

(47)

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\delta }_i}{\partial }_{\overline{j}}={\mathrm{\Gamma }}_{i\overline{j}}^r{\delta }_r+{\mathrm{\Gamma }}_{i\overline{j}}^{\overline{r}}{\partial }_{\overline{r}},{\tilde{\nabla }}_{{\partial }_{\overline{i}}}{\delta }_j={\mathrm{\Gamma }}_{\overline{i}j}^r{\delta }_r+{\mathrm{\Gamma }}_{\overline{i}j}^{\overline{r}}{\partial }_{\overline{r}},\end{array}$$

(48)

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\delta }_i}^{*}{\delta }_j={\mathrm{\Gamma }}_{ij}^{*r}{\delta }_r+{\mathrm{\Gamma }}_{ij}^{*\overline{r}}{\partial }_{\overline{r}},{\tilde{\nabla }}_{{\partial }_{\overline{i}}}^{*}{\partial }_{\overline{j}}={\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*r}{\delta }_r+{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*\overline{r}}{\partial }_{\overline{r}},\end{array}$$

(49)

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\delta }_i}^{*}{\partial }_{\overline{j}}={\mathrm{\Gamma }}_{i\overline{j}}^{*r}{\delta }_r+{\mathrm{\Gamma }}_{i\overline{j}}^{*\overline{r}}{\partial }_{\overline{r}},{\tilde{\nabla }}_{{\partial }_{\overline{i}}}^{*}{\delta }_j={\mathrm{\Gamma }}_{\overline{i}j}^{*r}{\delta }_r+{\mathrm{\Gamma }}_{\overline{i}j}^{*\overline{r}}{\partial }_{\overline{r}},\end{array}$$

(50)

where ${\mathrm{\Gamma }}_{BC}^A,{\mathrm{\Gamma }}_{BC}^{*A}\in C^{\infty }(TM)$, and $A,B,C\in \{1,\cdots n,\overline{1},\cdots \overline{n}\}$. If we use torsion-freeness of $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ and applying (47)–(50), then we obtain the following:

Lemma 3.

Let $(M,g)$ be a Riemannian manifold and let $\stackrel{g}{\nabla }$ be an affine connection on it. Let $\stackrel{g}{{\mathrm{\Gamma }}_{ij}^r},\stackrel{g}{T_{ij}^r}$, and $\stackrel{g}{R_{ijs}^r}$ be the connection coefficients, torsion components, and curvature components of $\stackrel{g}{\nabla }$, respectively. If $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ are two affine torsion-free connections on the $TM$, then the following equations hold:

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{i\overline{j}}^r={\mathrm{\Gamma }}_{\overline{j}i}^r,{\mathrm{\Gamma }}_{i\overline{j}}^{\overline{r}}-{\mathrm{\Gamma }}_{\overline{j}i}^{\overline{r}}=\stackrel{g}{{\mathrm{\Gamma }}_{ij}^r}-\stackrel{g}{T_{ij}^r},{\mathrm{\Gamma }}_{ij}^r={\mathrm{\Gamma }}_{ji}^r,\end{array}$$

(51)

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{ij}^{\overline{r}}-{\mathrm{\Gamma }}_{ji}^{\overline{r}}=-y^s\stackrel{g}{R_{ijs}^r},{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^r={\mathrm{\Gamma }}_{\overline{j}\overline{i}}^r,{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{r}}={\mathrm{\Gamma }}_{\overline{j}\overline{i}}^{\overline{r}},\end{array}$$

(52)

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{i\overline{j}}^{*r}={\mathrm{\Gamma }}_{\overline{j}i}^{*r},{\mathrm{\Gamma }}_{i\overline{j}}^{*\overline{r}}-{\mathrm{\Gamma }}_{\overline{j}i}^{*\overline{r}}=\stackrel{g}{{\mathrm{\Gamma }}_{ij}^r}-\stackrel{g}{T_{ij}^r},{\mathrm{\Gamma }}_{ij}^{*r}={\mathrm{\Gamma }}_{ji}^{*r},\end{array}$$

(53)

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{ij}^{*\overline{r}}-{\mathrm{\Gamma }}_{ji}^{*\overline{r}}=-y^s\stackrel{g}{R_{ijs}^r},{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*r}={\mathrm{\Gamma }}_{\overline{j}\overline{i}}^{*r},{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*\overline{r}}={\mathrm{\Gamma }}_{\overline{j}\overline{i}}^{*\overline{r}}.\end{array}$$

(54)

Proof. 

Using (5) and (47)–(50) and the torsion-freeness of $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ completes the proof. □

Now, if we put elements of $\{{\delta }_i,{\partial }_{\overline{i}}\}$ in (6), and use the above relations, we derive the following:

Proposition 2.

Let $(M,g)$ be a Riemannian manifold and let $(TM,\overline{g})$ be its tangent bundle equipped with the Riemannian metric $\overline{g}$ (defined as above). If $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ are two affine torsion-free connections on the $TM$, then $(\overline{g},\tilde{\nabla },{\tilde{\nabla }}^{*})$ is a dualistic structure on the $TM$ if and only if (51)–(54) and the following equations hold:

(1) 

${\partial }_{\overline{i}}{\gamma }_{jk}=2{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^r{\beta }_{rk}+{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{r}}{\gamma }_{rk}+2{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*r}{\beta }_{jr}+{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*\overline{r}}{\gamma }_{jr}$;

(2) 

${\delta }_i{\alpha }_{jk}={\mathrm{\Gamma }}_{ij}^r{\alpha }_{rk}+2{\mathrm{\Gamma }}_{ij}^{\overline{r}}{\beta }_{rk}+{\mathrm{\Gamma }}_{ik}^{*r}{\alpha }_{jr}+2{\mathrm{\Gamma }}^{*\overline{r}}{\beta }_{jr}$;

(3) 

$2{\delta }_i{\beta }_{jk}=2{\mathrm{\Gamma }}_{ij}^r{\beta }_{rk}+{\mathrm{\Gamma }}_{ij}^{\overline{r}}{\gamma }_{rk}+{\mathrm{\Gamma }}_{i\overline{k}}^{*r}{\alpha }_{jr}+2{\mathrm{\Gamma }}_{i\overline{k}}^{*\overline{r}}{\beta }_{jr}$;

(4) 

${\partial }_{\overline{k}}{\alpha }_{ij}={\mathrm{\Gamma }}_{\overline{k}i}^r{\alpha }_{rj}+2{\mathrm{\Gamma }}_{\overline{k}i}^{\overline{r}}{\beta }_{rj}+{\mathrm{\Gamma }}_{\overline{k}j}^{*r}{\alpha }_{ir}+2{\mathrm{\Gamma }}_{\overline{k}j}^{*\overline{r}}{\beta }_{ir}$;

(5) 

$2{\partial }_{\overline{i}}{\beta }_{jk}={\mathrm{\Gamma }}_{\overline{i}\overline{j}}^r{\alpha }_{rk}+2{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{r}}{\beta }_{rk}+2{\mathrm{\Gamma }}_{\overline{i}k}^{*r}{\beta }_{jr}+{\mathrm{\Gamma }}_{\overline{i}k}^{*\overline{r}}{\gamma }_{jr}$;

(6) 

${\delta }_k{\gamma }_{ij}=2{\mathrm{\Gamma }}_{k\overline{i}}^r{\beta }_{rj}+{\mathrm{\Gamma }}_{k\overline{i}}^{\overline{r}}{\gamma }_{rj}+2{\mathrm{\Gamma }}_{k\overline{j}}^{*r}{\beta }_{ir}+{\mathrm{\Gamma }}_{k\overline{j}}^{*\overline{r}}{\gamma }_{ir}$.

Now, we assume that $(\overline{g},\tilde{\nabla },{\tilde{\nabla }}^{*})$ is a dualistic structure on the $TM$, then we show under which conditions the $TM$ is a conjugate symmetric space whenever $\tilde{R}$ and ${\tilde{R}}^{*}$ are curvature tensors of $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$. We do this using a direct computation of the curvature components of $\tilde{R}$ and ${\tilde{R}}^{*}$ on the $TM$. According to the basis $\{{\delta }_i,{\partial }_{\overline{i}}\}$ for $T_v(TM)$, we imply that the curvature components of $\tilde{R}$ and ${\tilde{R}}^{*}$ are as follows:

$$\begin{array}{c}\hfill \tilde{R}({\delta }_i,{\delta }_j){\delta }_k={\tilde{R}}_{ijk}^l{\delta }_l+{\tilde{R}}_{ijk}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\delta }_i,{\delta }_j){\delta }_k={\tilde{R}}_{ijk}^{*l}{\delta }_l+{\tilde{R}}_{ijk}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(55)

$$\begin{array}{c}\hfill \tilde{R}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}={\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^l{\delta }_l+{\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}={\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^{*l}{\delta }_l+{\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(56)

$$\begin{array}{c}\hfill \tilde{R}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={\tilde{R}}_{ij\overline{k}}^l{\delta }_l+{\tilde{R}}_{ij\overline{k}}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={\tilde{R}}_{ij\overline{k}}^{*l}{\delta }_l+{\tilde{R}}_{ij\overline{k}}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(57)

$$\begin{array}{c}\hfill \tilde{R}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j={\tilde{R}}_{i\overline{k}j}^l{\delta }_l+{\tilde{R}}_{i\overline{k}j}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j={\tilde{R}}_{i\overline{k}j}^{*l}{\delta }_l+{\tilde{R}}_{i\overline{k}j}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(58)

$$\begin{array}{c}\hfill \tilde{R}({\partial }_{\overline{k}},{\delta }_i){\delta }_j={\tilde{R}}_{\overline{k}ij}^l{\delta }_l+{\tilde{R}}_{\overline{k}ij}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\partial }_{\overline{k}},{\delta }_i){\delta }_j={\tilde{R}}_{\overline{k}ij}^{*l}{\delta }_l+{\tilde{R}}_{\overline{k}ij}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(59)

$$\begin{array}{c}\hfill \tilde{R}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k={\tilde{R}}_{\overline{i}\overline{j}k}^l{\delta }_l+{\tilde{R}}_{\overline{i}\overline{j}k}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k={\tilde{R}}_{\overline{i}\overline{j}k}^{*l}{\delta }_l+{\tilde{R}}_{\overline{i}\overline{j}k}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(60)

$$\begin{array}{c}\hfill \tilde{R}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}={\tilde{R}}_{\overline{i}k\overline{j}}^l{\delta }_l+{\tilde{R}}_{\overline{i}k\overline{j}}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}={\tilde{R}}_{\overline{i}k\overline{j}}^{*l}{\delta }_l+{\tilde{R}}_{\overline{i}k\overline{j}}^{*\overline{l}}{\partial }_{\overline{l}},\end{array}$$

(61)

$$\begin{array}{c}\hfill \tilde{R}({\delta }_k,{\partial }_{\overline{i}}){\partial }_{\overline{j}}={\tilde{R}}_{k\overline{i}\overline{j}}^l{\delta }_l+{\tilde{R}}_{k\overline{i}\overline{j}}^{\overline{l}}{\partial }_{\overline{l}},{\tilde{R}}^{*}({\delta }_k,{\partial }_{\overline{i}}){\partial }_{\overline{j}}={\tilde{R}}_{k\overline{i}\overline{j}}^{*l}{\delta }_l+{\tilde{R}}_{k\overline{i}\overline{j}}^{*\overline{l}}{\partial }_{\overline{l}}.\end{array}$$

(62)

According to (55)–(62), we obtain the following:

Proposition 3.

Let $(\overline{g},\tilde{\nabla },{\tilde{\nabla }}^{*})$ be a dualistic structure on the $TM$ and let $\tilde{R}$ and ${\tilde{R}}^{*}$ be the curvature tensors of $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ on the $TM$, respectively. Then, the $TM$ is a conjugate symmetric space if and only if the following identities hold:

$$\begin{array}{c}\hfill {\tilde{R}}_{ijk}^l={\tilde{R}}_{ijk}^{*l},{\tilde{R}}_{ijk}^{\overline{l}}={\tilde{R}}_{ijk}^{*\overline{l}},{\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^l={\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^{*l},{\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^{\overline{l}}={\tilde{R}}_{\overline{i}\overline{j}\overline{k}}^{*\overline{l}},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{R}}_{ij\overline{k}}^l={\tilde{R}}_{ij\overline{k}}^{*l},{\tilde{R}}_{ij\overline{k}}^{\overline{l}}={\tilde{R}}_{ij\overline{k}}^{*\overline{l}},{\tilde{R}}_{i\overline{k}j}^l={\tilde{R}}_{i\overline{k}j}^{*l},{\tilde{R}}_{i\overline{k}j}^{\overline{l}}={\tilde{R}}_{i\overline{k}j}^{*\overline{l}},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{R}}_{\overline{k}ij}^l={\tilde{R}}_{\overline{k}ij}^{*l},{\tilde{R}}_{\overline{k}ij}^{\overline{l}}={\tilde{R}}_{\overline{k}ij}^{*\overline{l}},{\tilde{R}}_{\overline{i}\overline{j}k}^l={\tilde{R}}_{\overline{i}\overline{j}k}^{*l},{\tilde{R}}_{\overline{i}\overline{j}k}^{\overline{l}}={\tilde{R}}_{\overline{i}\overline{j}k}^{*\overline{l}},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{R}}_{\overline{i}k\overline{j}}^l={\tilde{R}}_{\overline{i}k\overline{j}}^{*l},{\tilde{R}}_{\overline{i}k\overline{j}}^{\overline{l}}={\tilde{R}}_{\overline{i}k\overline{j}}^{*\overline{l}},{\tilde{R}}_{k\overline{i}\overline{j}}^l={\tilde{R}}_{k\overline{i}\overline{j}}^{*l},{\tilde{R}}_{k\overline{i}\overline{j}}^{\overline{l}}={\tilde{R}}_{k\overline{i}\overline{j}}^{*\overline{l}}.\end{array}$$

6. Mutual Curvatures of the Tangent Bundle

This section is concerned with the mutual curvatures of the $TM$. In this part we study the components of the mutual curvature ${\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}$ of the $TM$ whenever $(\overline{g},\tilde{\nabla },{\tilde{\nabla }}^{*})$ is a dualistic structure on the $TM$.

We assume that $(M,g)$ is a Riemannian manifold with affine connection $\stackrel{g}{\nabla }$ and $\stackrel{g}{{\mathrm{\Gamma }}_{ij}^r},\stackrel{g}{T_{ij}^r}$, and $\stackrel{g}{R_{ijs}^r}$ are the connection coefficients, torsion components, and curvature components of $\stackrel{g}{\nabla }$. According to (47)–(50) and using (34), we have

Theorem 3.

Let $(M,g)$ be a Riemannian manifold and let $(TM,\overline{g})$ be its tangent bundle equipped with the Riemannian metric $\overline{g}$ (defined as above), and let $(\overline{g},\tilde{\nabla },{\tilde{\nabla }}^{*})$ be a dualistic structure on the $TM$. If ${\mathrm{\Gamma }}_{ij}^r$ and ${\mathrm{\Gamma }}_{ij}^{*r}$ are the connection coefficients of $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$ on the $TM$, respectively, then the components of the mutual curvature ${\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}$ on the $TM$ are as follows:

$$\begin{array}{cc}\hfill {\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}({\delta }_i,{\delta }_j){\delta }_k& =\frac{1}{2}\{({\mathrm{\Gamma }}_{jk}^{*r}{\mathrm{\Gamma }}_{ir}^t+{\delta }_i({\mathrm{\Gamma }}_{jk}^{*t})+{\mathrm{\Gamma }}_{jk}^{*\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^t-{\mathrm{\Gamma }}_{ik}^{*r}{\mathrm{\Gamma }}_{jr}^t-{\delta }_j({\mathrm{\Gamma }}_{ik}^{*t})-{\mathrm{\Gamma }}_{ik}^{*\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^t+y^r{\stackrel{g}{R}}_{ijr}^l{\mathrm{\Gamma }}_{\overline{l}k}^t\hfill \\ \hfill & +{\delta }_i({\mathrm{\Gamma }}_{jk}^t)+{\mathrm{\Gamma }}_{jk}^r{\mathrm{\Gamma }}_{ir}^{*t}+{\mathrm{\Gamma }}_{jk}^{\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{*t}-{\delta }_j({\mathrm{\Gamma }}_{ik}^t)-{\mathrm{\Gamma }}_{ik}^r{\mathrm{\Gamma }}_{jr}^{*t}-{\mathrm{\Gamma }}_{ik}^{\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^{*t}+y^l{\stackrel{g}{R}}_{ijl}^s{\mathrm{\Gamma }}_{\overline{s}k}^{*t})\}{\delta }_t\hfill \\ \hfill & +\frac{1}{2}\{({\mathrm{\Gamma }}_{jk}^{*r}{\mathrm{\Gamma }}_{ir}^{\overline{t}}+{\mathrm{\Gamma }}_{jk}^{*\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{\overline{t}}+{\delta }_i({\mathrm{\Gamma }}_{jk}^{*\overline{t}})-{\mathrm{\Gamma }}_{ik}^{*r}{\mathrm{\Gamma }}_{jr}^{\overline{t}}-{\mathrm{\Gamma }}_{ik}^{*\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^{\overline{t}}-{\delta }_j({\mathrm{\Gamma }}_{ik}^{*\overline{t}})+y^r{\stackrel{g}{R}}_{ijr}^l{\mathrm{\Gamma }}_{\overline{l}k}^{\overline{t}}\hfill \\ \hfill & +{\mathrm{\Gamma }}_{jk}^r{\mathrm{\Gamma }}_{ir}^{*\overline{t}}+{\delta }_i({\mathrm{\Gamma }}_{jk}^{\overline{t}})+{\mathrm{\Gamma }}_{jk}^{\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{*\overline{t}}-{\mathrm{\Gamma }}_{ik}^r{\mathrm{\Gamma }}_{jr}^{*\overline{t}}-{\delta }_j({\mathrm{\Gamma }}_{ik}^{\overline{t}})-{\mathrm{\Gamma }}_{ik}^{\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^{*\overline{t}}+y^l{\stackrel{g}{R}}_{ijl}^s{\mathrm{\Gamma }}_{\overline{s}k}^{*\overline{t}})\}{\partial }_{\overline{t}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}& =\frac{1}{2}\{({\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{*r}{\mathrm{\Gamma }}_{\overline{i}r}^t+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{*t})+{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^t-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*r}{\mathrm{\Gamma }}_{\overline{j}r}^t-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*t})-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^t\hfill \\ \hfill & +{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}\overline{k}}^t)+{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^r{\mathrm{\Gamma }}_{\overline{i}r}^{*t}+{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{*t}-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}\overline{k}}^t)-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^r{\mathrm{\Gamma }}_{\overline{j}r}^{*t}-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^{*t})\}{\delta }_t\hfill \\ \hfill & +\frac{1}{2}\{({\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{*r}{\mathrm{\Gamma }}_{\overline{i}r}^{\overline{t}}+{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{\overline{t}}+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{*\overline{t}})-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*r}{\mathrm{\Gamma }}_{\overline{j}r}^{\overline{t}}-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^{\overline{t}}-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{*\overline{t}})\hfill \\ \hfill & +{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^r{\mathrm{\Gamma }}_{\overline{i}r}^{*\overline{t}}+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{\overline{t}})+{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{*\overline{t}}-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^r{\mathrm{\Gamma }}_{\overline{j}r}^{*\overline{t}}-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{\overline{t}})-{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^{*\overline{t}})\}{\partial }_{\overline{t}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}& =\frac{1}{2}\{({\mathrm{\Gamma }}_{j\overline{k}}^{*r}{\mathrm{\Gamma }}_{ir}^t+{\delta }_i({\mathrm{\Gamma }}_{j\overline{k}}^{*t})+{\mathrm{\Gamma }}_{j\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^t-{\mathrm{\Gamma }}_{i\overline{k}}^{*r}{\mathrm{\Gamma }}_{jr}^t-{\delta }_j({\mathrm{\Gamma }}_{i\overline{k}}^{*t})-{\mathrm{\Gamma }}_{i\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^t+y^r{\stackrel{g}{R}}_{ijr}^l{\mathrm{\Gamma }}_{\overline{l}\overline{k}}^t\hfill \\ \hfill & +{\delta }_i({\mathrm{\Gamma }}_{j\overline{k}}^t)+{\mathrm{\Gamma }}_{j\overline{k}}^r{\mathrm{\Gamma }}_{ir}^{*t}+{\mathrm{\Gamma }}_{j\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{*t}-{\delta }_j({\mathrm{\Gamma }}_{i\overline{k}}^t)-{\mathrm{\Gamma }}_{i\overline{k}}^r{\mathrm{\Gamma }}_{jr}^{*t}-{\mathrm{\Gamma }}_{i\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^{*t}+y^r{\stackrel{g}{R}}_{ijr}^s{\mathrm{\Gamma }}_{\overline{s}\overline{k}}^{*t})\}{\delta }_t\hfill \\ \hfill & +\frac{1}{2}\{({\mathrm{\Gamma }}_{j\overline{k}}^{*r}{\mathrm{\Gamma }}_{ir}^{\overline{t}}+{\mathrm{\Gamma }}_{j\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{\overline{t}}+{\delta }_i({\mathrm{\Gamma }}_{j\overline{k}}^{*\overline{t}})-{\mathrm{\Gamma }}_{i\overline{k}}^{*r}{\mathrm{\Gamma }}_{jr}^{\overline{t}}-{\mathrm{\Gamma }}_{i\overline{k}}^{*\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^{\overline{t}}-{\delta }_j({\mathrm{\Gamma }}_{i\overline{k}}^{*\overline{t}})+y^r{\stackrel{g}{R}}_{ijr}^l{\mathrm{\Gamma }}_{\overline{l}\overline{k}}^{\overline{t}}\hfill \\ \hfill & +{\mathrm{\Gamma }}_{j\overline{k}}^r{\mathrm{\Gamma }}_{ir}^{*\overline{t}}+{\delta }_i({\mathrm{\Gamma }}_{j\overline{k}}^{\overline{t}})+{\mathrm{\Gamma }}_{j\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{*\overline{t}}-{\mathrm{\Gamma }}_{i\overline{k}}^r{\mathrm{\Gamma }}_{jr}^{*\overline{t}}-{\delta }_j({\mathrm{\Gamma }}_{i\overline{k}}^{\overline{t}})-{\mathrm{\Gamma }}_{i\overline{k}}^{\overline{r}}{\mathrm{\Gamma }}_{j\overline{r}}^{*\overline{t}}++y^r{\stackrel{g}{R}}_{ijr}^l{\mathrm{\Gamma }}_{\overline{l}\overline{k}}^{*\overline{t}})\}{\partial }_{\overline{t}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j& =\frac{1}{2}\{({\mathrm{\Gamma }}_{\overline{k}j}^{*r}{\mathrm{\Gamma }}_{ir}^t+{\delta }_i({\mathrm{\Gamma }}_{\overline{k}j}^{*t})+{\mathrm{\Gamma }}_{\overline{k}j}^{*\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^t-{\mathrm{\Gamma }}_{ij}^{*r}{\mathrm{\Gamma }}_{\overline{k}r}^t-{\partial }_{\overline{k}}({\mathrm{\Gamma }}_{ij}^{*t})-{\mathrm{\Gamma }}_{ij}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{k}\overline{r}}^t-({\stackrel{g}{\mathrm{\Gamma }}}_{ik}^r-{\stackrel{g}{T}}_{ik}^r){\mathrm{\Gamma }}_{\overline{r}j}^t\hfill \\ \hfill & +{\delta }_i({\mathrm{\Gamma }}_{\overline{k}j}^t)+{\mathrm{\Gamma }}_{\overline{k}j}^r{\mathrm{\Gamma }}_{ir}^{*t}+{\mathrm{\Gamma }}_{\overline{k}j}^{\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{*t}-{\partial }_{\overline{k}}({\mathrm{\Gamma }}_{ij}^t)-{\mathrm{\Gamma }}_{ij}^r{\mathrm{\Gamma }}_{\overline{k}r}^{*t}-{\mathrm{\Gamma }}_{ij}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{k}\overline{r}}^{*t}-({\stackrel{g}{\mathrm{\Gamma }}}_{ik}^r-{\stackrel{g}{T}}_{ik}^r){\mathrm{\Gamma }}_{\overline{r}j}^{*t})\}{\delta }_t\hfill \\ \hfill & +\frac{1}{2}\{({\mathrm{\Gamma }}_{\overline{k}j}^{*r}{\mathrm{\Gamma }}_{ir}^{\overline{t}}+{\mathrm{\Gamma }}_{\overline{k}j}^{*\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{\overline{t}}+{\delta }_i({\mathrm{\Gamma }}_{\overline{k}j}^{*\overline{t}})-{\mathrm{\Gamma }}_{ij}^{*r}{\mathrm{\Gamma }}_{\overline{k}r}^{\overline{t}}-{\mathrm{\Gamma }}_{ij}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{k}\overline{r}}^{\overline{t}}-{\partial }_{\overline{k}}({\mathrm{\Gamma }}_{ij}^{*\overline{t}})-({\stackrel{g}{\mathrm{\Gamma }}}_{ik}^r-{\stackrel{g}{T}}_{ik}^r){\mathrm{\Gamma }}_{\overline{r}j}^{\overline{t}}\hfill \\ \hfill & +{\mathrm{\Gamma }}_{\overline{k}j}^r{\mathrm{\Gamma }}_{ir}^{*\overline{t}}+{\delta }_i({\mathrm{\Gamma }}_{\overline{k}j}^{\overline{t}})+{\mathrm{\Gamma }}_{\overline{k}j}^{\overline{r}}{\mathrm{\Gamma }}_{i\overline{r}}^{*\overline{t}}-{\mathrm{\Gamma }}_{ij}^r{\mathrm{\Gamma }}_{\overline{k}r}^{*\overline{t}}-{\partial }_{\overline{k}}({\mathrm{\Gamma }}_{ij}^{\overline{t}})-{\mathrm{\Gamma }}_{ij}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{k}\overline{r}}^{*\overline{t}}-({\stackrel{g}{\mathrm{\Gamma }}}_{ik}^r-{\stackrel{g}{T}}_{ik}^r){\mathrm{\Gamma }}_{\overline{r}j}^{*\overline{t}})\}{\partial }_{\overline{t}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k& =\frac{1}{2}\{({\mathrm{\Gamma }}_{\overline{j}k}^{*r}{\mathrm{\Gamma }}_{\overline{i}r}^t+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}k}^{*t})+{\mathrm{\Gamma }}_{\overline{j}k}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^t-{\mathrm{\Gamma }}_{\overline{i}k}^{*r}{\mathrm{\Gamma }}_{\overline{j}r}^t-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}k}^{*t})-{\mathrm{\Gamma }}_{\overline{i}k}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^t\hfill \\ \hfill & +{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}k}^t)+{\mathrm{\Gamma }}_{\overline{j}k}^r{\mathrm{\Gamma }}_{\overline{i}r}^{*t}+{\mathrm{\Gamma }}_{\overline{j}k}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{*t}-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}k}^t)-{\mathrm{\Gamma }}_{\overline{i}k}^r{\mathrm{\Gamma }}_{\overline{j}r}^{*t}-{\mathrm{\Gamma }}_{\overline{i}k}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^{*t})\}{\delta }_t\hfill \\ \hfill & +\frac{1}{2}\{({\mathrm{\Gamma }}_{\overline{j}k}^{*r}{\mathrm{\Gamma }}_{\overline{i}r}^{\overline{t}}+{\mathrm{\Gamma }}_{\overline{j}k}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{\overline{t}}+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}k}^{*\overline{t}})-{\mathrm{\Gamma }}_{\overline{i}k}^{*r}{\mathrm{\Gamma }}_{\overline{j}r}^{\overline{t}}-{\mathrm{\Gamma }}_{\overline{i}k}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^{\overline{t}}-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}k}^{*\overline{t}})\hfill \\ \hfill & +{\mathrm{\Gamma }}_{\overline{j}k}^r{\mathrm{\Gamma }}_{\overline{i}r}^{*\overline{t}}+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{\overline{j}k}^{\overline{t}})+{\mathrm{\Gamma }}_{\overline{j}k}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{*\overline{t}}-{\mathrm{\Gamma }}_{\overline{i}k}^r{\mathrm{\Gamma }}_{\overline{j}r}^{*\overline{t}}-{\partial }_{\overline{j}}({\mathrm{\Gamma }}_{\overline{i}k}^{\overline{t}})-{\mathrm{\Gamma }}_{\overline{i}k}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{j}\overline{r}}^{*\overline{t}})\}{\partial }_{\overline{t}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{R}}_{\tilde{\nabla },{\tilde{\nabla }}^{*}}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}& =\frac{1}{2}\{({\mathrm{\Gamma }}_{k\overline{j}}^{*r}{\mathrm{\Gamma }}_{\overline{i}r}^t+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{k\overline{j}}^{*t})+{\mathrm{\Gamma }}_{k\overline{j}}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^t-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*r}{\mathrm{\Gamma }}_{kr}^t-{\delta }_k({\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*t})-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*\overline{r}}{\mathrm{\Gamma }}_{k\overline{r}}^t+({\stackrel{g}{\mathrm{\Gamma }}}_{ki}^r-{\stackrel{g}{T}}_{ki}^r){\mathrm{\Gamma }}_{\overline{r}\overline{j}}^t\hfill \\ \hfill & +{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{k\overline{j}}^t)+{\mathrm{\Gamma }}_{k\overline{j}}^r{\mathrm{\Gamma }}_{\overline{i}r}^{*t}+{\mathrm{\Gamma }}_{k\overline{j}}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^t-{\delta }_k({\mathrm{\Gamma }}_{\overline{i}\overline{j}}^t)-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^r{\mathrm{\Gamma }}_{kr}^t-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{r}}{\mathrm{\Gamma }}_{k\overline{r}}^{*t}+({\stackrel{g}{\mathrm{\Gamma }}}_{ki}^r-{\stackrel{g}{T}}_{ki}^r){\mathrm{\Gamma }}_{\overline{r}\overline{j}}^{*t})\}{\delta }_t\hfill \\ \hfill & +\frac{1}{2}\{({\mathrm{\Gamma }}_{k\overline{j}}^{*r}{\mathrm{\Gamma }}_{\overline{i}r}^{\overline{t}}+{\mathrm{\Gamma }}_{k\overline{j}}^{*\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{\overline{t}}+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{k\overline{j}}^{*\overline{t}})-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*r}{\mathrm{\Gamma }}_{kr}^{\overline{t}}-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*\overline{r}}{\mathrm{\Gamma }}_{k\overline{r}}^{\overline{t}}-{\delta }_k({\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{*\overline{t}})+({\stackrel{g}{\mathrm{\Gamma }}}_{ki}^r-{\stackrel{g}{T}}_{ki}^r){\mathrm{\Gamma }}_{\overline{r}\overline{j}}^{\overline{t}}\hfill \\ \hfill & +{\mathrm{\Gamma }}_{k\overline{j}}^r{\mathrm{\Gamma }}_{\overline{i}r}^{*\overline{t}}+{\partial }_{\overline{i}}({\mathrm{\Gamma }}_{k\overline{j}}^{\overline{t}})+{\mathrm{\Gamma }}_{k\overline{j}}^{\overline{r}}{\mathrm{\Gamma }}_{\overline{i}\overline{r}}^{\overline{t}}-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^r{\mathrm{\Gamma }}_{kr}^{\overline{t}}-{\delta }_k({\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{t}})-{\mathrm{\Gamma }}_{\overline{i}\overline{j}}^{\overline{r}}{\mathrm{\Gamma }}_{k\overline{r}}^{*\overline{t}}+({\stackrel{g}{\mathrm{\Gamma }}}_{ki}^r-{\stackrel{g}{T}}_{ki}^r){\mathrm{\Gamma }}_{\overline{r}\overline{j}}^{*\overline{t}})\}{\partial }_{\overline{t}}.\hfill \end{array}$$

Now, we consider the pair $(\stackrel{H}{\nabla },\stackrel{C}{\nabla })$ of connections on the $TM$ and study the mutual curvature ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ on the $TM$ and its relation with the curvature of M. Equations (34) and (4) imply the following:

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}={\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j={\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k={\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}=0,\end{array}$$

(63)

$$\begin{array}{cc}\hfill {\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}({\delta }_i,{\delta }_j){\delta }_k& ={(R({\partial }_i,{\partial }_j){\partial }_k)}^H+\frac{1}{2}{({\nabla }_{{\partial }_i}R(y,{\partial }_j){\partial }_k-{\nabla }_{{\partial }_j}R(y,{\partial }_i){\partial }_k)}^V\hfill \\ \hfill & +\frac{1}{2}{(R(y,{\partial }_i){\nabla }_{{\partial }_j}{\partial }_k-R(y,{\partial }_j){\nabla }_{{\partial }_i}{\partial }_k)}^V,\hfill \end{array}$$

(64)

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={(R({\partial }_i,{\partial }_j){\partial }_k)}^V.\end{array}$$

(65)

If ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ vanishes, then (64) and (65) imply that M is a flat space and ${\nabla }_{{\partial }_i}R(y,{\partial }_j){\partial }_k={\nabla }_{{\partial }_j}R(y,{\partial }_i){\partial }_k$, for all $i,j,k$. Conversely, if M is flat, then ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ vanishes.

Here, we investigate the mutual curvature ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ on the $TM$ and its relation with the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$, where $({\nabla }^1,{\nabla }^2)$ is a pair of connections on M such that $R_1$ and $R_2$ are curvature tensors of ${\nabla }^1$ and ${\nabla }^2$, respectively. From (34) and (4) we conclude the following:

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}={\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j={\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}({\partial }_{\overline{i}},{\partial }_{\overline{i}}){\delta }_k={\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}=0,\end{array}$$

(66)

$$\begin{array}{cc}\hfill & {\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}({\delta }_i,{\delta }_j){\delta }_k\hfill \\ \hfill & ={(R_{{\nabla }^1,{\nabla }^2}({\partial }_i,{\partial }_j){\partial }_k)}^H+\frac{1}{2}{(R_1(y,{\partial }_i){\nabla }_{{\partial }_j}^2{\partial }_k-R_1(y,{\partial }_j){\nabla }_{{\partial }_i}^2{\partial }_k)}^V\hfill \\ \hfill & +\frac{1}{2}{({\nabla }_{{\partial }_i}^1{R}_2(y,{\partial }_j){\partial }_k-{\nabla }_{{\partial }_j}^1{R}_2(y,{\partial }_i){\partial }_k)}^V+\frac{1}{2}{(R_2(y,{\partial }_i){\nabla }_{{\partial }_j}^1{\partial }_k-R_2(y,{\partial }_j){\nabla }_{{\partial }_i}^1{\partial }_k)}^V\hfill \\ \hfill & +\frac{1}{2}{({\nabla }_{{\partial }_i}^2{R}_1(y,{\partial }_j){\partial }_k-{\nabla }_{{\partial }_j}^2{R}_1(y,{\partial }_i){\partial }_k)}^V,\hfill \end{array}$$

(67)

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={(R_{{\nabla }^1,{\nabla }^2}({\partial }_i,{\partial }_j){\partial }_k)}^V.\end{array}$$

(68)

According to (66)–(68), we derive that if the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes and M is a flat space with respect to ${\nabla }^1$ and ${\nabla }^2$, then the mutual curvature ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ vanishes. Conversely, if the mutual curvature ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ vanishes, then from (68) we see that the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes. Furthermore, if we consider the pair $\{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}\}$ of connections on the $TM$, then, the same as in the above discussion, we obtain that all of the components of the mutual curvature ${\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}$ are zero except for

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}({\delta }_i,{\delta }_j){\delta }_k={(R_{{\nabla }^1,{\nabla }^2}({\partial }_i,{\partial }_j){\partial }_k)}^H,{\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={(R_{{\nabla }^1,{\nabla }^2}({\partial }_i,{\partial }_j){\partial }_k)}^V.\end{array}$$

(69)

Therefore, from (69) we deduce that ${\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}$ vanishes if and only if $R_{{\nabla }^1,{\nabla }^2}$ vanishes. As mentioned in [14], if ∇ is a flat and torsion-free connection, then $\stackrel{C}{\nabla }=\stackrel{H}{\nabla }$. It follows that, if M is a flat space and ∇ is the Levi-Civita connection, then the mutual curvature ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ reduces to the Riemannian curvature of the Levi-Civita connection ${\nabla }^S$, because $\stackrel{C}{\nabla }$ and $\stackrel{H}{\nabla }$ reduce to the ${\nabla }^S$. Now, we consider a pair of Levi-Civita connections $({\nabla }^1,{\nabla }^2)$ such that M is a flat space with respect to the ${\nabla }^i$ for $i=1,2$. Since $\stackrel{C}{{\nabla }^i}=\stackrel{H}{{\nabla }^i}$ for $i=1,2$, thus, the mutual curvatures ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ and ${\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}$ coincide and they reduce to the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$. So, according to above discussion we have the following:

Lemma 4.

Let $(M,g)$ be a Riemannian manifold with an affine connection ∇ and curvature tensor R and let $({\nabla }^1,{\nabla }^2)$ be a pair of connections on M such that $R_1$ and $R_2$ are curvature tensors of ${\nabla }^1$ and ${\nabla }^2$, respectively. Then, the following statements hold:

(1) 

If M is a flat space, then the mutual curvature ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ is zero.

(2) 

If M is a flat space and ∇ is the Levi-Civita connection, then the mutual curvature ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ reduces to the Riemannian curvature of the Levi-Civita connection ${\nabla }^S$.

(3) 

If the mutual curvature ${\tilde{R}}_{\stackrel{H}{\nabla },\stackrel{C}{\nabla }}$ vanishes, then M is flat.

(4) 

If the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes and M is a flat space with respect to ${\nabla }^1$ and ${\nabla }^2$, then the mutual curvature ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ vanishes.

(5) 

If the mutual curvature ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ vanishes, then the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes. Moreover, the mutual curvature ${\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}$ vanishes if and only if the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes.

(6) 

If ${\nabla }^1$ and ${\nabla }^2$ are Levi-Civita connections and M is a flat space with respect to ${\nabla }^i$ for $i=1,2$, then ${\tilde{R}}_{\stackrel{C}{{\nabla }^1},\stackrel{C}{{\nabla }^2}}$ and ${\tilde{R}}_{\stackrel{H}{{\nabla }^1},\stackrel{H}{{\nabla }^2}}$ are equal and reduce to the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$.

6.1. Mutual Curvature with Respect to a Pair of Levi-Civita Connections in the Tangent Bundle

Let M be a smooth manifold and $g_1$ and $g_2$ be two non-isometric Riemannian metrics with the Levi-Civita connections ${\nabla }^1$ and ${\nabla }^2$ and the Riemannian curvature tensors $R_1$ and $R_2$, respectively. We consider the pair $(\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S})$ on the $TM$ such that $\stackrel{1}{{\nabla }^S}$ and $\stackrel{2}{{\nabla }^S}$ are the Levi-Civita connections of the Sasaki lift metrics $g_1^S$ and $g_2^S$, respectively. Now, we study on the components of the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$ on the $TM$ and its relation with the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ on M. Equations (34), (5), (2), and (3) give us the following:

$$\begin{array}{cc}\hfill & {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}({\delta }_i,{\delta }_j){\delta }_k={(R_{{\nabla }^1,{\nabla }^2}({\partial }_i,{\partial }_j){\partial }_k)}^H\hfill \\ \hfill & +{\{\frac{1}{4}{\stackrel{1}{\nabla }}_{{\partial }_j}{R}_2({\partial }_i,{\partial }_k)y+\frac{1}{4}{R}_1({\partial }_j,{\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_k)y-\frac{1}{4}{\stackrel{1}{\nabla }}_{{\partial }_i}{R}_2({\partial }_j,{\partial }_k)y-\frac{1}{4}{R}_1({\partial }_i,{\stackrel{2}{\nabla }}_{{\partial }_j}{\partial }_k)y\}}^V\hfill \\ \hfill & +{\{\frac{1}{4}{R}_1(y,R_1({\partial }_i,{\partial }_j)y){\partial }_k+\frac{1}{8}{R}_1(y,R_2({\partial }_i,{\partial }_k)y){\partial }_j-\frac{1}{8}{R}_1(y,R_2({\partial }_j,{\partial }_k)y){\partial }_i\}}^H\hfill \\ \hfill & +{\{\frac{1}{4}{\stackrel{2}{\nabla }}_{{\partial }_j}{R}_1({\partial }_i,{\partial }_k)y+\frac{1}{4}{R}_2({\partial }_j,{\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k)y-\frac{1}{4}{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1({\partial }_j,{\partial }_k)y-\frac{1}{4}{R}_2({\partial }_i,{\stackrel{1}{\nabla }}_{{\partial }_j}{\partial }_k)y\}}^V\hfill \\ \hfill & +{\{\frac{1}{4}{R}_2(y,R_2({\partial }_i,{\partial }_j)y){\partial }_k+\frac{1}{8}{R}_2(y,R_1({\partial }_i,{\partial }_k)y){\partial }_j-\frac{1}{8}{R}_2(y,R_1({\partial }_j,{\partial }_k)y){\partial }_i\}}^H.\hfill \end{array}$$

(70)

If we consider $y=0$ in (70), then, except the first term, all of the terms on the right-hand side of the equation are zero. This implies that if ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$ vanishes, then $R_{{\nabla }^1,{\nabla }^2}$ vanishes. To continue, we also have

$$\begin{array}{cc}\hfill & {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={(R_{{\nabla }^1,{\nabla }^2}({\partial }_i,{\partial }_j){\partial }_k)}^V\hfill \\ \hfill & +{\{\frac{1}{4}{R}_1(y,{\stackrel{2}{\nabla }}_{{\partial }_j}{\partial }_k){\partial }_i+\frac{1}{4}{\stackrel{1}{\nabla }}_{{\partial }_i}{R}_2(y,{\partial }_k){\partial }_j-\frac{1}{4}{R}_1(y,{\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j-\frac{1}{4}{\stackrel{1}{\nabla }}_{{\partial }_j}{R}_2(y,{\partial }_k){\partial }_i\}}^H\hfill \\ \hfill & -{\{\frac{1}{8}{R}_1({\partial }_j,R_2(y,{\partial }_k){\partial }_i)y+\frac{1}{8}{R}_1({\partial }_i,R_2(y,{\partial }_k){\partial }_j)y\}}^V,\hfill \\ \hfill & +{\{\frac{1}{4}{R}_2(y,{\stackrel{1}{\nabla }}_{{\partial }_j}{\partial }_k){\partial }_i+\frac{1}{4}{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1(y,{\partial }_k){\partial }_j-\frac{1}{4}{R}_2(y,{\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j-\frac{1}{4}{\stackrel{2}{\nabla }}_{{\partial }_j}{R}_1(y,{\partial }_k){\partial }_i\}}^H\hfill \\ \hfill & -{\{\frac{1}{8}{R}_2({\partial }_j,R_1(y,{\partial }_k){\partial }_i)y+\frac{1}{8}{R}_2({\partial }_i,R_1(y,{\partial }_k){\partial }_j)y\}}^V.\hfill \end{array}$$

(71)

Furthermore, we have

$$\begin{array}{cc}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j& =\frac{1}{4}{\{{\stackrel{1}{\nabla }}_{{\partial }_i}{R}_2(y,{\partial }_k){\partial }_j-R_1(y,{\partial }_k){\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_j-R_1(y,{\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j\}}^H\hfill \\ \hfill & -\frac{1}{8}{(R_1({\partial }_i,R_2(y,{\partial }_k){\partial }_j)y)}^V,\hfill \\ \hfill & +\frac{1}{4}{\{{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1(y,{\partial }_k){\partial }_j-R_2(y,{\partial }_k){\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_j-R_2(y,{\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j\}}^H\hfill \\ \hfill & -\frac{1}{8}{(R_2({\partial }_i,R_1(y,{\partial }_k){\partial }_j)y)}^V,\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k& =\frac{1}{8}{\{R_1(y,{\partial }_i)R_2(y,{\partial }_j){\partial }_k-R_1(y,{\partial }_j)R_2(y,{\partial }_i){\partial }_k\}}^H\hfill \\ \hfill & +\frac{1}{8}{\{R_2(y,{\partial }_i)R_1(y,{\partial }_j){\partial }_k-R_2(y,{\partial }_j)R_1(y,{\partial }_i){\partial }_k\}}^H,\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill & {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}=\frac{1}{8}{\{R_1(y,{\partial }_i)R_2(y,{\partial }_j){\partial }_k+R_2(y,{\partial }_i)R_1(y,{\partial }_j){\partial }_k\}}^H,\hfill \\ & {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}=0.\hfill \end{array}$$

(74)

Using (70)–(74), we obtain the following:

Theorem 4.

Let M be a smooth manifold and $g_1$ and $g_2$ be two non-isometric Riemannian metrics with the Levi-Civita connections ${\nabla }^1$ and ${\nabla }^2$ and the Riemannian curvature tensors $R_1$ and $R_2$ such that $\stackrel{1}{{\nabla }^S}$ and $\stackrel{2}{{\nabla }^S}$ are the Levi-Civita connections of the Sasaki lift metrics $g_1^S$ and $g_2^S$, respectively. Then, the following statements hold:

(1) 

The mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$ vanishes if the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes and M is a flat space with respect to ${\nabla }^1$ and ${\nabla }^2$.

(2) 

If the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^S}}$ vanishes, then the mutual curvature $R_{{\nabla }^1,{\nabla }^2}$ vanishes.

6.2. Mutual Curvatures in the Tangent Bundle with Different Connections

In this part, we study the mutual curvatures ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}$ and ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}$ on the $TM$ and their geometric consequences, where ${\nabla }^1$ is the Levi-Civita connection of the metric g.

Now, we consider Riemannian manifold $(M,g)$ with the Levi-Civita connection $\stackrel{1}{\nabla }$ and Riemannian curvature $R_1$. Let $\stackrel{2}{\nabla }$ be an arbitrary affine connection on M. If we denote the Levi-Civita connection of the Sasaki lift metric $g^S$ by $\stackrel{1}{{\nabla }^S}$ and the horizontal lift connection of $\stackrel{2}{\nabla }$ by $\stackrel{2}{{\nabla }^H}$, then from (34) and (4), we deduce that the components of the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}$ are as follows:

$$\begin{array}{cc}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}({\delta }_i,{\delta }_j){\delta }_k& ={(R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}({\partial }_i,{\partial }_j){\partial }_k)}^H+\frac{1}{4}{\{R_1({\partial }_j,{\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_k)y-R_1({\partial }_i,{\stackrel{2}{\nabla }}_{{\partial }_j}{\partial }_k)y\}}^V\hfill \\ \hfill & +\frac{1}{4}{\{R_1(y,R_1({\partial }_i,{\partial }_j)y){\partial }_k\}}^H+\frac{1}{4}{\{{\stackrel{2}{\nabla }}_{{\partial }_j}{R}_1({\partial }_i,{\partial }_k)y-{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1({\partial }_j,{\partial }_k)y\}}^V.\hfill \end{array}$$

(75)

Setting $y=0$ in (75), the second, third, and fourth terms on the right-hand side of Equation (75) are zero. In this case, if we assume that the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}$ vanishes, then the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ vanishes. We have

$$\begin{array}{cc}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}& ={(R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}({\partial }_i,{\partial }_j){\partial }_k)}^V+\frac{1}{4}{\{R_1(y,{\stackrel{2}{\nabla }}_{{\partial }_j}{\partial }_k){\partial }_i-R_1(y,{\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j\}}^H\hfill \\ \hfill & +\frac{1}{4}{\{{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1(y,{\partial }_k){\partial }_j-{\stackrel{2}{\nabla }}_{{\partial }_j}{R}_1(y,{\partial }_k){\partial }_i\}}^H,\hfill \end{array}$$

(76)

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j=\frac{1}{4}{\{{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1(y,{\partial }_k){\partial }_j-R_1(y,{\partial }_k){\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_j-R_1(y,{\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j\}}^H,\end{array}$$

(77)

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}={\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k={\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}=0.\end{array}$$

(78)

According to (75)–(78) we derive that if $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ vanishes and M is a flat space with respect to $\stackrel{1}{\nabla }$, then ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}$ vanishes. As a special case, if $\stackrel{1}{\nabla }=\stackrel{2}{\nabla }=\nabla$, where ∇ is the Levi-Civita connection of g, then from (75)–(78) we derive that the mutual curvature ${\tilde{R}}_{{\nabla }^S,{\nabla }^H}$ vanishes if M is a flat space with respect to ∇. Moreover, in this case, ${\tilde{R}}_{{\nabla }^S,{\nabla }^H}$ reduces to the Riemannian curvature of the Levi-Civita connection ${\nabla }^S$. Furthermore, if ${\tilde{R}}_{{\nabla }^S,{\nabla }^H}$ vanishes, then M is a flat space. So, as a consequence of the above discussion we have the following:

Theorem 5.

Let $(M,g)$ be a Riemannian manifold with the Levi-Civita connection $\stackrel{1}{\nabla }$ and let $\stackrel{2}{\nabla }$ be an affine connection on M. If we denote the Levi-Civita connection of the Sasaki lift metric $g^S$ by $\stackrel{1}{{\nabla }^S}$ and the horizontal lift connection of $\stackrel{2}{\nabla }$ by $\stackrel{2}{{\nabla }^H}$, then the following assertions hold:

(1) 

If the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}$ vanishes, then the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ vanishes. Moreover, if the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ vanishes and M is a flat space with respect to $\stackrel{1}{\nabla }$, then the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^H}}$ vanishes.

(2) 

The mutual curvature ${\tilde{R}}_{\stackrel{S}{\nabla },\stackrel{H}{\nabla }}$ vanishes if and only if M is a flat space with respect to ∇, where ∇ is the Levi-Civita connection of metric g. Furthermore, in this case, the mutual curvature ${\tilde{R}}_{\stackrel{S}{\nabla },\stackrel{H}{\nabla }}$ reduces to the Riemannian curvature of the Levi-Civita connection ${\nabla }^S$.

Now, we consider a Riemannian manifold $(M,g)$ with the Levi-Civita connection $\stackrel{1}{\nabla }$. Let $\stackrel{2}{\nabla }$ be an affine connection on M, where $R_1$ and $R_2$ are curvature tensors of $\stackrel{1}{\nabla }$ and $\stackrel{2}{\nabla }$, respectively. If we denote the Levi-Civita connection of the Sasaki lift metric $g^S$ by $\stackrel{1}{{\nabla }^S}$ and the complete lift connection of $\stackrel{2}{\nabla }$ by $\stackrel{2}{{\nabla }^C}$, then from (34) and (4) we conclude that the components of the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}$ are as follows:

$$\begin{array}{cc}\hfill & {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}({\delta }_i,{\delta }_j){\delta }_k={(R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}({\partial }_i,{\partial }_j){\partial }_k)}^H\hfill \\ \hfill & +\frac{1}{4}{\{R_1(y,R_1({\partial }_i,{\partial }_j)y){\partial }_k+R_1(y,R_2(y,{\partial }_j){\partial }_k){\partial }_i-R_1(y,R_2(y,{\partial }_i){\partial }_k){\partial }_j\}}^H\hfill \\ \hfill & +{\{\frac{1}{4}{R}_1({\partial }_j,{\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_k)y-\frac{1}{4}{R}_1({\partial }_i,{\stackrel{2}{\nabla }}_{{\partial }_j}{\partial }_k)y+\frac{1}{2}{\stackrel{1}{\nabla }}_{{\partial }_i}{R}_2(y,{\partial }_j){\partial }_k-\frac{1}{2}{\stackrel{1}{\nabla }}_{{\partial }_j}{R}_2(y,{\partial }_i){\partial }_k\}}^V\hfill \\ \hfill & +{\{\frac{1}{2}{R}_2(y,{\partial }_i){\stackrel{1}{\nabla }}_{{\partial }_j}{\partial }_k-\frac{1}{2}{R}_2(y,{\partial }_j){\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k+\frac{1}{4}{\stackrel{2}{\nabla }}_{{\partial }_j}{R}_1({\partial }_i,{\partial }_k)y-\frac{1}{4}{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1({\partial }_j,{\partial }_k)y\}}^V.\hfill \end{array}$$

(79)

If we put $y=0$ in (79), then we derive that the second, third and fourth terms on the right-hand side of Equation (79) are zero. Thus, if ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}=0$, then from (79) we obtain $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}=0$. Furthermore, we have

$$\begin{array}{cc}\hfill & {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}({\delta }_i,{\delta }_j){\partial }_{\overline{k}}={(R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}({\partial }_i,{\partial }_j){\partial }_k)}^V\hfill \\ & +\frac{1}{4}{\{R_2(y,{\partial }_i)R_1(y,{\partial }_k){\partial }_j-R_2(y,{\partial }_j)R_1(y,{\partial }_k){\partial }_i\}}^V\hfill \\ \hfill & +\frac{1}{4}{\{R_1(y,{\stackrel{2}{\nabla }}_{{\partial }_j}{\partial }_k){\partial }_i-R_1(y,{\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j+{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1(y,{\partial }_k){\partial }_j-{\stackrel{2}{\nabla }}_{{\partial }_j}{R}_1(y,{\partial }_k){\partial }_i\}}^H,\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}({\delta }_i,{\partial }_{\overline{k}}){\delta }_j& =\frac{1}{4}{\{{\stackrel{2}{\nabla }}_{{\partial }_i}{R}_1(y,{\partial }_k){\partial }_j-R_1(y,{\partial }_k){\stackrel{2}{\nabla }}_{{\partial }_i}{\partial }_j-R_1(y,{\stackrel{1}{\nabla }}_{{\partial }_i}{\partial }_k){\partial }_j\}}^H\hfill \\ \hfill & +\frac{1}{4}{\{R_2(y,{\partial }_i)R_1(y,{\partial }_k){\partial }_j\}}^V,\hfill \end{array}$$

(81)

$$\begin{array}{c}\hfill {\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\delta }_k={\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}({\partial }_{\overline{i}},{\delta }_k){\partial }_{\overline{j}}={\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}({\partial }_{\overline{i}},{\partial }_{\overline{j}}){\partial }_{\overline{k}}=0.\end{array}$$

(82)

From (79)–(82), we deduce that if $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}=0$ and M is a flat space with respect to $\stackrel{1}{\nabla }$ and $\stackrel{2}{\nabla }$, then ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}=0$. In the special case where $\stackrel{1}{\nabla }=\stackrel{2}{\nabla }=\nabla$, where ∇ is the Levi-Civita connection of metric g, then from (79)–(82) we obtain that if the mutual curvature ${\tilde{R}}_{{\nabla }^S,{\nabla }^C}$ vanishes, then M is a flat space. Moreover, if M is a flat space, then ${\tilde{R}}_{{\nabla }^S,{\nabla }^C}$ vanishes and in this case, ${\tilde{R}}_{{\nabla }^S,{\nabla }^C}$ reduces to the Riemannian curvature of the Levi-Civita connection ${\nabla }^S$. Thus, as a result of the above discussion we derive the following:

Theorem 6.

Let $(M,g)$ be a Riemannian manifold with the Levi-Civita connection $\stackrel{1}{\nabla }$ and let $\stackrel{2}{\nabla }$ be an affine connection on M with the curvature tensors $R_1$ and $R_2$, respectively. If $\stackrel{1}{{\nabla }^S}$ is the Levi-Civita connection of the Sasaki lift metric $g^S$ and $\stackrel{2}{{\nabla }^C}$ is the complete lift connection of $\stackrel{2}{\nabla }$, then the following assertions hold:

(1) 

If the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}$ vanishes, then the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ vanishes. Moreover, if the mutual curvature $R_{\stackrel{1}{\nabla },\stackrel{2}{\nabla }}$ vanishes and M is a flat space with respect to $\stackrel{1}{\nabla }$ and $\stackrel{2}{\nabla }$, then the mutual curvature ${\tilde{R}}_{\stackrel{1}{{\nabla }^S},\stackrel{2}{{\nabla }^C}}$ vanishes.

(2) 

The mutual curvature ${\tilde{R}}_{\stackrel{S}{\nabla },\stackrel{C}{\nabla }}$ vanishes if and only if M is a flat space with respect to ∇, where ∇ is the Levi-Civita connection of metric g. Moreover, in this case the mutual curvature ${\tilde{R}}_{\stackrel{S}{\nabla },\stackrel{C}{\nabla }}$ reduces to the Riemannian curvature of the Levi-Civita connection ${\nabla }^S$.