Statistical connections on decomposable Riemann manifold

1.   Introduction

Statistical structures in modern differential geometry have studied by many authors in recent years. One of them is Lauritzen. In ^[1], Lauritzen created a statistical manifold by defining a totally symmetric tensor field $C$ (cubic form) of type $(0, 3)$ on a Riemann manifold $(M_{n}, g)$. He has shown that there is a torsion-free linear connection $^{(\alpha)}\nabla$ such that $^{(\alpha)}\nabla g = \alpha C$, where $g$ is Riemann metric and $\alpha = \pm 1$. Then, he examined some properties of the curvature tensor field and defined the dual connection of this connection. Also, he showed the relationship between curvature and dual curvature tensor field of that connection and presented examples on the statistical manifold.

In this paper, we create a special connection inspired by statistical manifold on locally product Riemann manifold $(M_{n}, \varphi, g)$. We call this new connection as statistical $(\alpha, \varphi)$-connection. We investigate the decomposable condition for cubic form $C$ expressed by the product structure $\varphi$. Then, we calculate the curvature tensor field of that connection and examine its some properties. We give two examples that support this connection. Finally, we define the dual of the new connection and investigate its curvature tensor field.

2.   Preliminaries

Let $M_{n}$ be a $n$-dimensional manifold. Throughout this article, all tensor fields, linear connections, and manifolds will always be regarded as differentiable of class $C^{\infty }$. The class of $(p, q)$-type tensor fields will also be denoted $\Im _{q}^{p}(M_{n})$. For example if the tensor field $V$ is of type $(1, 2)$, then $V\in \Im _{2}^{1}(M_{n})$.

The tensor field $K$ of type $(0, q)$ is called pure with respect to the $\varphi$ if the following equation holds:

where $\varphi$ is endomorphism, namely, $\varphi \in \Im _{1}^{1}(M_{n})$ and $Y_{1}, Y_{2}, ..., Y_{q}\in \Im _{0}^{1}(M_{n})$ [2,p.208]. Then, the $\Phi$ operator (or Tachibana operator) applied to pure tensor field $K$ of type $(0, q)$ is given by

where $L_{Y}$ is the Lie differentiation according to a vector field $Y$ [2,p.211].

In the equation (2.1), if $\Phi _{\varphi }K = 0$, then $K$ is named $\Phi$-tensor field. Especially, if $\varphi$ is product structure, that is, $\varphi ^{2} = I$ and $\Phi _{\varphi }K = 0$, then $K$ is called a decomposable tensor field [2,p.214].

The almost product Riemann manifold $(M_{n}, \varphi, g)$ is a manifold that satisfies

and $\varphi ^{2} = I$, where $g$ is Riemann metric. In ^[3], the authors (in Theorem 1) show that in almost product Riemann manifold, if $\Phi _{\varphi }g = 0$, then $\varphi$ is integrable. Then, it is clear that the condition $\Phi _{\varphi }g = 0$ is equivalent $\nabla \varphi = 0$, where $\nabla$ is Riemann (or Levi-Civita) connection of Riemann metric $g$. It is well-known that if $\varphi$ is integrable, then the triplet $(M_{n}, \varphi, g)$ is named locally product Riemann manifold. Besides, locally product Riemann manifold $(M_{n}, \varphi, g)$ is a locally decomposable if and only if the product structure $\varphi$ is parallel according to the Riemann connection $\nabla$, in other words, $\nabla \varphi = 0$ [4,p.420]. Thus, it is easily said that the $(M_{n}, \varphi, g)$ is a locally decomposable Riemann manifold if and only if $\Phi _{\varphi }g = 0$ ^[3] (in Theorem 2).

In adition, in ^[5], the authors (in Proposition 4.2) examined properties of the Riemann curvature tensor field $R$ of the locally product Riemann manifold $(M_{n}, \varphi, g)$ and showed that $\Phi _{\varphi }R = 0$, that is, the Riemann curvature tensor field $R$ is decomposable.

3.   Statistical ($\alpha, \varphi)$-connections

Let $(M_{n}, g, \varphi)$ be a locally decomposable Riemann manifold and $^{(\alpha)}\nabla$ be a torsion-free linear connection on this manifold that provides the following equation:

Then, the connection $^{(\alpha)}\nabla$ is named statistical $\alpha$ -connection If $(^{(\alpha)}\nabla _{X}g)(Y, Z) = \alpha C(X, Y, Z)$ is satisfied and $C$ is totally symmetric, where $\alpha = \pm 1$, $C$ is a the cubic form such that $C(X, Y, Z) = g(\overline{C}(X, Y), Z)$ and $\nabla$ is Riemann connection of metric $g$ [1,p.179–180].

In this paper, we will study a special version of cubic form $\overline{C}$, which is expressed as follows:

where $\varphi$ is a product structure, that is, $\varphi ^{2}(X) = I(X)$, $\eta$ is a covector field (or 1-form) and $U$ is a vector field such that $U = g^{\sharp }(\eta) = \eta ^{\sharp }$, where $g^{\sharp }:\Im _{1}^{0}(M_{n})\longrightarrow \Im _{0}^{1}(M_{n})$, that is $g^{\sharp }$ is a musical isomorphisms. From the equation (3.2) and $C(X, Y, Z) = g(\overline{C}(X, Y), Z)$, we have

Then, it is clear that

and $C(X, Y, Z)$ is completely symmetric, that is,

From the (3.2), we get

Proposition 3.1. The product structure $\varphi$ is parallel according to the $\alpha$ -connection $^{(\alpha)}\nabla$ on locally decomposable Riemann manifold $(M_{n}, g, \varphi)$, that is, $^{(\alpha)}\nabla \varphi = 0$.

Throughout this paper, the $\alpha$-connection $^{(\alpha)}\nabla$ on locally decomposable Riemann manifold $(M_{n}, g, \varphi)$ is called "statistical $(\alpha, \varphi)$-connection".

We easily say that the cubic form $C$ is pure with regard to the product structure $\varphi$, that is,

Therefore, we get

i.e., the connection $^{(\alpha)}\nabla$ is pure according to product structure $\varphi$. Also, in [6,p.19], the author has already shown that any $\varphi$-connection $\overline{\nabla }$ is pure if and only if its torsion tensor is pure. Then, we write

Theorem 3.1. In locally decomposable Riemann manifold $(M_{n}, g, \varphi)$, if the covector field $\eta$ in (3.3) is a decomposable tensor field, then the cubic form $C$ is a decomposable tensor field.

Proof. For the cubic form $C$ given by the (3.3), from the (2.1), we obtain

Substituting (3.3) into the last equation, we get

and for covector field $\eta$, we have

From the last two equations, we get

It is clear that if $\Phi _{\varphi }\eta = 0$, then $\Phi _{\varphi }C = 0$.

Corollary 3.1. From the equation (3.4) in Theorem 3.1, we can write

that is, the covariant derivation of the cubic form $C$ is pure with respect to product structure $\varphi$.

In the following sections of the paper, we will assume that the covector field $\eta$ is decomposable tensor field, i.e., the following equation always applies:

The $\alpha$-curvature tensor field $^{(\alpha)}\overline{R}$ of the statistical $(\alpha, \varphi)$-connection $^{(\alpha)}\nabla$ is given by

Substituting (3.1) into the last equation, we obtain

where $g(^{(\alpha)}\overline{R}(X, Y, Z), W) = \ ^{(\alpha)}R(X, Y, Z, W)$ and $R$ is Riemann curvature tensor field of Riemann metric $g$,

and

From the last two equations, we get

where $d$ is exterior derivate operator applied to the covector field $\eta$. Then, we can write the following proposition and corollary.

Proposition 3.2. The covector field $\eta$ is closed, that is, $d\eta = 0$ if and only if

Corollary 3.2. For the differentiable function $f$ on locally decomposable Riemann manifold $(M_{n}, g, \varphi)$, it is well-known that $d^{2}f = 0$. So, if $\eta = df$ $= \frac{\partial f}{\partial x^{i}}dx^{i}$, then $d\eta = 0$ is directly obtained and we can write the equation (3.10).

The tensor fields $\rho$ and $q$ are given by Eqs (3.8) and (3.9), respectively, is pure according to the product structure $\varphi$. Then, we write

In addition, from the Eqs (2.1) and (3.8), we get

Substituting (3.8) into the Eq (3.11), we obtain

For the Ricci identity of the covector field $\eta$, we have

and

From the last equations, we write

and in the same way $(\Phi _{\varphi X}q) = 0$. Then, we have

Proposition 3.3. The tensor fields $\rho$ and $q$ are given by Eqs (3.8) and (3.9), respectively are a decomposable tensor fields and because of the equation (3.11), we can write

and

that is, the covariant derivation of the tensor fields $\rho$ and $q$ are pure with respect to the product structure $\varphi$.

With the simple calculation, we can say that the $\alpha$-curvature tensor field$\; ^{(\alpha)}R$ is pure with regard to the product structure $\varphi$, namely,

Then, from the Eq (2.1), we have

If the expression of the $\alpha$-curvature tensor field $^{(\alpha)}R$ is written in the last equation, then we obtain

Furthermore, from Proposition 3, the last equation becomes the following form:

and

Then, we obtain

Theorem 3.2. The $\alpha$-curvature tensor field ${}^{(\alpha)}R$ of the statistical $(\alpha, \varphi)$-connection ${}^{(\alpha)}\nabla$ is decomposable tensor field and due to the equation (3.14), we can say that

namely, the covariant derivation of the $\alpha$-curvature tensor field ${}^{(\alpha)}R$-is pure with respect to the product structure $\varphi$.

4.   Some example for Statistical ($\alpha, \varphi)$-connections

Example 4.1. Let $M_{2} = \{(x, y)\in \texttt{R} ^{2}, x > 0\}$ be a manifold with the metric $g$ such that

Then, $(M_{2}, g)$ is a Riemann manifold. The component of the Riemann connection of this manifold is as follow:

and the others are zero. In addition, we say that $(M_{2}, g)$ is a flat manifold, that is, Riemann curvature tensor $R$ of that manifold is vanishing. The equation system satisfying the conditions $\varphi _{i}^{m}g_{mj} = \varphi _{j}^{m}g_{im}$ (purity) and $\varphi _{i}^{m}\varphi _{m}^{j} = \delta _{i}^{j}$ (product structure) is

where

Then, a general solution of the equation system (4.1) is

In the last equation, a special solution for $d = 0$ is as follow:

Here, because of $\ \nabla \varphi = 0$, the triplet $(M_{2}, g, \varphi)$ is locally decomposable Riemann manifold.

The expression of the cubic form in local coordinates given by (3.2) is

where $\eta ^{k} = \eta _{i}g^{ik}$ and $\varphi _{ij} = \varphi _{i}^{k}g_{kj}$. For $\eta (x, y) = \left(\eta _{1}(x, y), \eta _{2}(x, y)\right)$, the matrix shape of the cubic form is as follows:

for example, $\overline{C}_{11}^{1} = 3\eta _{1}$, $\overline{C}_{11}^{2} = \frac{3}{x}\eta _{2}$, ..., etc. In adition, the statistical $(\alpha, \varphi)$-connection is given by

and these components are

For $i, j, m = 1, 2$, the components of the $\Phi _{\varphi }\eta$ are

where $\frac{\partial }{\partial x_{1}} = \frac{\partial }{\partial x}$ and $\frac{\partial }{\partial x_{2}} = \frac{\partial }{\partial y}$. Then, we have

and because of $\Phi _{\varphi }\eta = 0$, we obtain

Then, the components of the $\alpha$-curvature tensor field $^{(\alpha)}R$ are

Example 4.2. Let $N_{2} = \{(x, y)\in \texttt{R} ^{2}, x < 0\}$ be a manifold with the metric $g$ such that

Then, $(N_{2}, g)$ is a Riemann manifold and the component of the Riemann connection of this manifold is the following form:

and the others are zero. Also, $(N_{2}, g)$ is a flat manifold. The equation system satisfying the conditions $\varphi _{i}^{m}g_{mj} = \varphi _{j}^{m}g_{im}$ and $\varphi _{i}^{m}\varphi _{m}^{j} = \delta _{i}^{j}$ is as follow:

where

Then, a general solution of the equation system (4.2) is

In the equation (4.4), a special solution for $d = 1$ is

where for $x < 0$, $(N_{2}, g, \varphi)$ is locally decomposable Riemann manifold because of $\nabla \varphi = 0$. Then, we get

and

From the equation (3.6), we have

and because of $\Phi _{\varphi }\eta = 0$,

Then, we easily say that the components of the $\alpha$-curvature tensor field $^{(\alpha)}R$ are

Then, we get

Corollary 4.1. The $\alpha$-curvature tensor field $^{(\alpha)}R$ of $(N_{2}, g, \varphi)$ is vanishing, that is, $^{(\alpha)}R = 0$ if and only if

where $c_{1}$, $c_{2}$, and $c_{3}$ are scalars.

5.   Dual statistical ($\alpha, \varphi)$-connections

In [1,p.181], the author defined the dual of the $\alpha$ -connection $^{(\alpha)}\nabla$ of given by the equation (3.1) as follows:

We easily say that $^{D(\alpha)}\nabla = \ ^{-(\alpha)}\nabla$. Furthermore,

and

that is, the dual $\alpha$-connection $^{D(\alpha)}\nabla$ is statistical ($\alpha, \varphi)$-connection and is named "dual statistical ($\alpha, \varphi)$-connection". Also, in [1,p.182] (in Proposition 3.5), the author shows that the dual $\alpha$-curvature tensor field $^{D(\alpha)}R$ of $^{D(\alpha)}\nabla$ is as follow:

Then, we have

Theorem 5.1. The dual $\alpha$-curvature tensor field $^{D(\alpha)}R$ is a decomposable tensor field, i.e.,

6.   Conclusions

In this study, we define a special connection using the cubic form $C$ on locally product Riemann manifold. We name this new connection as statistical $(\alpha, \varphi)$-connection. We examine the curvature properties and give examples of this new connection. However, the cubic form $C$ customized for the locally product Riemann manifold is made only for the product structure. This cubic form can also be studied on different special Riemann manifolds such that Kahler (or Anti-Kahler) manifold with complex structure $E$, $E^{2} = -I$, Tangent (dual) manifold with tangent structure $F$, $F^{2} = 0$ and Golden Riemann manifold with golden structure $\varphi$, $\varphi ^{2} = \varphi +I$, which is the most interesting structure lately.

Acknowledgments

The author sincerely thank the reviewers for their careful reading and constructive comments.

Conflict of interest

The author declares no conflict of interest in this paper.