^{1}

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we analyze the geometrical structures of statistical
manifold

So far, more and more geometric approaches have been applied to various fields such as in statistics, physics, and control [

Entropic dynamics is a theoretical framework constructed on statistical manifolds to explore the possibility of laws of physics [

In probability theory and directional statistics [

Assume that all information relevant to dynamical evolution of the model can be obtained from the associated probability distribution, which is the wrapped Cauchy distribution in our consideration. We will call this model the WCED (wrapped Cauchy entropic dynamical) model for simplicity.

The remainder of the paper is organized as follows. In Section

It is well known that the upper plane

From the theory of information geometry, one can define an

If

The Ricci curvature and the sectional curvature are defined by

A curve

Considering the wrapped Cauchy entropic dynamical model, the corresponding statistical manifold is

The nonzero components of the Ricci curvature are

A direct calculation of (

The following theorem is obtained by substituting (

The topology, as well as the smooth structure, of

If

Let

(1) Suppose that

(2) Suppose every divergent curve has infinite length. Assume that

Then

Suppose that

Although

Let

For any fixed

Consider

As a result,

Let

The codimension of the orbit space is closely related to the homogeneous properties. For example, if

Actually, cohomogeneity one manifolds are natural generalizations of homogeneous manifolds. The systematic study of cohomogeneity one manifolds was started by Bergery, who successfully constructed new invariant Einstein metrics on cohomogeneity one manifolds. In addition, Bryant and Salamon constructed special metrics with exceptional holonomy groups

We state a well known result without proof before Theorem

Every homogeneous space is complete.

Let

On the other hand,

In this section, we will calculate some geodesics and investigate the instability of the geodesics by the Jacobi vector fields.

Combining (

All lines parallel to the y-axis are geodesics.

It is well known that the fixed-points set of any isometry is a geodesic. Since

According to Lemma

Consider the parameter family of geodesics

The stability of the geodesics is completely determined by the curvature of manifold. Studying the stability of dynamics means determining the evolution of perturbations of geodesics. For isotropic manifolds, the geodesic spread is unstable only if their constant sectional curvatures are negative. As long as the curvatures are negative, the geodesic spread is unstable even if the manifold is no longer isotropic [

The Jacobi vector field

In this paper, we investigate the manifold of wrapped Cauchy distributions. By considering the geometric structures of the WCED, we conclude that the WCED is a constant negative curvature space with

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final paper.

This work is supported by the National Natural Science Foundations of China (nos. 11126161 and NSFC61440058), Beijing Higher Education Young Elite Teacher Project (no. YETP 0388), and the Fundamental Research Funds for the Central Universities (no. FRF-BR-12-005).