# A Statistical Cohomogeneity One Metric on the Upper Plane with Constant Negative Curvature

Authors: Limei Cao; Didong Li; Erchuan Zhang; Zhenning Zhang; Huafei Sun
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

we analyze the geometrical structures of statistical
manifoldS consisting of all the wrapped Cauchy distributions. We prove that S is a simply
connected manifold with constant negative curvature K=-2. However, it is not isometric to
the hyperbolic space because S is noncomplete. In fact, S is approved to be a cohomogeneity
one manifold. Finally, we use several tricks to get the geodesics and explore the divergence
performance of them by investigating the Jacobi vector field.


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## Body

## 1. Introduction

So far, more and more geometric approaches have been applied to various fields such as in statistics, physics, and control [1–6], by analyzing their complexity characterizations or giving their solutions.Entropic dynamics is a theoretical framework constructed on statistical manifolds to explore the possibility of laws of physics [7]. Until now, researchers have tried to study a variety of entropic dynamic models from the viewpoint of information geometry [4–6, 8]. In [8], two entropic dynamical models are considered, in which the stability of the Jacobi vector fields on Riemannian manifolds is investigated. In [4], authors explored the geometric structures and instability of entropic dynamical models. Information geometric characterization of fractional Brownian motions was considered in [5]. Hamiltonian dynamics are studied from the viewpoint of geometry in [6].In probability theory and directional statistics [9], a wrapped probability distribution is a continuous probability distribution, whose state can be considered as lying on a unit n-sphere Sn. Examples include the Bingham distribution [10], the Kent distribution (or the Fisher-Bingham distribution) [11], and the von Mises-Fisher distribution [9, 12]. Particularly for one-dimensional cases, a wrapped distribution consists of points on the unit circle. For instance, a wrapped Cauchy distribution [9, 13, 14]
(1)p(x;μ,γ)=12πsinh⁡⁡γcosh⁡⁡γ-cos⁡(x-μ)
is a wrapped probability distribution that results from the “wrapping” of the Cauchy distribution around the unit circle, where apparently x∈[-π,π) which is parameterized from the unit circle, γ∈R, and μ>0. The functions sinh⁡ and cosh⁡ are, respectively, the hyperbolic sine function and the hyperbolic cosine function. As the random variable x∈[-π,π], the wrapped probability distribution can be considered as an angular probability to measure the directional probability. Therefore, it has been widely applied in the field of random walk for studying animal movements [15, 16], such as identifying fishing trip behavior and estimating fishing effort [17], and ringing recoveries of pied flycatchers [18].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 Section2, the geometric structures, including the Fisher metric and the sectional curvature of the WCED, are calculated. In Section 3, this manifold is proved to be simply connected but not complete, hence not isometric to the hyperbolic space. However, it is proved to be a cohomogeneity one manifold by Theorem 14. In Section 4, several tricks are used based on some special properties to get geodesics and the Jacobi vector fields. The geodesics are the sets of fixed points of isometric transformations, which include symmetries to lines paralleled to y-axis. Meanwhile, the Jacobi vector fields are calculated. Then the instability of the geodesic spreads is analyzed via the behaviors of the the Jacobi vector field.Remark 1.
It is well known that the upper planeR+2 admits a hyperbolic metric gH with constant negative curvature. That is,
(2)gH=1y2001y2.
However, in this paper, we give another metric on R+2 with constant negative curvature but not isometric to the hyperbolic space.


## 2. Geometric Structures of the WCED Model

From the theory of information geometry, one can define anm-dimensional statistical manifold M over unit spheres, which is a set of probability densities; namely,
(3)M=px;θ.
The parameter θ=(θ1,θ2,…,θm) plays the role of coordinate systems. The Fisher information matrix g, acting as a Riemannian metric, is then given by the expectation
(4)gij=E∂il∂jl,
where l(x;θ)∶=ln⁡p(x;θ) and ∂i∶=∂/∂θi. On a statistical manifold, the Riemannian connection coefficient Γijk is (see, e.g., [1, 2, 19–21])
(5)Γijk=E∂i∂jl∂kl+1-α2E∂il∂jl∂kl,
which can be expressed as a classical formula Γijk=(1/2)(∂igjk+∂jgki-∂kgij) equivalently.If(θ1,θ2,…,θm) is a local coordinate of M, then for any vector field X on M, one has X=Xi∂i, where {Xi} are smooth functions with respect to (θ1,θ2,…,θm). The Einstein summation convention is used here and all through the paper. With the Riemannian connection, the curvature tensor is defined by [19, 20]
(6)R(X,Y)Z=∇X∇YZ-∇Y∇XZ-∇[X,Y]Z,
and the Riemannian curvature tensor is
(7)R(X,Y,Z,W)=g(R(Z,W)X,Y),
where X, Y, Z, W are smooth vector fields on M and [·,·] represents the Lie bracket; namely, [X,Y]=X∘Y-Y∘X. The local representation of the curvature tensor satisfies
(8)Rmjkl=∂mΓjkt-∂jΓmktgtl+ΓmtlΓjkt-ΓjtlΓmkt,
where Γmjk=Γmjtgtk and (gtk) is the inverse of the metric matrix (gtk).The Ricci curvature and the sectional curvature are defined by(9)Rmk=Rmjlkgjl,(10)Kij=-Rijijgiigjj-gij2,
respectively.A curveξ(t) on M is said to be a geodesic if its tangent vector ξ˙(t) is displaced parallel along the curve ξ(t); that is,
(11)∇ξ˙(t)ξ˙(t)=0,
and it has the following local form
(12)d2θkdt2+Γjmkdθjdtdθmdt=0.
Suppose that the solution of (12) is θ(t;β), where β is a vector whose components are the integration constants.Considering the wrapped Cauchy entropic dynamical model, the corresponding statistical manifold is(13)S∶=px;θ∣θ=θ1,θ2∈R+2,
where p(x;θ) is given by (1) with parameters θ1=μ and θ2=γ. The Fisher information matrix can be easily calculated as
(14)g=diag⁡12sinh⁡2θ2,12sinh⁡2θ2.Proposition 2.
The nonzero components of the Ricci curvature are(15)R11=R22=-1sinh⁡2θ2.Proof.
A direct calculation of (5) gives the nonvanishing Riemannian connection coefficients
(16)Γ112=-Γ121=-Γ211=-Γ222=cosh⁡θ22sinh⁡3θ2.
From the relation Γmjk=Γmjtgtk, one can get another type of connection coefficients
(17)Γ112=-Γ121=-Γ211=-Γ222=cosh⁡θ2sinh⁡θ2.
From (8), the following nonzero component of the curvature tensor is obtained:
(18)R1212=12sinh⁡4θ2.
Therefore, the proof can be finished by a simple substitution of (18) into (9).The following theorem is obtained by substituting (18) and (14) into (10).Theorem 3.
S is a manifold with constant negative curvature K=-2.


## 3. Topological and Geometric Properties ofS

Lemma 4.
S is diffeomorphic to R+2 and hence simply connected.Proof.
The topology, as well as the smooth structure, ofS is induced by those on R+2; hence S is diffeomorphic to R+2. The fact that R+2 is simply connected implies that S is also simply connected.IfS were also complete, it would be a space form with constant negative curvature, which was isometric to the hyperbolic space H2(-2). However, S is not complete. We begin with a necessary and sufficient condition of the completeness.Definition 5 (see [19]).
Let(M,g) be a Riemannian manifold. A smooth curve γ:[0,+∞)→M is called a divergent curve if for any compact subset K⊂M, there exists t0>0, s.t., γ(t0)∉K. If γ is a divergent curve, its length is defined by
(19)L(γ)∶=lim⁡s→+∞∫0s‍γ′tdt.Lemma 6 (see [22]).
(
M
,
g
,
d
) is a connected Riemannian manifold; then M is complete if and only if every divergent curve has infinite length.Proof.
(1) Suppose thatM is complete and γ is a divergent curve. Consider
(20)V-n∶=p∈M∣dp,γ0≤n,∀n∈N,
where for each n, V-n is bounded and closed, hence compact. The divergence of γ implies that, for all n∈N*, ∃tn≥0, s.t., γ(tn)∉V-n. As a result,
(21)∫0tn‍γ′tdt≥dγ0,γtn≥n;
hence L(γ)=+∞.
(2) Suppose every divergent curve has infinite length. Assume thatM is noncomplete, which implies that ∃p∈M, ∃v0∈TpM, and γ(t)=exp⁡p(tv0) is only defined on t∈[0,1).
L
(
γ
)
<
|
v
0
|
<
+
∞ means γ is not divergent; that is, there exists a compact subset K⊂M s.t. γ(t)∈K,  for all t. Choose sequence {tn}n=1+∞, s.t., tn→1 and tn<tn+1, for all n∈N. It is obvious that d(γ(tn),γ(tm))→0 when n,m are sufficiently large. Hence {γ(tn)} is a Cauchy sequence. The completeness of M implies that there exists q∈K s.t. γ(tn)→q.
Thenγ(1)∶=q is a extension of γ from [0,1) to [0,1] which contradicts to the assumption, which implies that M is in fact complete.Theorem 7.
S is noncomplete.Proof.
Suppose thatγ(t)∶=(0,t)∈R+2, t∈[1,+∞) is a curve along the y-axis. It is obvious that γ is divergent. However, the length of γ satisfies
(22)Lγ=∫1+∞‍γ′tdt=∫1+∞‍12sinh⁡(t)dt=12ln⁡tanht21+∞=ln⁡(e+1)-ln⁡(e-1)2<+∞.
This implies S is noncomplete by Lemma 6.AlthoughS is neither isometric to the hyperbolic space nor a symmetric space, it is “partly symmetric,” which is presented in Lemma 10.Definition 8.
LetI(S) be the group of isometric transformations on S. A family of transformations on S are defined as
(23)φα:S⟶S,(θ1,θ2)⟼(2α-θ1,θ2),∀α∈R.Remark 9.
φ
α is just the reflection about the line θ1=α, which is a line parallel to the y-axis.Lemma 10.
φ
α
∈
I
(
S
), for all α∈R.Proof.
For any fixedα∈R, for all p=(θ1,θ2)∈S, q∶=φα(p)=(2α-θ1,θ2).
Consider(24)γ1(t)∶=(θ1+t,θ2),γ2t∶=θ1,θ2+t,φα*p∂∂θ1=ddtt=0φα∘γ1t=ddtt=0(2α-θ1-t,θ2)=-∂∂θ1q,φα*p∂∂θ2=ddtt=0(φα∘γ2(t))=ddtt=0(2α-θ1,θ2+t)=∂∂θ2q,gqφα*p∂∂θ1,φα*p∂∂θ1=gq(-∂∂θ1,-∂∂θ1)=12sinh⁡2θ2=gp(∂∂θ1,∂∂θ1),gqφα*p∂∂θ1,φα*p∂∂θ2=gq(-∂∂θ1,∂∂θ2)=0=gp(∂∂θ1,∂∂θ2),gqφα*p∂∂θ2,φα*p∂∂θ2=gq(∂∂θ2,∂∂θ2)=12sinh⁡2θ2=gp(∂∂θ2,∂∂θ2).
Hence φα is an isometry.As a result,G∶=φα∣α∈R is a subgroup of I(S). Lemma 10 implies that S is “almost” homogeneous, although not truly homogeneous. In fact, S is a cohomogeneity one manifold, defined in Definition 11.Definition 11.
Let(M,g) be a Riemannian manifold and I(M) is its group of isometric transformations. Then I(M) has a natural action Φ on M:
(25)Φ:IM×M⟶M,φ,p⟼φp,∀φ∈IM,p∈M.M is a cohomogeneity one manifold if the codimension of orbit space of this action is 1.Remark 12.
The codimension of the orbit space is closely related to the homogeneous properties. For example, ifM is a homogeneous space, then the orbit space is whole manifold M, and hence its codimension is 0. The less that codimension is, the more homogeneous the manifold is. In other words, a cohomogeneity one manifold is “the most homogeneous manifold” except for the real ones.
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 groupsG2 and Spin(7) on these manifolds [23–30].We state a well known result without proof before Theorem14.Lemma 13.
Every homogeneous space is complete.Theorem 14.
S is a cohomogeneity one manifold.Proof.
Letp=(θ1,θ2)∈S and I(S)p∶=φp∣φ∈IS be its orbit space. If dim⁡(I(S)p)=2, S will be homogeneous. Lemma 13 implies that S is complete, which contradicts to Theorem 7. This contradiction implies dim⁡(I(S)p)≤1.
On the other hand,(26)Gp∶=φp∣φ∈G=φαp∣α∈R=2α-θ1,θ2∣α∈R=x,θ2∣x∈R.
Since G is a subgroup of I(S), Gp⊂I(S)p. This implies that
(27)dim⁡(ISp)≥dim⁡(Gp)=1.
Based on the two inequalities, we conclude that dim⁡(I(S)p)=1; hence its codimension is 1, which means that S is a cohomogeneity one manifold.


## 4. Instability Analysis

In this section, we will calculate some geodesics and investigate the instability of the geodesics by the Jacobi vector fields.Combining (12) and (17), the geodesic equations of the WCED model are given by the following differential system:
(28)d2θ1dt2-2cosh⁡θ2sinh⁡θ2dθ1dtdθ2dt=0,d2θ2dt2+cosh⁡θ2sinh⁡θ2dθ1dt2-cosh⁡θ2sinh⁡θ2dθ2dt2=0.
Generally, these equations are quite hard to be solved; however we can get some special geodesics based on the cohomogeneity one property.Lemma 15.
All lines parallel to the y-axis are geodesics.Proof.
It is well known that the fixed-points set of any isometry is a geodesic. Sinceφα is an isometry and its fixed-points set is the line x=θ1, the lemma follows.According to Lemma15, the equation of such geodesic satisfies
(29)θ1=θ01.
Hence (28) can be reduced to
(30)dθ1dt=0,d2θ2dt2-cosh⁡θ2sinh⁡θ2dθ2dt2=0.
Then solution is expressed as
(31)θ1=a3,θ2=ln⁡a2exp⁡{a1t}+1a2exp⁡{a1t}-1.Consider the parameter family of geodesics(32)F(C)=θlt,Cl=1,2,
where θl is a geodesic satisfying (28) and C is an integration constant vector. The length of geodesics in F(C) is defined as
(33)Θ(t;β)∶=∫gijdθidθj1/2.
Therefore, the length of geodesics on WCED model is obtained as
(34)Θ(t;C)=∫0tg11dθ1dt2+g22dθ2dt2dτ=2C12t,
where C1 is constant. In order to investigate the behavior of two neighboring geodesics labeled by the parameter C1, we consider the following difference:
(35)ΔΘ=2(C1+δC1)2-2C12t.
It is clear that ΔΘ diverges; that is, the length of two neighboring geodesics with slightly different parameters C1 and C1+δC1 differs in a remarkable way as t→∞.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 [19]. However, it is more attractive to know the divergent degree of a geodesic.The Jacobi vector fieldJ, that is, the evolution perturbation vector, satisfies the following (Jacobi) equation:
(36)∇ξ˙(t)∇ξ˙(t)J=R(ξ˙(t),J)ξ˙(t),
where ξ(t) is a geodesic on manifold M. It is also called the geodesic derivation equation, as it has close connections with the geodesic. In general, the Jacobi equations are difficult to solve, but in particular this manifold has a constant negative curvature, which is of great help. Given a normal geodesic ξ and supposing J is a normal Jacobi vector field along ξ(t), then
(37)Rξ˙t,Jξ˙t=2(g(ξ˙(t),ξ˙(t))J-g(ξ˙(t),J)ξ˙(t))=2J.
Thus the Jacobi equation is reduced to
(38)d2J(t)dt2-2J(t)=0.
Choose a unit orthogonal frame {ei(t)}, s.t., e2(t)=ξ˙(t) and suppose
(39)J(t)=Ji(t)ei(t).
Then the Jacobi equation satisfies
(40)d2J1tdt2-2J1t=0,J2t=0.
The solutions are
(41)J1(t)=C1e2t+C2e-2t,J2(t)=0.
This means that the geodesic spread on manifold is described by means of an exponential order divergent Jacobi vector field.


## 5. Conclusion and Remarks

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 withK=-2. By a series of lemmas and theorems, we prove that it is a cohomogeneity one manifold. This example is interesting because of its statistical background, while some other examples are deliberately constructed. In addition, we calculate the geodesics and the Jacobi vector field by some tricks based on some special properties of S. As a result, the divergent behavior of geodesics can be described as an exponential order divergent Jacobi vector field.



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*Source: 832683-2014-12-30.xml*