Geometry of Fisher Information Metric and the Barycenter Map ^†

Abstract

Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold X. We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesic. Using Fisher metric and thus obtained properties of geodesics, a fibre space structure of barycenter map and geodesical properties of each fibre are discussed. Moreover, an isometry problem on an Hadamard manifold X and its ideal boundary ∂X—for a given homeomorphism Φ of ∂X find an isometry of X whose ∂X-extension coincides with Φ—is investigated in terms of the barycenter map.

1. Introduction

The aim of this article is to deal with two subjects related with information geometry. One is Fisher metric G defined on a space $\mathcal{P}(M)$ of probability measures having continuous positive density function over a connected, compact manifold M, and another one is barycenter map from $\mathcal{P}(\partial X)$ to an Hadamard manifold X, where ∂X is the ideal boundary of X. This article is an extended version of [1] presented at MaxEnt 2014, Amboise, France.

The Fisher metric G, remarkably important in information geometry, is defined in a natural way. The metric G is push-forward invariant, and has an explicit formula of Levi–Civita connection and its sectional curvature is constant 1/4, as shown in [2] by T. Friedrich.

Before introducing main results, we will explain motivation and background of our study.

An n-dimensional Hadamard manifold (X, g) is diffeomorphic to Rⁿ, and hence to an open ball Dⁿ, whose actual boundary is Sⁿ⁻¹. X admits also the ideal boundary ∂X as a quotient space of oriented geodesics on X. Then, we are able to consider Dirichlet problem at boundary ∂X; given a f ∈ C⁰(∂X), find a solution u = u(x) on X satisfying Δu = 0, u|_∂X = f. Using the fundamental solution P = P (x, θ), called Poisson kernel, when its existence is guaranteed, the solution is described as

$$u(x)={\displaystyle {\int }_{\theta \in \partial X}P(x,\theta )d\theta ,x\in X.}$$

(1)

Refer to [3] for precise definition of Poisson kernel. We obtain then a probability measure P (x, θ)dθ on ∂X parametrized in x ∈ X and have a map, called Poisson kernel map $\mathrm{\Theta }:X\to \mathcal{P}(\partial X)$.

Theorem 1 ([4–6]). Let (X, g) be an n-dimensional Damek-Ricci space. Then the map Θ is homothetic with respect to the Fisher metric G and g;$\mathrm{\Theta }*G=\frac{Q}{n}g$ where Q is volume entropy of (X, g). Further Θ is a harmonic map.

Here, for volume entropy Q refer to §4. The quantity Q is an invariant of Riemannian geometry which is closely related to the topological entropy of geodesic flow ([7,8]). Refer to [9] with respect to volume entropy treated in a framework of information geometry.

In the theorem a Damek-Ricci space is a solvable Lie group of a left invariant metric, one dimensional extension of a generalized Heisenberg group. Refer to [10] for details. A Damek-Ricci space is a harmonic, Einstein Hadamard manifold and any rank one symmetric space of non-compact type, namely hyperbolic spaces over the real numbers R, the complex numbers C, the quaternions H and 16-dimensional one over Cayley numbers O are also Damek-Ricci spaces. With respect to Theorem 1, we have the following theorem.

Theorem 2 ([5]). Let (X, g) be an Hadamard manifold which is equipped with Poisson kernel P (x, θ). Assume that the map$\mathrm{\Theta }:X\to \mathcal{P}(\partial X)$ is homothetic; Θ^*G = C g, C > 0, and harmonic with respect to the metrics G and g. Then (X, g) is asymptotically harmonic and satisfies visibility axiom. Moreover, C = Q/n and the Poisson kernel has the form P (x, θ) = exp{−Q B_θ(x)} in terms of Busemann function B_θ on X.

The terminology with respect to asymptotical harmonicity, visibility axiom and Busemann function will be explained in the subsequent sections.

Remark that the equality C = Q/n is derived from asymptotical formula related with mean curvature of geodesic spheres and mean curvature of corresponding horospheres, level hypersurfaces of Busemann function ([11]).

With respect to these theorems we are interested in characterization of Damek-Ricci space from information geometry, especially from a viewpoint of Fisher metric G, since a Damek-Ricci space is a counterexample of Lichnerowicz conjecture of non-compact version ([12]) and its characterization is only given by Heber in [13] by Lie group theory argument. By approaching from a viewpoint of the ideal boundary ∂X, we focus on barycenter of probability measures on ∂X with respect to Busemann function and shed a light on information geometry of barycenter map bar : $\mathcal{P}(\partial X)\to X$.

2. Main Results and Conclusive Remarks

Before entering into the detailed argument, we give an outline of main results and remarks.

In Section 3 we deal with several results on Fisher metric G and also on geodesic, a basic notion of geometry, defined on $(\mathcal{P}(M),G)$. We give a formula of geodesic μ(t) = exp_μ tτ on $\mathcal{P}(M)$ in a simple form (Theorem 9);

$$\mu (t)={\left(\cos \frac{t}{2}+\sin \frac{t}{2}\frac{d\tau }{d\mu }(x)\right)}^2\mu ,x\in M$$

for an initial condition; μ(0) = μ, $\dot{\mu }(0)=\tau (|\tau {|}_{G,\mu }=1)$. Here, (dτ/dμ)(x) denotes Radon-Nikodym derivative of τ with respect to μ. From this, it is concluded in Corollary 2 that any geodesic is periodic, of period 2π, while not definable over R. Moreover, from this formula which is an improvement of the formula given by T. Friedrich ([2]) we obtain

Theorem 3. Let μ and μ^* be arbitrary distinct probability measures in$\mathcal{P}(M)$. Then, a curve$t\in \mathbf{R}\mapsto \mu (t)\in \mathcal{P}(M)$defined by

$$\mu (t)={\exp }_{\mu }t\tau ={\left(\cos \frac{t}{2}+\sin \frac{t}{2}\frac{d\tau }{d\mu }(x)\right)}^2\mu$$

(2)

is a unique geodesic such that μ(0) = μ and μ(ℓ) = μ. Here ℓ = ℓ (μ, μ*) is defined by (4) and τ is a unit tangent vector at μ given by

$$\tau =\frac{1}{\sin \frac{\ell }{2}}\left(\sqrt{\frac{d\mu *}{d\mu }}(x)-{\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)}d\mu (y)\right)\mu (x).$$

(3)

This theorem asserts that any μ, μ^*, μ ≠ μ^* can be joined by a unique geodesic. The quantity ℓ = ℓ(μ, μ*), 0< ℓ _< π _is defined as

$$\cos \frac{\ell }{2}={\displaystyle {\int }_{x\in M}\sqrt{\frac{d\mu *}{d\mu }}(x)d\mu (x),}$$

(4)

giving an apparent length of a geodesic joining μ and μ*. Notice that the RHS is an f-divergence-like quantity with respect to $f(u)=\sqrt{u}$ (refer to [14]).

Another main subject is information geometry of barycenter map by applying results of Fisher information geometry, thus obtained in Section 3. Related results on barycenter map will be explained in Section 4 and Section 5.

Let (X, g) be an Hadamard manifold with a Riemannian metric g, a simply connected, complete Riemannian manifold of non-positive curvature. Then, it admits the ideal boundary ∂X and by using a probability measure defined on ∂X, we consider a function, a μ-average Busemann function B_μ : X → R;

$${\mathrm{B}}_{\mu }(x)={\displaystyle {\int }_{\theta \in \partial X}{B}_{\theta }(x)d\mu (\theta )}$$

whose critical point is called a barycenter of a probability measure μ so that we have a map, barycenter map, from a space $\mathcal{P}(\partial X)$ of probability measures on ∂X to an Hadamard manifold X. Here, the integrand is a normalized Busemann function (for its detailed argument see Section 4).

Recall, here, an original definition of a barycenter, a center of mass, as follows. Let y₁, …, y_n be points of a Euclidean space R³ and μ₁, …, μ_n be non-negative real numbers satisfying ${\displaystyle {\sum }_i{\mu }_i=1}$. A point p of R³ is called a barycenter of y_i, i = 1, …, n of weights μ_i, i = 1, …, n, when p satisfies

$$p={\displaystyle \sum _{i=1}^n{\mu }_i{y}_i}\mathrm{or}{\displaystyle \sum _{i=1}^n{\mu }_i(y_i-p)=0.}$$

A barycenter is defined also by a critical point of a function on R³; f : R³ → R; $f(q)={\displaystyle {\sum }_{i=1}^n{\mu }_i{d}^2(q,y_i)}$.

This definition of barycenter for a finite points of R³ with weights with respect to the square-distance can be generalized as one for points of R³ distributed continuously over a bounded set D of R³;

$$f:{\mathbf{R}}^3\to \mathbf{R};f(q)={\displaystyle {\int }_{{\mathbf{R}}^3}{d}^2(q,x)\mu (x)dx},$$

where μ = μ(x) is a non-negative function with supp(μ) ⊂ D satisfying ${\displaystyle {\int }_{{\mathbf{R}}^3}\mu (x)dx=1}$. A critical point of f can be regarded as a barycenter of a probability measure μ(x) dx. A famous theorem of E. Cartan is regarded as a barycenter theorem ([15]). A choice of testing function d²(x, y) is not essential. Convexity of testing function is crucial in a theory of barycenter. In our study we deal with barycenter with respect to Busemann function, a convex testing function, by following the idea of Douady, Earle ([16]) and Besson, Courtois and Gallot ([8,17]). Refer to [18,19] for studies and results on barycenter of square-distance and of distance over a Riemannian manifold. Refer also to [20] in this direction, which is a reference comment due to Professor M. Gromov at the conference.

In our situation, the existence of barycenter for any $\mu \in \mathcal{P}(\partial X)$ is assured in Theorem 12, when (X, g) satisfies visibility axiom (for precise definition see Definition 4 and refer to [21]) and Busemann function B_θ(x) on X is continuous with respect to θ ∈ ∂X. Uniqueness of barycenter for any μ is assured, when, for some μ₀, average Hessian $\nabla d{\mathrm{B}}_{{\mu }_0}$of ${\mathrm{B}}_{{\mu }_0}$ is positive definite everywhere on X (Proposition 6). Thus, we have the barycenter map bar : $\mathcal{P}(\partial X)\to X;\mu \mapsto y$, by assigning to μ a barycenter point y of μ. This map turns out to be surjective, when (X, g) admits Busemann-Poisson kernel P (x, θ) = exp{−Q B_θ(x)}, Poisson kernel of Busemann type. Denote by μ_x the probability measure P (x, θ) dθ. Then, bar(μ_x) = x for any x ∈ X. Busemann-Poisson kernel ensures also the uniqueness of barycenter for any μ from the identity (see Theorem 14)

$${(\nabla d{\mathrm{B}}_{{\mu }_x})}_x(u,v)=Q{G}_{{\mu }_x}\left(v_x^{{\mu }_x}u,v_x^{{\mu }_x}u\right),u,v\in T_{x}X,x\in X$$

where, $G_{{\mu }_x}$ is Fisher metric at the tangent space $T_{{\mu }_x}\mathcal{P}(\partial X)$, and $v_x^{{\mu }_x}:T_{x}X\to T_{{\mu }_x}\mathcal{P}(\partial X)$ is an injective linear map associated to μ and a point x = bar(μ) (for its definition see Section 4).

The map bar : $\mathcal{P}(\partial X)\to X$, being surjective gives us a projection of a fibre space whose total space is $\mathcal{P}(\partial X)$ and base space is X with fibres {bar⁻¹(x); x ∈ X}. The image of the linear map $v_x^{\mu }$ for μ and x = bar(μ) yields a subspace of $T_{\mu }\mathcal{P}(\partial X)$, normal to T_μbar⁻¹(x), the subspace tangent to a fibre bar⁻¹(x) so that $T_{\mu }\mathcal{P}(\partial X)$ splits into a G-orthogonal direct sum (Theorem 15);

$$T_{\mu }\mathcal{P}(\partial X)=T_{\mu }{\text{bar}}^{-}{}^1(x)\oplus \text{Im}{v}_x^{\mu }.$$

T_μbar⁻¹(x) and ${\text{Im v}}_x^{\mu }$ are the vertical, horizontal subspaces of $T_{\mu }\mathcal{P}(\partial X)$, respectively. Here, dim ${\text{dim Im v}}_x^{\mu }=\dim X$. Remark that the fibration asserted here is infinitesimal.

Each fibre bar⁻¹(x), x ∈ X is a path-connected submanifold of $\mathcal{P}(\partial X)$. Its geometry is investigated in terms of the second fundamental form;

$$H_{\mu }:T_{\mu }{\text{bar}}^{-1}(x)\times T_{\mu }{\text{bar}}^{-1}(x)\to \mathrm{Im}{v}_x^{\mu };H_{\mu }(\tau ,{\tau }_1)={({\nabla }_{\tau }{\tau }_1)}^{\perp },$$

(5)

which is the normal component of ∇_ττ₁, the covariant derivative of τ₁ in direction to τ with respect to the Levi–Civita connection ∇. Refer to [22,23] for definition of the second fundamental form. Applying the results concerning geodesics on $\mathcal{P}(\partial X)$ given in Section 3 to a submanifold bar⁻¹(x), we are able to determine a geodesic μ(t) = exp_μ tτ which is entirely contained in bar⁻¹(x) as

Theorem 4. Let μ(t) = exp_μ tτ be a unit speed geodesic, of μ(0) = μ, μ˙(0)=τ, |τ|_G,μ = 1. Then, μ(t) lies entirely on fibre bar⁻¹(x) if and only if, μ ∈ bar⁻¹(x), τ ∈ T_μ bar⁻¹ (x) and H_μ(τ, τ) = 0.

Remark that the equation H_μ(τ, τ) = 0 on τ is written down in a manner of information geometry as ${\displaystyle {\int }_{\theta \in \partial X}{(d{B}_{\theta })}_x(u){(d\tau /d\mu )}^2(\theta )d\mu (\theta )=0}$.

Moreover, by applying Theorem 11 in Section 3, it is possible to assert the following theorem.

Theorem 5. Let μ, μ* ∈ bar⁻¹(x). Then, a geodesic joining μ and μ* is contained completely in the same fibre bar⁻¹(x) if and only if μ and μ* fulfill

$${\displaystyle {\int }_{\theta \in \partial X}{(d{B}_{\theta })}_x(u)\sqrt{\frac{d\mu *}{d\mu }}(\theta )d\mu =0}$$

for any u ∈ T_xX.

Let ϕ be an isometry of an Hadamard manifold (X, g). Then, a μ-average Busemann function B_μ satisfies a cocycle formula with respect to ϕ;

$${\mathrm{B}}_{\mu }({\varphi }^{-1}x)={\mathrm{B}}_{({\widehat{\varphi }}_{\#})\mu }(x)+{\mathrm{B}}_{\mu }({\varphi }^{-1}{x}_o),x\in X,\mu \in \mathcal{P}(\partial X),$$

where ϕ⁻¹ is the inverse of ϕ, and $\widehat{\varphi }:\partial X\to \partial X$is a ∂X-extension of ϕ and ${\widehat{\varphi }}_{\#}$ is a push-forward induced by $\widehat{\varphi }$. See Theorem 18. From this formula we have

$$bar\left({\widehat{\varphi }}_{\#}\mu \right)=\varphi (bar(\mu )),\mu \in \mathcal{P}(\partial X)$$

from which each fibre bar⁻¹(x) is mapped by ${\widehat{\varphi }}_{\#}$ to a fibre bar⁻¹(ϕx) over ϕx.

By the aid of information geometry we are able to apply above results to an isometry problem; given a homeomorphism Φ of ∂X, find an isometry ϕ of (X, g) whose ∂X-extension coincides with Φ. With respect to this problem we consider a bijective map ϕ of X satisfying bar(Φ_#μ) = ϕ(bar(μ)) for any $\mu \in \mathcal{P}(\partial X)$ (we call such a map ϕ as barycentrically associated to Φ).

The following theorem gives us an answer to this problem, even partial, provided there exists a cross section of the fibre space bar : $\mathcal{P}(\partial X)\to X$ enjoying commutativity properties.

Theorem 6. Let φ : X → X be a C¹-map barycentrically associated to a homeomorphism Φ : ∂X → ∂X. Assume that there exists a cross section$\mathrm{\Sigma }:X\to \mathcal{P}(\partial X)$of the fibre space bar : $\mathcal{P}(\partial X)\to X$, a map satisfying bar ○ Σ = id_X such that a fibrewise diagram commutes

$$\begin{array}{ccc}\mathcal{P}(\partial X)& \stackrel{{\mathrm{\Phi }}_{\#}}{\to }& \mathcal{P}(\partial X)\\ \uparrow \mathrm{\Sigma }& & \uparrow \mathrm{\Sigma }\\ X& \stackrel{\phi }{\to }& X\end{array}$$

and a diagram of tangent space level commutes

$$\begin{array}{ccc}{T}_{x}X& \stackrel{v_x^{{\mu }_x}}{\to }& T_{{\mu }_x}\mathcal{P}(\partial X)\\ \downarrow {({\phi }_{*})}_x& & \downarrow {\mathrm{\Phi }}_{\#}\\ T_{\phi x}X& \stackrel{v_{\phi x}^{{\mu }_{\phi x}}}{\to }& T_{{\mu }_{\phi x}}\mathcal{P}(\partial X)\end{array}$$

(6)

where we denote by${\mu }_x:=\mathrm{\Sigma }(x)\in \mathcal{P}(\partial X)$. Then, φ is an isometry of (X, g) and ∂X-extension$\widehat{\phi }$ of φ coincides with Φ.

A particularly significant cross section Σ is given by a Poisson kernel map; $\mathrm{\Theta }:X\to \mathcal{P}(\partial X)$; x ↦ P (x, θ)dθ = exp{−QB_θ(x)} dθ, where P (x, θ) is a Busemann-Poisson kernel on (X, g). The differential map (Θ_*)_x of Θ fulfills ${({\mathrm{\Theta }}_{*})}_x(u)=-Q{v}_x^{{\mu }_x}(u),u\in T_{x}X$ in terms of the linear map $v_x^{{\mu }_x}$, μ_x := P (x, θ)dθ, so that we have

Corollary 1 ([24]). Let Φ : ∂X → ∂X be a homeomorphism of ∂X and φ : X → X be a C¹-map. Assume the following diagram commutes with respect to Poisson kernel map Θ;

$$\begin{array}{ccc}\mathcal{P}(\partial X)& \stackrel{{\mathrm{\Phi }}_{\#}}{\to }& \mathcal{P}(\partial X)\\ \uparrow \mathrm{\Theta }& & \uparrow \mathrm{\Theta }\\ X& \stackrel{\phi }{\to }& X\end{array}$$

(7)

Then, φ is an isometry of (X, g) and its ∂X-extension coincides with Φ of ∂X.

Theorem 6 is a generalization of Corollary 1, a main result of [24,25].

The article is organized as follows. In Section 3 we introduce basic notions of information geometry of a space $\mathcal{P}(M)$ of probability measures on a compact manifold M and define Fisher metric G on it. We show several useful theorems on the Levi–Civita connection and geodesics on $\mathcal{P}(M)$ with detailed proofs. We derive in Section 4 fundamental properties of Busemann function on an Hadamard manifold, preliminarily. By using them, we investigate existence and uniqueness of barycenter of a probability measure, following a proof given in [8]. We define the barycenter map and develop information geometry of this map. A fibration theorem is similar to our earlier paper [25]. However, geodesical arguments on fibres develop further the arguments of [25], by applying the results of geodesics on $\mathcal{P}(M)$ in Section 3. Finally, in Section 5, we treat an isometry problem for an Hadamard manifold and give a proof of Theorem 6.

3. A Space of Probability Measures and Fisher Metric

3.1. A Space of Probability Measures

Let M be a connected, compact smooth manifold. Let $\mathcal{B}(M)$ be the family of Borel sets of M. Here $\mathcal{B}(M)$ is a family of subsets of M which satisfies the following; (i) $\mathcal{B}(M)$ is a σ-family of M, (ii) every open subset of M is an element of $\mathcal{B}(M)$ and (iii) if $\mathcal{F}$ is a family of subsets of M satisfying (i), (ii), then $\mathcal{F}\subset \mathcal{B}(M)$.

A function $P:\mathcal{B}(M)\to \mathbf{R}$ is called a probability measure of a measurable space $(M,\mathcal{B}(M))$, or a probability measure on M, when P satisfies

• P (A) ≥ 0 for any $A\in \mathcal{B}(M)$, P (M) = 1 and P (∅) = 0.
• Let {E_j | j = 1, …, } be a countable sequence of sets of $\mathcal{B}(M)$ satisfying E_i ∩ E_j = ∅ for any i, j, i ≠ j. Then

  $$P\left({\displaystyle \underset{i=1}{\overset{\infty }{\cup }}{E}_i}\right)={\displaystyle \sum _{i=1}^{\infty }P(E_i).}$$

A smooth manifold M, even in an unorientable case, admits a measure induced by the volume measure of M. We normalize this measure and denote this normalized measure by dθ. The measure dθ is a probability measure on M.

For example, let M = Sn⁻¹ = {x ∈ Rⁿ ; |x| = 1} be a unit (n−1)-sphere and let dω be the standard (n − 1)-spherical volume measure on Sⁿ⁻¹. Then dθ = (1/A_n−1) dω is the normalized measure, where A_n−1 is the volume of Sⁿ⁻¹.

Let μ, μ₁ be probability measures on M. μ is called to be absolutely continuous with respect to μ₁, if μ(A) = 0 for any $A\in \mathcal{B}(M)$, whenever μ₁(A) = 0.

Let μ be a probability measure on M, absolutely continuous with respect to dθ. Then, from Radon-Nikodym theorem ([26]) there exists a p ∈ L¹(M, dθ) such that μ is represented as μ = p dθ, namely μ satisfies

$$\mu (A)={\displaystyle {\int }_{x\in A}p(x)d\theta (x),\forall A\in \mathcal{B}(M).}$$

The probability density function p = p(x) is called Radon-Nikodym derivative of μ with respect to dθ, written by p = dμ/dθ.

We denote by $\mathcal{P}(M)$ a space of probability measures μ on M, dθ-absolutely continuous (denoted by μ ≪ dθ) such that μ has a positive continuous density function p = p(x), i.e., p ∈ C⁰(M), p(x) > 0 for any x ∈ M.

A manifold M admits an L²-function space L² (M; dθ) as

$$L^2(M,d\theta )=\left\{h:M\to \mathbf{R};{\displaystyle {\int }_M{h}^2(x)d\theta (x)<\infty }\right\}.$$

We notice that there exists a natural embedding

$$\rho :\mathcal{P}(M)\to L^2(M,d\theta );\mu =pd\theta \mapsto \sqrt{p}=\sqrt{\frac{d\mu }{d\theta }}.$$

(8)

By using this embedding we induce a topology on $\mathcal{P}(M)$. Remark that a sequence {μ_i} of $\mathcal{P}(M)$ does not necessarily admit a limit inside $\mathcal{P}(M)$.

Let μ, μ₁ be probability measures in $\mathcal{P}(M)$. Then we can join μ and μ₁ by a path μ(t) = (1−t)μ+tμ₁, t ∈ [0, 1] inside $\mathcal{P}(M)$.

Differentiate μ(t) as a curve in $\mathcal{P}(M)$ to have

$$\frac{d}{dt}((1-t)\mu +t{\mu }_1))={\mu }_1-\mu ,$$

which is a measure on M, represented as μ₁ − μ = (p₁ − p) dθ satisfying

$${\displaystyle {\int }_{M}d({\mu }_1-\mu )=0}.$$

Based on this fact, a tangent space $T_{\mu }\mathcal{P}(M)$ of $\mathcal{P}(M)$ at μ is defined as

$$T_{\mu }\mathcal{P}(M)=\left\{\tau =q(x)d\theta (x);q\in C^0(M),{\displaystyle {\int }_{M}q(x)d\theta (x)=0}\right\}$$

(9)

Notice that the RHS (right hand side) of (9) is independent of μ. So, if we denote by V the RHS of (9), then V is an infinite dimensional vector space and an arbitrary τ ∈ V induces at any $\mu \in \mathcal{P}(M)$ a curve $\mu +t\tau \in \mathcal{P}(M)$ for t ∈ (−ε, ε) with a sufficiently small ε.

We define a curve $c:(a,b)\to \mathcal{P}(M)$ as

$$c(t)=p(x,t)d\theta ,$$

where p(x, t) is of C¹ in t for any fixed x ∈ M. So, c = c(t) has velocity vector field along c

$$\frac{dc}{dt}(t)=\frac{\partial }{\partial t}p(x,t)d\theta \in T_{c(t)}\mathcal{P}(M),t\in (a,b).$$

3.2. Fisher Metric

Definition 1. A positive definite inner product G_μ on$T_{\mu }\mathcal{P}(M)$ at$\mu \in \mathcal{P}(M)$ is defined as

$$G_{\mu }:T_{\mu }\mathcal{P}(M)\times T_{\mu }\mathcal{P}(M)\to \mathbf{R};G_{\mu }(\tau ,{\tau }_1)={\displaystyle {\int }_{x\in M}\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)d\mu (x).}$$

A family$G=\{G_{\mu }|\mu \in \mathcal{P}(M)\}$is called Fisher metric.$\sqrt{G_{\mu }(\tau ,\tau )}$ is denoted by |τ|_{G, μ}.

The Fisher metric is a generalization of Fisher information matrices appeared in parametric models.

The following is one of remarkable properties of the Fisher metric G.

Let Φ : M → M be a homeomorphism of M. Then, Φ induces a push-forward

$${\mathrm{\Phi }}_{\#}:\mathcal{P}(M)\to \mathcal{P}(M);({\mathrm{\Phi }}_{\#}(\mu ))(A):=\mu ({\mathrm{\Phi }}^{-1}(A)),A\in \mathcal{B}(M).$$

(10)

Here μ(A) = ^x A dμ(x). See [27]. We represent (10) in an integral form as

$${\displaystyle {\int }_{x\in A}q(x)d({\mathrm{\Phi }}_{\#}\mu )(x)}={\displaystyle {\int }_{x\in {\mathrm{\Phi }}^{-1}(A)}q(\mathrm{\Phi }(x))d\mu (x)}$$

for any measurable function q : M → R.

When Φ is self-diffeomorphism of M, Φ_#μ coincides with (Φ⁻¹)^*μ, the pull-back of μ by the inverse diffeomorphism Φ⁻¹. Notice that

$${(\mathrm{\Phi }\circ \mathrm{\Psi })}_{\#}={\mathrm{\Phi }}_{\#}\circ {\mathrm{\Psi }}_{\#}$$

for homeomorphisms Φ, Ψ of M and that the push-forward Φ_# : $\mathcal{P}(M)\to \mathcal{P}(M)$ has differential map

$${(d{\mathrm{\Phi }}_{\#})}_{\mu }:T_{\mu }\mathcal{P}(M)\to T_{\mathrm{\Phi }\#\mu }\mathcal{P}(M);{(d{\mathrm{\Phi }}_{\#})}_{\mu }(\tau )={\mathrm{\Phi }}_{\#}(\tau ).$$

Here Φ_#τ is defined similarly as (10). In fact, we have

$${(d{\mathrm{\Phi }}_{\#})}_{\mu }(\tau )=\frac{d}{dt}{|}_{t=0}({\mathrm{\Phi }}_{\#}(\mu +t\tau ))=\frac{d}{dt}{|}_{t=0}({\mathrm{\Phi }}_{\#}(\mu )+t{\mathrm{\Phi }}_{\#}(\tau ))={\mathrm{\Phi }}_{\#}(\tau ).$$

Theorem 7 ([2]). Let Φ_# be a push-forward. Then it acts on$\mathcal{P}(M)$ isometrically with respect to the Fisher metric G. Namely,

$$G_{\mathrm{\Phi }\#\mu }({\mathrm{\Phi }}_{\#}\tau ,{\mathrm{\Phi }}_{\#}{\tau }_1)=G_{\mu }(\tau ,{\tau }_1),\forall \tau ,{\tau }_1\in T_{\mu }\mathcal{P}(M),\forall \mu \in \mathcal{P}(M).$$

Proof. We write μ = p(x) dθ(x) and τ = q(x) dθ(x), τ₁ = q₁(x) dθ(x). Set σ = Φ_#μ. We have then from definition of push-forward

$$\mathrm{\sigma }=p({\mathrm{\Phi }}^{-1}(x)){\mathrm{\Phi }}_{\#}d\theta (x).$$

(11)

This follows in fact from

$$\begin{array}{l}{\displaystyle {\int }_{M}h(x)d({\mathrm{\Phi }}_{\#}\mu )(x)}={\displaystyle {\int }_{M}h(\mathrm{\Phi }(x))d\mu (x)}={\displaystyle {\int }_{M}h(\mathrm{\Phi }(x))p(x)d\theta (x)}\\ ={\displaystyle {\int }_M(h\times (p\circ {\mathrm{\Phi }}^{-1}))(\mathrm{\Phi }(x))d\theta (x)}\end{array}$$

(12)

for any measurable function h on M, which, by definition of push-forward, coincides with ${\displaystyle {\int }_{M}h(x)(p\circ {\mathrm{\Phi }}^{-1})(x){\mathrm{\Phi }}_{\#}d\theta (x)}$and the above is obtained.

In a similar way to (11) we have

$${\mathrm{\Phi }}_{\#}\tau =q({\mathrm{\Phi }}^{-1}(x)){\mathrm{\Phi }}_{\#}d\theta (x),{\mathrm{\Phi }}_{\#}{\tau }_1=q_1({\mathrm{\Phi }}^{-1}(x)){\mathrm{\Phi }}_{\#}d\theta (x),$$

so that

$$\begin{array}{l}{G}_{\mathrm{\sigma }}({\mathrm{\Phi }}_{\#}\tau ,{\mathrm{\Phi }}_{\#}{\tau }_1)={\displaystyle {\int }_{x\in M}\left(\frac{d{\mathrm{\Phi }}_{\#}\tau }{d\mathrm{\sigma }}\right)(x)\left(\frac{d{\mathrm{\Phi }}_{\#}{\tau }_1}{d\mathrm{\sigma }}\right)(x)d\mathrm{\sigma }(x)}\\ ={\displaystyle {\int }_{x\in M}\frac{q({\mathrm{\Phi }}^{-1}(x))}{p({\mathrm{\Phi }}^{-1}(x))}\frac{q_1({\mathrm{\Phi }}^{-1}(x))}{p({\mathrm{\Phi }}^{-1}(x))}}d({\mathrm{\Phi }}_{\#}\mu )(x)\\ ={\displaystyle {\int }_{x\in M}\frac{d\tau }{d\mu }({\mathrm{\Phi }}^{-1}(x))\frac{d{\tau }_1}{d\mu }({\mathrm{\Phi }}^{-1}(x))d({\mathrm{\Phi }}_{\#}\mu )(x).}\end{array}$$

Set $F(x)=\frac{d\tau }{d\mu }({\mathrm{\Phi }}^{-1}(x))\frac{d{\tau }_1}{d\mu }({\mathrm{\Phi }}^{-1}(x))$ and write the above as

$${\displaystyle {\int }_{M}F(x)d({\mathrm{\Phi }}_{\#}\mu )(x)}={\displaystyle {\int }_{M}F(\mathrm{\Phi }(x))d\mu (x)}={\displaystyle {\int }_M\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)d\mu (x)}$$

which is the inner product G_μ(τ, τ₁) of τ and τ₁ at μ.□

Note. The following is known. For any $\mu \in \mathcal{P}(M)$ there exists a homeomorphism Φ : M → M satisfying μ = Φ_#dθ (refer to [28,29]). This fact implies that the action of Homeo(M), which is the group of homeomorphisms on M, on the space $\mathcal{P}(M)$ is isometric and transitive.

Remark 1. The embedding ρ given in (8) satisfies$\rho *{\langle \cdot ,\cdot \rangle }_{L^2}=\frac{1}{4}{G}_{\mu }$, that is,

$${\langle ({\rho }_{*}){\mu }^{\tau },({\rho }_{*}){\mu }^{{\tau }_1}\rangle }_{L^2}=\frac{1}{4}G\mu (\tau ,{\tau }_1);\forall \tau ,{\tau }_1\in T\mu \mathcal{P}(M),\mu \in \mathcal{P}(M).$$

where${\langle \cdot ,\cdot \rangle }_{L^2}$ is the L²-inner product of L²(M; dθ);

$${\langle f_{1,}{f}_2\rangle }_{L^2}:={\displaystyle {\int }_{x\in M}{f}_1(x)f_2(x)d\theta (x),f_{1,}{f}_2\in L^2(M,d\theta ).}$$

3.3. Levi–Civita Connection

The Fisher metric provides the space $\mathcal{P}(M)$ a Riemannian metric as above and then induces on P(M) the Levi–Civita connection ∇ and the Riemannian curvature tensor R. To derive their formulae we will introduce a constant vector fields on P(M).

Let τ ∈ V. Then, τ is considered as a constant vector field on $\mathcal{P}(M)$ by defining a vector field $\{{\tau }_{\mu }|\mu \in \mathcal{P}(M)\}$, ${\tau }_{\mu }=\frac{d}{dt}{|}_{t=0}\mu (t)\in T_{\mu }\mathcal{P}(M)$ that is, τ is a velocity vector of a curve μ (t) = μ + tτ at t = 0. Notice that an integral curve of a constant vector field τ passing through $\mu \in \mathcal{P}(M)$ is given by the curve μ (t).

Theorem 8. Let τ, τ₁ be constant vector fields on P(M). Then, the Levi–Civita connection of the Fisher metric G at μ ∈ P(M) is represented as

$$\begin{array}{l}{\nabla }_{\tau }{\tau }_1=-\frac{1}{2}\left(\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)-G_{\mu }(\tau ,{\tau }_1)\right)\mu \\ =-\frac{1}{2}\left(\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)-{\displaystyle {\int }_{y\in M}\frac{d\tau }{d\mu }(y)\frac{d{\tau }_1}{d\mu }(y)d\mu (y)}\right)\mu .\end{array}$$

(13)

For this formula see [2].

Proof. Recall that on a Riemannian manifold N with a Riemannian metric g the Levi–Civita connection ∇ of g is an affine connection on N, that is, ∇ is a bilinear map; (Y, Z) ↦ ∇_Y Z satisfying

$$\begin{array}{l}{\nabla }_{fY}Z=f{\nabla }_{Y}Z,\\ {\nabla }_Y(fZ)=Yf\cdot Z+f{\nabla }_{Y}Z\end{array}$$

(14)

for smooth vector fields Y, Z on N and a smooth function f on N, which satisfies (∇_Y g)(Z, W ) = 0 together with a symmetry condition, that is, the torsion tensor T (Y, Z) := ∇_Y Z − ∇_ZY −[Y, Z] vanishes. Then, the Levi–Civita connection ∇ exists uniquely and one has Koszul’s formula for ∇

$$g({\nabla }_{Y}Z,W)=\frac{1}{2}\{Y_g(Z,W)+Z_g(W,Y)-W_g(Y,Z)+g([Y,Z],W)-g([Z,W],Y)-g([Y,W],Z)\}.$$

Here Y, Z, W are smooth vector fields on N ([22]).

We give a reference comment on a metric connection with non-trivial torsion, appeared in information geometry. A non-trivial torsion T implies geometrically a breaking of the symmetry in connection coefficients; ${\mathrm{\Gamma }}_{ij}^k={\mathrm{\Gamma }}_{ji}^k$. In a framework of classical parametric model there are very few study of a metric connection with non-trivial torsion. However, as far as the authors know, the e-connection developed in a quantum model has non-trivial torsion. Refer to Chapter 7 of [14] and references cited there.

Now we return back to our situation, that is, to the space $(\mathcal{P}(M),G)$ in which we have for constant vector fields τ, τ₁ and τ₂

$$G({\nabla }_{{\tau }_2}\tau ,{\tau }_1)=\frac{1}{2}\{{\tau }_{2}G(\tau ,{\tau }_1)+\tau G({\tau }_1,{\tau }_2)-{\tau }_{1}G(\tau ,{\tau }_2)\},$$

since [τ, τ₁] = [τ, τ₂] = [τ₁, τ₂] = 0.

Let μ(t) = μ + t τ₂ be a curve in $\mathcal{P}(M)$ of μ(0) = μ and $\dot{\mu }(t)={\tau }_2$. We have then

$$\begin{array}{l}{({\tau }_2)}_{\mu }G(\tau ,{\tau }_1)=\frac{d}{dt}{|}_{t=0}{G}_{\mu (t)}(\tau ,{\tau }_1)\\ =\frac{d}{dt}{|}_{t=0}{\displaystyle {\int }_M\frac{d\tau }{d\mu (t)}}(x)\frac{d{\tau }_1}{d\mu (t)}(x)d\mu (t)(x)\\ ={\displaystyle {\int }_M\frac{\partial }{\partial t}{|}_{t=0}}\left(\frac{d\tau }{d\mu (t)}(x)\frac{d{\tau }_1}{d\mu (t)}(x)d\mu (t)(x)\right)\end{array}$$

in which the integrand is

$$\begin{array}{l}\frac{\partial }{\partial t}{|}_{t=0}\left(\frac{d\tau }{d\mu (t)}(x)\frac{d{\tau }_1}{d\mu (t)}(x)d\mu (t)(x)\right)\\ =\frac{\partial }{\partial t}\left(\frac{d\tau }{d\mu (t)}(x)\right){|}_{t=0}\frac{d{\tau }_1}{d\mu }(x)d\mu (x)+\frac{d\tau }{d\mu }(x)\frac{\partial }{\partial t}\left(\frac{d{\tau }_1}{d\mu (t)}(x)\right){|}_{t=0}d\mu (x)\\ +\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)\frac{\partial }{\partial t}(d\mu (t)(x)){|}_{t=0}.\end{array}$$

The partial derivative term becomes

$$\frac{\partial }{\partial t}\left(\frac{d\tau }{d\mu (t)}(x)\right){|}_{t=0}=-\frac{d\tau }{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x),$$

since by calculation

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\frac{d\tau }{d\mu (t)}(x)\right){|}_{t=0}=\frac{\partial }{\partial t}\left(\frac{q(x)}{p(x)+t{q}_2(x)}\right){|}_{t=0}=-\frac{q(x)q_2(x)}{{(p(x)+t{q}_2(x))}^2}{|}_{t=0}\\ =-\frac{q(x)q_2(x)}{p^2(x)}=-\frac{d\tau }{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x).\end{array}$$

Similarly

$$\frac{\partial }{\partial t}\left(\frac{d{\tau }_1}{d\mu (t)}(x)\right){|}_{t=0}=-\frac{d{\tau }_1}{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x).$$

Since

$$\frac{\partial }{\partial t}(d\mu (t)(x)){|}_{t=0}=d{\tau }_2(x),$$

one obtains

$$\begin{array}{l}{({\tau }_2)}_{\mu }G(\tau ,{\tau }_1)={\displaystyle {\int }_M\left(-\frac{d\tau }{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x)\right)\frac{d{\tau }_1}{d\mu }(x)}d\mu (x)\\ +{\displaystyle {\int }_M\frac{d\tau }{d\mu }(x)\left(-\frac{d{\tau }_1}{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x)\right)}d\mu (x)+{\displaystyle {\int }_M\left(\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)\right)d{\tau }_2}(x).\end{array}$$

Here $d{\tau }_2(x)=\frac{d{\tau }_2}{d\mu }(x)d\mu (x)$. So,

$${({\tau }_2)}_{\mu }G(\tau ,{\tau }_1)=-{\displaystyle {\int }_M\frac{d\tau }{d\mu }(x)\frac{d{\tau }_1}{d\mu }(x)\frac{d{\tau }_2}{d\mu }}(x)d\mu (x).$$

One obtains similar formulae for the terms τ_μG(τ₁, τ₂), (τ₁)_μG(τ, τ₂) and then finally

$$G_{\mu }({\nabla }_{{\tau }_2}\tau ,{\tau }_1)=-\frac{1}{2}{\displaystyle {\int }_M\frac{d\tau }{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x)\frac{d{\tau }_1}{d\mu }}(x)d\mu (x).$$

On the other hand, one observes

$${\displaystyle {\int }_M{G}_{\mu }(\tau ,{\tau }_2)}\frac{d{\tau }_1}{d\mu }(x)d\mu (x)=G_{\mu }(\tau ,{\tau }_2){\displaystyle {\int }_{M}d{\tau }_1(x)=0}$$

and hence

$$\begin{array}{l}{G}_{\mu }({\nabla }_{{\tau }_2}\tau ,{\tau }_1)=-\frac{1}{2}{\displaystyle {\int }_M\left(\frac{d\tau }{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x)-G_{\mu }(\tau ,{\tau }_2)\right)\frac{d{\tau }_1}{d\mu }(x)}d\mu (x)\\ =G_{\mu }\left(-\frac{1}{2}\left(\frac{d\tau }{d\mu }(x)\frac{d{\tau }_2}{d\mu }(x)-G_{\mu }(\tau ,{\tau }_2)\right)\mu ,{\tau }_1\right).\end{array}$$

Since τ₁ is arbitrary, (13) is derived. □

Theorem 9. The Riemannian curvature tensor R of the Fisher metric G satisfies

$$R_{\mu }({\tau }_1,{\tau }_2)\tau =\frac{1}{4}(G_{\mu }(\tau ,{\tau }_2){\tau }_1-(G_{\mu }(\tau ,{\tau }_1){\tau }_2$$

for constant vector fields τ, τ₁, τ₂. Hence, sectional curvature of any section τ ∧ τ₁ is$K(\tau \wedge {\tau }_1)=\frac{1}{4}$.

Refer to [2] for this theorem. We omit proving this theorem. In general, a finite dimensional Riemannian manifold of constant sectional curvature 1/4 is (locally) isometric to a sphere of radius 2. So, the space $\mathcal{P}(M)$ with the metric G is considered to be isometrically an infinite dimensional sphere of radius 2.

As is shown in the next section, this infinite dimensional Riemannian manifold $(\mathcal{P}(M),G)$ is not geodesically complete, in other words, every geodesic is not necessarily extended over R.

3.4. Geodesics

Theorem 10. Let μ ∈ P(M) and$\tau \in T_{\mu }\mathcal{P}(M)$. Assume τ is a unit tangent vector at μ, i.e., |τ|_G,μ = 1. Then, the geodesic μ(t), denoted by exp_μ tτ, with μ(0) = μ, $\dot{\mu }\left(0\right)=\tau$has the form represented by

$$\mu (t)={\left(\cos \frac{t}{2}+\sin \frac{t}{2}\frac{d\tau }{d\mu }(x)\right)}^2\mu ,$$

(15)

in other words,

$$\mu (t)={\left(\cos \frac{t}{2}+\sin \frac{t}{2}\frac{q(x)}{p(x)}\right)}^{2}p(x)d\theta (x)$$

(16)

where μ = p(x) dθ and τ = q(x) dθ are density function representation of μ, τ.

Note. Set t = π into (15). Then $\mu (\mathrm{\pi })={\left(\frac{d\tau }{d\mu }(x)\right)}^2\mu =\frac{q{(x)}^2}{p(x)}d\theta$. However, τ is a tangent vector to P(M) so τ satisfies ${\displaystyle {\int }_{x\in M}q(x)d\theta (x)=0}$ from which there exists a point x_o ∈ M with q(x_o) = 0 and then the density function of μ(π) vanishes at point x_o and then $\mu (\mathrm{\pi })\notin \mathcal{P}(M)$.

To prove Theorem 10, we will show the following lemma, obtained by T. Friedrich ([2]).

Lemma 1. Let μ(t) = p(x, t) dθ be a geodesic such that μ(0) = μ = p₀(x) dθ, $\dot{\mu }\left(0\right)=\tau =q(x)d\theta \in T_{\mu }{\mathcal{P}}^{+}(M)$, |τ|G; μ = 1. Then,

$$p(x,t)=\frac{1}{1+{\tan }^2\frac{t}{2}}\left\{p_0(x)+2\tan \frac{t}{2}q(x)+{\tan }^2\frac{t}{2}\frac{q^2(x)}{p_0(x)}\right\}.$$

(17)

From this lemma it is easy to see that μ(t) = p(x, t) dθ has the above form (15).

Proof of Lemma 1. A proof is given in [2]. However, we will give a proof for a later convenience in proving Theorems 16 and 17. So, for simplicity we write by abbreviation μ(t) = p(t) dθ and $\dot{\mu }\left(t\right)=\dot{p}\left(t\right)d\theta$. Letting τ be a constant vector field, we have

$$G({\nabla }_{\dot{\mu }(t)}\dot{\mu }(t),\tau )=\dot{\mu }(t)(G(\dot{\mu }(t),\tau )-G(\dot{\mu }(t),{\nabla }_{\dot{\mu }(t)}\tau ),$$

(18)

since ∇ preserves the metric G.

Notice that from the rule (14) of Levi–Civita connection, the tangent vector $\dot{\mu }(t)\in T_{\mu }{}_{(t)}\mathcal{P}(M)$of ${\nabla }_{\dot{\mu }(t)}$, appeared in the second term of (18) can be extended as a constant vector field, denoted by the same symbol. So, we can apply (13) to the above and have

$${\nabla }_{\dot{\mu }(t)}\tau =-\frac{1}{2}\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\frac{d\tau }{d\mu (t)}-G_{\mu (t)}(\dot{\mu }(t),\tau )\right)\mu (t).$$

Thus, the Radon-Nikodym derivative of ∇_μ˙(t)τ with respect to μ(t) is

$$\frac{d{\nabla }_{\dot{\mu }(t)}\tau }{d\mu (t)}=-\frac{1}{2}\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\frac{d\tau }{d\mu (t)}-G_{\mu (t)}(\dot{\mu }(t),\tau )\right).$$

Therefore, we have

$$\begin{array}{l}G(\dot{\mu }(t),{\nabla }_{\dot{\mu }(t)},\tau )={\displaystyle {\int }_M\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\right)}\left\{-\frac{1}{2}\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\frac{d\tau }{d\mu (t)}-G_{\mu (t)}(\dot{\mu }(t),\tau )\right)\right\}d\mu (t)\\ =-\frac{1}{2}{{\displaystyle {\int }_M\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\right)}}^2\frac{d\tau }{d\mu (t)}d\mu (t)+\frac{1}{2}{G}_{\mu (t)}(\dot{\mu }(t),\tau ){\displaystyle {\int }_M\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\right)}d\mu (t)\\ =-\frac{1}{2}{{\displaystyle {\int }_M\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\right)}}^{2}d\tau +\frac{1}{2}{G}_{\mu (t)}(\dot{\mu }(t),\tau ){\displaystyle {\int }_M}d\dot{\mu }(t)\\ =-\frac{1}{2}{{\displaystyle {\int }_M\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\right)}}^{2}d\tau =-\frac{1}{2}{{\displaystyle {\int }_M\left(\frac{\dot{p}(t)}{p(t)}\right)}}^{2}d\tau .\end{array}$$

On the other hand, the first term of (18) is

$$\dot{\mu }(t)G(\dot{\mu }(t),\tau )=\frac{d}{dt}{G}_{\mu (t)}(\dot{\mu }(t),\tau )=\frac{d}{dt}{\displaystyle {\int }_M\left(\frac{\dot{p}(t)}{p(t)}\right)}d\tau ={\displaystyle {\int }_M\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)}d\tau .$$

Then (18) becomes

$$G({\nabla }_{\dot{\mu }(t)}\dot{\mu }(t),\tau )={\displaystyle {\int }_M\left\{\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2\right\}}d\tau .$$

Here, $\left\{\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2\right\}\mu (t)$ is not necessarily a tangent vector at μ(t). We choose a real valued function C(t) of t, independent of x ∈ M satisfying

$$\left\{\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2+C(t)\right\}\mu (t)\in T_{\mu (t)}{\mathcal{P}}^{+}(M).$$

In fact, we define C(t) as

$$\begin{array}{l}C(t)=-{\displaystyle {\int }_M\left\{\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2\right\}p(t)d\theta }\\ =-{\displaystyle {\int }_M\frac{\partial }{\partial t}\left(\frac{d\dot{\mu }(t)}{d\mu (t)}\right)d\mu (t)-\frac{1}{2}G(\dot{\mu }(t),\dot{\mu }(t)).}\end{array}$$

Hence, we have

$$G({\nabla }_{\dot{\mu }(t)}\dot{\mu }(t)\tau )=G\left(\left\{\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2+C(t)\right\}\mu (t),\tau \right)$$

for an arbitrary constant vector field τ.

Therefore we have

$${\nabla }_{\dot{\mu }(t)}\dot{\mu }(t)=\left\{\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2+C(t)\right\}\mu (t)$$

Thus, it is concluded that μ(t) = p(t) dθ is a geodesic if and only if p(t) = p(x, t) satisfies

$$\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2+C(t)=0,{\displaystyle {\int }_M\dot{p}(t)d\theta }=0.$$

(19)

Remark 2. Equation (19) is expressed as

$$\frac{\partial }{\partial t}\left(\frac{\dot{\mu }(t)}{\mu (t)}\right)+\frac{1}{2}{\left(\frac{\dot{\mu }(t)}{\mu (t)}\right)}^2+C(t)=0,\dot{\mu }(t)\in T_{\mu (t)}\mathcal{P}(M).$$

Now we will solve these equations.

Notice that if μ(t) is a geodesic, $G_{\mu }{}_{(t)}(\dot{\mu }(t),\dot{\mu }(t))$is constant along μ(t) so from the initial condition $G_{\mu }{}_{(t)}(\dot{\mu }(t),\dot{\mu }(t))\equiv 1$. Therefore, we can write (19) as

$$\begin{array}{c}\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2-{\displaystyle {\int }_M\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)p(t)d\theta }-\frac{1}{2}=0,\\ {\displaystyle {\int }_M\frac{{\dot{p}}^2(t)}{p(t)}d\theta =1,}{\displaystyle {\int }_M\dot{p}(t)d\theta =0.}\end{array}$$

(20)

The second equation means that $|\dot{\mu }(t){|}_{G,\mu (t)}=1$. Set $g(t):={\displaystyle {\int }_M\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)}p(t)d\theta$. We have then

$$\begin{array}{l}g(t)=\frac{d}{dt}\left({\displaystyle {\int }_M\left(\frac{\dot{p}(t)}{p(t)}\right)p(t)d\theta }\right)-{\displaystyle {\int }_M\left(\frac{\dot{p}(t)}{p(t)}\right)\dot{p}(t)d\theta }\\ =\frac{d}{dt}\left({\displaystyle {\int }_{M}d\dot{\mu }(t)}\right)-G_{\mu (t)}(\dot{\mu }(t),\dot{\mu }(t)=0-1=-1.\end{array}$$

So Equations (20) reduce to

$$\frac{\partial }{\partial t}\left(\frac{\dot{p}(t)}{p(t)}\right)+\frac{1}{2}{\left(\frac{\dot{p}(t)}{p(t)}\right)}^2+\frac{1}{2}=0,{\displaystyle {\int }_M{\left(\frac{\dot{p}(t)}{p(t)}\right)}^{2}p(t)d\theta }=1,{\displaystyle {\int }_M\dot{p}(t)d\theta =0.}$$

To solve these, we set $w(t)=\frac{\dot{p}(t)}{p(t)}$. Then w(t) satisfies

$$\dot{w}(t)+\frac{1}{2}(w^2(t)+1)=0.$$

(21)

By solving (21), we have $w(t)=\tan \left(-\frac{1}{2}t+A(x)\right)$, x ∈ M, where A(x) is an integral constant depending on x. By integrating $w(t)=\dot{p}(t)/p(t)$, where p(t) = p(x, t),

$$\log p(x,t)=2\log \cos \left(-\frac{1}{2}t+A(x)\right)+B_1(x),$$

and hence

$$p(x,t)=B(x){\cos }^2\left(-\frac{1}{2}t+A(x)\right).$$

Here B(x) is an integral constant. The constants A(x), B(x) are given as

$$\begin{array}{l}A(x)=\arctan \left(\frac{\dot{p}(x,0)}{p(x,0)}\right),\\ B(x)=\frac{p^2(x,0)+{\dot{p}}^2(x,0)}{p(x,0)},\end{array}$$

where p(x, 0) = p₀(x), p˙(x, 0) = q(x). □

Corollary 2 ([2]). Every geodesic on$(\mathcal{P}(M),G)$ is periodic, of period 2π.

In fact, from (15) a geodesic μ(t) is represented as

$$\mu (t)=\left({\cos }^2\frac{t}{2}+2\sin \frac{t}{2}\cos \frac{t}{2}\frac{d\tau }{d\mu }(x)+{\sin }^2\frac{t}{2}{\left(\frac{d\tau }{d\mu }\right)}^2(x)\right)\mu$$

and ${\cos }^2\frac{t}{2}=\frac{1}{2}(1+\cos t)$. We have then the corollary.

Definition 2. Define$\ell :\mathcal{P}(M)\times \mathcal{P}(M)\to [0,\mathrm{\pi });(\mu ,\mu *)\mapsto \ell =\ell (\mu ,\mu *)$ by

$$\cos \frac{\ell }{2}={\displaystyle {\int }_{x\in M}\sqrt{\frac{d\mu *}{d\mu }}(x)d\mu (x).}$$

(22)

One sees ℓ (μ, μ*) = ℓ (μ, μ*) and that ℓ = 0 if and only if μ = μ*. $\cos \frac{\ell }{2}$ is an f-divergence-like quantity with respect to $f(u)=\sqrt{u}$ (See [14]).

Remark 3. In [2] T. Friedrich remarked that the distance between μ and μ* in$\mathcal{P}(M)$ is given by ℓ = ℓ (μ, μ*).

Theorem 11. Let μ and μ^* be arbitrary probability measures in P(M), μ ≠ μ^*. Then, there exists a unique geodesic μ(t), i.e., a curve;$t\in I\subset \mathcal{R}\mapsto \mu (t)\in \mathcal{P}(M)$with μ(0) = μ, $\mu (\mathrm{l})=\mu *$, where ℓ = ℓ (μ, μ*) is given by (22) and I is an open interval;

$$\mu (t)={\exp }_{\mu }t\tau ={\left(\cos \frac{t}{2}+\sin \frac{t}{2}\frac{d\tau }{d\mu }(x)\right)}^2\mu ,$$

(23)

where τ is a unit tangent vector at μ represented by

$$\tau =\frac{1}{\sin \frac{\ell }{2}}\left(\sqrt{\frac{d\mu *}{d\mu }}(x)-{\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)\mu (x).$$

(24)

Proof. First we will show

Assertion 1. The measure τ given by (24) is a unit tangent vector to $\mathcal{P}(M)$ at μ.

In fact,

$$\begin{array}{l}{\displaystyle {\int }_{M}d\tau }=\frac{1}{\sin \frac{\ell }{2}}{\displaystyle {\int }_{x\in M}\left(\sqrt{\frac{d\mu *}{d\mu }}(x)-{\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)d\mu (x)}\\ =\frac{1}{\sin \frac{\ell }{2}}\left({\displaystyle {\int }_{x\in M}\sqrt{\frac{d\mu *}{d\mu }}(x)d\mu (x)-{\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}}\right)=0,\end{array}$$

so that τ is a tangent vector to P(M). Moreover, τ is unit, i.e., G_μ(τ, τ) = 1, as we compute straightforward

$$G_{\mu }(\tau ,\tau )={\displaystyle {\int }_M{\left(\frac{d\tau }{d\mu }(x)\right)}^{2}d\mu (x)}$$

(25)

and substitute (24) into (25) to have

$$\begin{array}{l}{G}_{\mu }(\tau ,\tau )=\frac{1}{{\sin }^2\frac{\ell }{2}}{\displaystyle {\int }_{x\in M}{\left(\sqrt{\frac{d\mu *}{d\mu }}(x)-{\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)}^{2}d\mu (x)}\\ =\frac{1}{{\sin }^2\frac{\ell }{2}}{\displaystyle {\int }_{x\in M}\left\{\frac{d\mu *}{d\mu }(x)-2\left({\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)\sqrt{\frac{d\mu *}{d\mu }}(x)+{\left({\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)}^2\right\}}d\mu (x)\\ =\frac{1}{{\sin }^2\frac{\ell }{2}}\left\{{\displaystyle {\int }_{x\in M}d\mu *(x)-2{\left({\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)}^2+{\left({\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)}\right)}^2}\right\}\\ =\frac{1}{{\sin }^2\frac{\ell }{2}}\left(1-{\cos }^2\frac{\ell }{2}\right)=1.\end{array}$$

Assertion 2. A geodesic defined by (23) satisfies μ(0) = μ and μ(ℓ) = μ*.

It is seen μ(0) = μ from (23). At t= ℓ.

$$\mu (\ell )={\left(\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tau }{d\mu }(x)\right)}^2\mu$$

and from (25)

$$\sin \frac{\ell }{2}\frac{d\tau }{d\mu }(x)=\sqrt{\frac{d\mu *}{d\mu }}(x)-{\displaystyle {\int }_{y\in M}\sqrt{\frac{d\mu *}{d\mu }}(y)d\mu (y)=}\sqrt{\frac{d\mu *}{d\mu }}(x)-\cos \frac{\ell }{2}.$$

Hence, we find μ(ℓ) = μ* as follows;

$$\mu (\ell )={\left\{\cos \frac{\ell }{2}+\left(\sqrt{\frac{d\mu *}{d\mu }}(x)-\cos \frac{\ell }{2}\right)\right\}}^2\mu ={\left(\sqrt{\frac{d\mu *}{d\mu }}(x)\right)}^2\mu =\frac{d\mu *}{d\mu }(x)\mu =\mu *.$$

Assertion 3. A geodesic joining μ and μ* is unique for μ ≠ μ.

To verify this assertion let μ(t) = exp_μ tτ and $\tilde{\mu }(t)={\exp }_{\mu }t\tilde{\tau }$be unit speed geodesics satisfying $\mu (0)=\tilde{\mu }(0)=\mu$and $\mu (\ell )=\tilde{\mu }(\ell )=\mu *$. From the latter condition we have by using (15)

$${\left(\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tilde{\tau }}{d\mu }(x)\right)}^2\mu ={\left(\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tau }{d\mu }(x)\right)}^2\mu ,\forall x\in M$$

from which

$$\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tilde{\tau }}{d\mu }(x)=\pm \left(\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tau }{d\mu }(x)\right)$$

for any x ∈ M.

To assert

$$\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tilde{\tau }}{d\mu }(x)=\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tau }{d\mu }(x),\forall x\in M$$

define subsets M_± of M by

$$M_{\pm }=\left\{x\in M;\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tilde{\tau }}{d\mu }(x)=\pm \left(\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tau }{d\mu }(x)\right)\right\}.$$

Then, M = M₊ ∪ M₋. Moreover M₊ ∩ M₋ = ∅. This is because if otherwise, M₊ ∩ M₋ ≠ ∅, then at a point x ∈ M₊ ∩ M₋ it holds

$$\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tilde{\tau }}{d\mu }(x)=0.$$

So, at x, $\tilde{\mu }(\ell )=\left(\cos \frac{\ell }{2}+\sin \frac{\ell }{2}\frac{d\tilde{\tau }}{d\mu }(x)\right)\mu =0$. However, it must coincide at x with μ* which has a positive density function. So M₊ ∩ M₋ = ∅.

We see that $M_{+}=\{x\in M;\tilde{\tau }(x)=\tau (x)\}$ and $M_{-}=\{x\in M;q(x)+\tilde{q}(x)=-2\cot \frac{\ell }{2}\}$. Here we write τ = q(x) dθ and $\tilde{\tau }=\tilde{q}(x)d\theta$. Then, M₊ and M₋ are closed in M.

Suppose M₋ ≠ Ø;. Then, M₋ must be a non-empty, proper subset of M. This is because, otherwise if M₋ = M is assumed, then, since $\tilde{\tau }$, τ are tangent to P(M) we see, from 0< ℓ < π.

$${\displaystyle {\int }_{M}d\tilde{\tau }}=-{\displaystyle {\int }_{M}d\tau }-2\cot \frac{\ell }{2}{\displaystyle {\int }_{M}d\theta }=-2\cot \frac{\ell }{2}\ne 0.$$

This is a contradiction. So, M₋ is a proper subset and hence M₊ = M \ M₋ is a non-empty closed, but open subset of M. Therefore, since M is connected, M₊ = M, namely $\tilde{\tau }(x)=\tau (x)$ for any x ∈ X, from which the assertion is proved.

From these assertions Theorem 11 is verified. □

Remark 4. For the ℓ of (22)

$$\mu *\ne \mu \iff \sin \frac{\ell }{2}\ne 0.$$

It suffices for this to show

$$\mu *=\mu \iff \ell =0,$$

since, for ℓ ∈ [0, π), sin ℓ/2 = 0 if and only if ℓ = 0. With respect to the embedding ρ : $\mathcal{P}(M)\to L^2(M,d\theta )$, given in (8), we have

$$||\rho (\mu *)-\rho (\mu )|{|}_{L^2}^2=2-2\cos \frac{\ell }{2}$$

from which it follows that ℓ = 0 implies$||\rho (\mu *)-\rho (\mu )|{|}_{L^2}=0$ and hence μ^* = μ, since ρ is an embedding. Conversely, if μ^* = μ, then$\sqrt{d\mu */d\mu }(x)=1$so$\cos \ell /2={\displaystyle {\int }_M\sqrt{d\mu */d\mu }(x)d\mu =1}$ and ℓ = 0.

To guarantee completeness of geodesics we must extend the space P(M), for example, to the space of probability measures on M, absolutely continuous with respect to dθ and with non-negative density function.

4. Hadamard Manifolds and Barycenter Map

4.1. Hadamard Manifolds and Ideal Boundary

Let (X, g) be an Hadamard manifold. Then the ideal boundary ∂X of (X, g) is defined by taking quotient of space of geodesics of X and is homeomorphic to an (n − 1)-sphere Sⁿ⁻¹. For any θ ∈ ∂X Busemann function B_θ normalized at some point and parametrized in θ ∈ ∂X provides a μ-average Busemann function B_μ on X in terms of a probability measure μ on ∂X. Under some geometrical assumptions which X fulfills, B_μ admits a unique critical point so that we have a barycenter map bar : $\mathcal{P}(\partial X)\to X$ by assigning to an arbitrary probability measure μ on ∂X a point in X as its barycenter, a critical point of B_μ.

Let (X, g) be an Hadamard manifold, a simply connected, complete Riemannian manifold with a metric g = 〈·,·〉 of non-positive curvature. By Cartan-Hadamard theorem an Hadamard manifold is diffeomorphic to a Euclidean space of same dimension. Refer, for this theorem, to [30,31]

A Euclidean space together with a real hyperbolic space Hⁿ(R) are typical examples of Hadamard manifold. Geometrical properties which an Hadamard manifold enjoys are the following;

• Any two points on X can be joined by a unique geodesic.
• Let Δ be a geodesic triangle in X with interior angles α₁, α₂, α₃ and lengths of opposite side, ℓ₁, ℓ₂, ℓ₃. Then, we have a law of cosines;

  $${\ell }_3^2\ge {\ell }_1^2+{\ell }_2^2-2{\ell }_1{\ell }_2\cos {\mathrm{\alpha }}_3.$$
• The distance function from a fixed point x_o ∈ X; $d_{x_o}:X\to \mathbf{R}$, x ↦d(x, x_o) is convex, i.e., for any geodesic γ in X; t ↦γ(t) the restricted function $d_{x_o}\circ \mathrm{\gamma }:t\mapsto d_{x_o}(\mathrm{\gamma }(t))$is convex on R.

Here a function f : R → R is convex, if it satisfies

$$f(a{t}_1+(1-a)t_2)\le af(t_1)+(1-a)f(t_2),\forall t_1,t_2\in \mathbf{R},0\le a\le 1.$$

(26)

Let us define for an Hadamard manifold (X, g) the ideal boundary ∂X, or a boundary at infinity.

Let γ, σ; R → X be unit speed geodesics on X. We say that γ is asymptotically equivalent with σ, denoted by γ ∼ σ, when there exists a constant C > 0 such that d(γ(t), σ(t)) ≤ C for any t ≥ 0. The relation ∼ is an equivalence relation on the space Geo(X) of all oriented, unit speed geodesics on X. The quotient space Geo(X)/∼ is called the ideal boundary of X, denoted by ∂X. An equivalence class represented by γ ∈ Geo(X) is called an asymptotic class, denoted by [γ] or γ(∞). Notice that all geodesics on X are assumed to be of unit speed and oriented.

Let x ∈ X be an arbitrary point of X and S_xX the space of unit tangent vectors at x;

$$S_{x}X=\{v\in T_{x}X;\Vert v\Vert =1\}.$$

Then we define a map

$$\mathit{\beta }={\mathit{\beta }}_x:S_{x}X\to \partial X;v\mapsto [\mathrm{\gamma }],$$

(27)

where γ ∈ Geo(X) is a geodesic given by γ(t) = exp_x(tv), t ∈ R.

Proposition 1 ([30]). The map β_x is bijective.

Moreover, we equip the space X ∪ ∂X with a topology, called a cone topology as follows. For x ∈ X and θ₁, θ₂ ∈ X ∪ ∂X (x ≠ θ₁, x ≠ θ₂) we define ${\angle }_x({\theta }_1,{\theta }_2)=\angle ({\dot{\mathrm{\gamma }}}_1(0),{\dot{\mathrm{\gamma }}}_2(0))$ angles between a geodesic γ₁ from x to θ₁ and a geodesic γ₂ from x to θ₂. For x ∈ X and θ ∈ ∂X, ε > 0 let C_x(θ, ε) = {θ₁ ∈ X ∪ ∂X ; θ₁ ≠ x, ∠ x(θ, θ₁) < ε} be a cone. Further, let T_x(θ, ε) = C_x(θ, ε) \ B(x, r) be a truncated cone (B(x, r) = {y ∈ X|d(y, x) ≤ r} is a closed geodesic ball). Then, a topology generated by open geodesic balls in X and such truncated cones is called a cone topology of X ∪ ∂X. Notice that thus defined cone topology when restricted to ∂X is homeomorphic to the usual topology on S_xX via the mapping β_x. Refer to [31] for the detail.

Let (dθ)_x be a standard volume measure on S_xX, normalized by (dθ)_x(S_xX) = 1. Through β_x we obtain a measure (β_x)_#(dθ)_x on ∂X, denoted by dθ.

4.2. Normalized Busemann Function

Busemann function on X is introduced to define an average Busemann function in terms of a probability measure on ∂X.

Let γ : R → X be a geodesic on X. Define a function f_t : X → R for t > 0 by

$$f_t(x)=d_x(\mathrm{\gamma }(t))-t=d(x,\mathrm{\gamma }(t))-d(\mathrm{\gamma }(0),\mathrm{\gamma }(t)).$$

For any x ∈ X a limit lim_t→∞ f_t(x) exists, as we will see in Proposition 2. We write this limit as f_∞(x) and define a function on X; x ↦ f_∞(x), called Busemann function, denoted by B_γ : X → R; x ↦ f_∞(x).

Each level set of B_γ is called a horosphere, important in studying geometry of Hadamard manifolds. See [11], for example.

Example 1. In a Euclidean space (X, g) = (Rⁿ, g_o) let γ be a geodesic, γ(t) = (t, 0, …, 0). Then, B_γ(x) = −x¹ for x = (x¹, …, xⁿ) ∈ Rⁿ from the following

$$f_t(\mathrm{x})=d(\text{x,}\mathrm{\gamma }(t))-t=\sqrt{{(x^1-t)}^2+{\displaystyle \sum _{i=2}^n{(x^i)}^2}}-t=\frac{-2t{x}^1+{\displaystyle {\sum }_{i=2}^n{(x^i)}^2}}{\sqrt{{(x^1-t)}^2+{\displaystyle {\sum }_{i=2}^n{(x^i)}^2+t}}}\to -x^1,$$

as t → ∞.

Example 2. Busemann function on Hⁿ(R), an n-dimensional real hyperbolic space with standard hyperbolic metric, normalized at o, is given

$$B_{\mathrm{\gamma }}(x)=-\log \frac{1-|x{|}^2}{|x-\theta {|}^2},$$

where θ = γ(∞) ∈ Sⁿ⁻¹.

Proposition 2. The functions f_t : X → R, t > 0, introduced above, have a limit lim_t→∞ f_t(x) for each x ∈ X.

From the triangle inequality we have

$$t_1<t_2=\Rightarrow f_{t_1}(x)\ge f_{t_2}(x),\forall x\in X.$$

In fact, since d(γ(t₁), γ(t₂)) = t₂ − t₁, we see

$$d(x,\mathrm{\gamma }(t_2))\le d(x,\mathrm{\gamma }(t_1))+d(\mathrm{\gamma }(t_1),\mathrm{\gamma }(t_2))=d(x,\mathrm{\gamma }(t_1))+(t_2-t_1)$$

from which the above is derived. On the other hand, we observe uniform boundedness of f_t, that is |f_t(x)| ≤ d(γ(0), x) for any x ∈ X and t > 0 as follows;

$$f_t(x)=d(x,\mathrm{\gamma }(t))-t\le d(x,\mathrm{\gamma }(0))+d(\mathrm{\gamma }(0),\mathrm{\gamma }(t))-t=d(x,\mathrm{\gamma }(0))$$

and

$$\begin{array}{l}t-d(x,\mathrm{\gamma }(t))=d(\mathrm{\gamma }(0),\mathrm{\gamma }(t))-d(x,\mathrm{\gamma }(t))\\ \le d(\mathrm{\gamma }(0),x)+d(x,\mathrm{\gamma }(t))-d(x,\mathrm{\gamma }(t))=d(x,\mathrm{\gamma }(0))\end{array}$$

so −d(x, γ(0)) ≤ f_t(x) ≤ d(x, γ(0)).

Therefore, the sequence {f_t(x)|t > 0} is bounded and decreasing and then has a limit as t → ∞.

Proposition 3. Let γ and σ be geodesics. If γ ∼ σ, then

$$B_{\mathrm{\gamma }}(x)-B_{\mathrm{\sigma }}(x)=c,\forall x\in X,$$

for a constant c.

See [30,31] for this proposition, from which, for θ ∈ ∂X Busemann function B_γ associated with a geodesic γ, [γ] = θ, gives us a same function on X modulo additive constant. So, let x_o ∈ X be an arbitrary point of X as a base point. Then, from non-positive curvature of X there exists a unique geodesic γ such that γ(0) = x_o and [γ] = θ.

Definition 3. Let x_o ∈ X and θ ∈ ∂X. Let γ : R → X be a geodesic satisfying γ(0) = x_o and [γ] = θ. The Busemann function B_γ associated to γ is called normalized Busemann function, denoted by B_θ.

Properties of (normalized) Busemann function:

• B_θ(x_o) = 0 for any θ ∈ ∂X,
• B_θ(γ(t)) = −t, t ∈ R, for any θ ∈ ∂X, where γ is a geodesic satisfying γ(0) = x_o, [γ] = θ.
• Busemann function is Lipschitz continuous;

  $$|B_{\theta }(x)-B_{\theta }(y)|\le d(x,y),x,y\in X.$$
• Busemann function is of class C² (refer to [32]).
• Gradient vector field ∇B_θ satisfies |(∇B_θ)_x| ≡ 1 for any x ∈ X and θ ∈ ∂X. Here (∇B_θ)_x ∈ T_xX is defined by h(∇B_θ)_x, vi = v(B_θ), directional derivative of B_θ with respect to v ∈ T_xX. An integral curve x(t) of ∇B_θ passing through a point x is obtained by x(t) = σ(−t), where σ is a geodesic of σ(0) = x and [σ] = θ. Moreover, for any x ∈ X and any vector v ∈ S_xX there exists a θ ∈ ∂X such that v = −(∇B_θ)_x so that β_x(v) = θ.
• Busemann function is convex (see (26)), since it is a limit of convex functions.
• From (vi), the Hessian of Busemann function (∇dB_θ)_x : T_x × T_x → R is positive semi-definite at any point x ∈ X, i.e., (∇dB_θ)_x(v, v) ≥ 0, for any v ∈ T_xX and x ∈ X, and satisfies

  $${(\nabla d{B}_{\theta })}_x({(\nabla B_{\theta })}_x,v)=0,v\in T_{x}X.$$

  Here, for a C²-function f on X the Hessian ∇df is a symmetric bilinear form, defined by

  $${(\nabla df)}_x(u,v)=u(Vf)-({\nabla }_{u}V)f,u,v\in T_{x}X,x\in X$$

  where V is a smooth vector field, an extension of v. Notice that for a unit vector u ∈ S_xX

  $${(\nabla df)}_x(u,u)=\frac{d^2}{d{t}^2}|{}_{t=0}f(\mathrm{\gamma }(t))$$

  with respect to a geodesic γ such that γ(0) = x, γ˙(0) = u.

Example 3. On a Euclidean space ∇dB_θ = 0 for any θ ∈ ∂X(due to Example 1).

Example 4. On a real hyperbolic space Hⁿ(R), n ≥ 2,

$${(\nabla d{B}_{\theta })}_x(u,v)=\langle u,v\rangle -\langle \nabla B_{\theta },u\rangle \langle \nabla B_{\theta },u\rangle ,u,v\in T_{x}X.$$

See for this [8].

Let (X, g) be an Hadamard manifold and ϕ : X → X be an isometry of X, i.e., a smooth transformation of X satisfying ϕ^*g = g. An isometry preserves the distance d of X, i.e., d(ϕ(x), ϕ(y)) = d(x, y), x, y ∈ X and transforms a geodesic σ into a new geodesic ϕ ○ σ so that, if σ ∼ γ, then ϕ ○ σ ∼ ϕ ○ γ. Therefore, ϕ induces a transformation $\widehat{\varphi }$ of ∂X, a ∂X-extension of ϕ as

$$\widehat{\varphi }:\partial X\to \partial X;\theta =[\mathrm{\gamma }]\mapsto [\varphi \circ \mathrm{\gamma }].$$

Notice that $({\widehat{\varphi }}^{-1})=\widehat{{\varphi }^{-1}}$ for the inverse ϕ⁻¹ of ϕ. $\widehat{\varphi }$ is a homeomorphism of ∂X in terms of cone topology.

Proposition 4 (Busemann cocycle formula [33]). Any normalized Busemann function enjoys a cocycle formula with respect to an isometry ϕ of X;

$$B_{\theta }(\varphi (x))=B_{{\widehat{\varphi }}^{-1}(\theta )}(x)+B_{\theta }(\varphi (x)).$$

(28)

Proof. Let γ : R → X be a geodesic, γ(0) = x_o, [γ] = θ. Notice that ϕ ○ γ is a geodesic with ϕ ○ γ(0) = ϕ(x_o), which, in general, does not coincide with the base point x_o. For the Busemann function $B_{{\varphi }^{-1}\circ \mathrm{\gamma }}(x)$ with respect to a geodesic ϕ⁻¹ ○ γ we have

$$\begin{array}{l}{B}_{{\varphi }^{-1}\circ \mathrm{\gamma }}(x)=\underset{t\to \infty }{\lim }\left(d(x,{\varphi }^{-1}\circ \mathrm{\gamma }(t))-t\right)\\ =\underset{t\to \infty }{\lim }\left(d(\varphi (x),\mathrm{\gamma }(t))-t\right)=B_{\mathrm{\gamma }}(\varphi (x)).\end{array}$$

(29)

On the other hand, ϕ⁻¹ ○ γ belongs to ${\widehat{\varphi }}^{-}{}^1(\theta )$ and (ϕ⁻¹ ○ γ)(0) = ϕ⁻¹(x_o). Let σ be a geodesic such that $[\mathrm{\sigma }]={\widehat{\varphi }}^{-}{}^1(\theta ),\mathrm{\sigma }(0)=x_o$ Then, the normalized Busemann function $B_{{\widehat{\varphi }}^{-1}(\theta )}$ is given by B_σ. Since ϕ⁻¹ ○ γ and σ belong to the same ${\widehat{\varphi }}^{-}{}^1(\theta )$, from (29) $B_{\mathrm{\sigma }}-B_{{\varphi }^{-1}\circ \mathrm{\gamma }}$ is a constant function on X. This constant is given from the above by $(B_{\mathrm{\sigma }}-B_{{\varphi }^{-1}\circ \mathrm{\gamma }})(x_o)=-B_{{\varphi }^{-1}\circ \mathrm{\gamma }}(x_o)=-B_{\mathrm{\gamma }}(\varphi (x_o))$ so that on X

$$B_{\mathrm{\sigma }}(x)-B_{{\varphi }^{-1}\circ \mathrm{\gamma }}(x)\equiv -B_{\mathrm{\gamma }}(\varphi (x_o)).$$

(30)

Since B_θ is given by B_γ, (28) is obtained from (30). □

In what follows, every normalized Busemann function B_θ on X is assumed to be continuous with respect to θ ∈ ∂X for each fixed point x ∈ X. This assumption is guaranteed by a real hyperbolic space. See Example 3. Rank one symmetric spaces of non-compact type and Damek-Ricci spaces satisfy this assumption, as is seen in [25].

Definition 4 ([21]). An Hadamard manifold (X, g) is said to satisfy visibility axiom, if for any distinct ideal point θ, θ₁ of ∂X there exists a geodesic γ : R → X such that γ(+∞) = θ and γ(−∞) = θ₁. Here γ(−∞) ∈ ∂X defined by [γ⁻], where γ⁻ is the geodesic of reversed orientation given by γ⁻(t) = γ(−t), t ∈ R.

A Euclidean space does not satisfy visibility axiom.

Proposition 5. Let (X, g) be an Hadamard manifold. (X, g) satisfies visibility axiom if and only if, for any θ ∈ ∂X

$$\underset{x\to {\theta }_1}{\lim }{B}_{\theta }(x)=+\infty ,$$

provided θ₁ ≠ θ. Refer to [31] for this.

Notice B_θ(x) = −∞, if x → θ, from property (i) of normalized Busemann function.

4.3. Average Busemann Function and Barycenter

In what follows, an Hadamard manifold satisfies visibility axiom and Busemann function B_θ(x) is continuous with respect to every θ.

Let ∂X be, as before, the ideal boundary of an Hadamard manifold (X, g), diffeomorphic to Sⁿ⁻¹, n = dim X and dθ a normalized standard measure on ∂X.

Denote by $\mathcal{P}(\partial X)$ a space of probability measures μ on ∂X which is absolutely continuous with respect to dθ (μ ⩽ dθ) whose density function p = p(θ) is of C⁰ and positive;

$$\mathcal{P}(\partial X)=\left\{\mu =p(\theta )d\theta ;{\displaystyle {\int }_{\partial X}p(\theta )d\theta =1,p\in C^0(\partial X),p(\theta )>(\forall \theta \in \partial X)}\right\}.$$

Definition 5. Let.$\mu \in \mathcal{P}(M)$ Then, a function B_μ : X → R, called μ-average Busemann function, is defined by

$${\mathrm{B}}_{\mu }(x)={\displaystyle {\int }_{\theta \in \partial X}{B}_{\theta }(x)d\mu (\theta ).}$$

Average Busemann function for any $\mu \in \mathcal{P}(M)$ fulfills the following;

• For any $\mu \in \mathcal{P}(M)$ each B_μ is convex on X and B_μ(x_o) = 0.
• B_μ(γ(t)) → +∞, as t → ∞, where γ : R → X is an arbitrary geodesic in X (Theorem 12).
• B_μ is Lipschitz continuous, in fact, |B_μ(x) − B_μ(y)| ≤ |d(x, y)| for x, y ∈ X.
• The gradient vector field ∇B_μ is defined on X as

  $${(\nabla d{\mathrm{B}}_{\mu })}_x={\displaystyle {\int }_{\partial X}{(\nabla B_{\theta })}_{x}d\mu (\theta ),}x\in X$$

  and satisfies |(∇B_μ)_x| ≤ 1, x ∈ X.
• the Hessian ∇dB_μ can be defined as μ-average Hessian;

  $${(\nabla d{\mathrm{B}}_{\mu })}_x(u,v)={\displaystyle {\int }_{\partial X}{(\nabla d{B}_{\theta })}_x(u,v)d\mu (\theta ),u,v\in T_{x}X}x\in X,$$

  provided (X, g) is of bounded Ricci curvature and moreover ΔB_θ, the Laplacian of B_θ together with d(ΔB_θ) are uniformly bounded with respect to x ∈ X and θ ∈ ∂X. Here Δf = −trace ∇df for a C²-function f on X. This is derived from Bochner formula (see [23]). If (X, g) is asymptotically harmonic ([34]), i.e., ΔB_θ ≡ c for any θ, and of bounded Ricci curvature, the average Hessian ∇dB_μ, $\mu \in \mathcal{P}(M)$ is defined.

Definition 6. Let$\mu \in \mathcal{P}(M)$. A critical point of μ-average Busemann function B_μ is called a barycenter of μ.

For a C¹-function f : X → R, y ∈ X is called a critical point of f, if one of the following equivalent conditions holds;

• the differential of f at y vanishes along all directional vector, i.e.,

  $$\frac{d}{dt}{|}_{t=0}f(x(t))=0$$

  for any C¹-curve x(t) of x(0) = y,
• the one-form df, or the gradient vector field ∇f vanishes at y.

Observation. For $\mu =p(\theta )d\theta \in \mathcal{P}(\partial X)$, x ∈ X is a barycenter of μ if and only if (dB_μ)_x(u) = 0 for any u ∈ T_xX, which is equivalent to stating that a measure τ defined by τ = (dB_θ)_x(u) dμ = h(∇B_θ)_x, ui p(θ) dθ is a tangent vector to $\mathcal{P}(\partial X)$ at μ for each u ∈ T_xX.

Theorem 12. If, as is assumed, an Hadamard manifold (X, g) satisfies visibility axiom and Busemann function is continuous with respect to any θ ∈ ∂X. Then, every$\mu \in \mathcal{P}(M)$ admits a barycenter.

Proof. This theorem is proved by Besson, Courtois and Gallot in [8] by showing B_μ(γ(t)) → ∞ as t → +∞ along any geodesic γ of X. However, they assume that all probability measures on ∂X are without atom and an Hadamard manifold (X, g) is of special type, i.e., a rank one symmetric space of non-compact type. We restrict the space of probability measures as $\mathcal{P}(\partial X)$. However, we relax the assumptions concerning an Hadamard manifold (X, g) and then, assume only that (X, g) satisfies visibility axiom and Busemann function is continuous with respect to any θ ∈ ∂X.

Let C > 0 be a constant and set A_C = {y ∈ X ; B_μ(y) ≤ C}. A_C is a convex set and x_o ∈ A_C, since B_μ is convex and B_μ(x_o) = 0. Note A_C contains a geodesic ball {y ∈ X ; d(y, x_o) ≤ C/2} since B_μ is Lipschitz. Let $\mu \in \mathcal{P}(M)$ and γ be a geodesic satisfying γ(0) = x_o and [γ] = θ. Then it is possible to verify lim_t→+∞ B_μ(γ(t)) = +∞ in the following steps;

Step I. Since Busemann function B_θ is convex and B_θ(x₀) = 0 for any θ, we have

$$d(\mathrm{\gamma }(t_1),x_0)B_{\theta }(\mathrm{\gamma }(t)\ge d(\mathrm{\gamma }(t),x_0)B_{\theta }(\mathrm{\gamma }(t_1)$$

(31)

in other words

$$t_1{B}_{\theta }(\mathrm{\gamma }(t))\ge t{B}_{\theta }(\mathrm{\gamma }(t_1))(0\le t_1\le t).$$

In fact, if we set a = t₁/t, then, 0 ≤ a ≤ 1 and we have t₁ = (1 − a) 0 + at. The Convex function B_θ(γ(t)) fulfills

$$B_{\theta }(\mathrm{\gamma }(t_1))\le (1-a)B_{\theta }(\mathrm{\gamma }(0))+a{B}_{\theta }(\mathrm{\gamma }(t))=a{B}_{\theta }(\mathrm{\gamma }(t)),$$

that is

$$B_{\theta }(\mathrm{\gamma }(t_1))\le a{B}_{\theta }(\mathrm{\gamma }(t))$$

(32)

which is just (31).

Step II. Fix t₁ > 0 of Step I. Take an arbitrary θ₀ ∈ ∂X and fix it. Let γ be a geodesic of γ(0) = x₀ and [γ] = θ₀. For any t > 0 set

$$J_{{\theta }_0}(t):=\{\theta \in \partial X;B_{\theta }(\mathrm{\gamma }(t))\le 0\}.$$

We show that there exists a t₁ ∈ (0, ∞) such that $\mu (J_{{\theta }_0}(t_1))<1$, as follows.

Since B_θ(x) is continuous with respect to θ, the set $J_{{\theta }_0}(t)$ is compact in ∂X. We see ${\theta }_0\in J_{{\theta }_0}(t)$. From (32) it holds

$$J_{{\theta }_0}(t)\subset J_{{\theta }_0}(t_1)(0\le t_1\le t).$$

(33)

It follows then

$${\displaystyle \underset{t\in [0,\infty )}{\cap }{J}_{{\theta }_0}(t)}=\{{\theta }_0\},$$

because from the visibility axiom, we have from Proposition 3.4 for any θ ∈ ∂X such that θ ≠ θ₀ lim_t→∞ B_θ(γ(t)) = +∞. Moreover, from (33) for the μ we find

$$\underset{t\to \infty }{\lim }\mu (J_{{\theta }_0}(t))=\mu \left({\displaystyle \underset{t\in [0,\infty )}{\cap }{J}_{{\theta }_0}(t)}\right)=\mu (\{{\theta }_0\})=0.$$

(34)

Therefore, we have a t₁ ∈ (0, ∞) such that $\mu (J_{{\theta }_0}(t_1))<1$. Here, notice $\mu (J_{{\theta }_0}(t))\le \mu (J_{{\theta }_0}(t_1))<1$ for any t ≥ t₁ > 0.

Step III. Let K be a compact subset of $\partial X\backslash J_{{\theta }_0}(t_1)$ satisfying μ(K) > 0. It is possible to choose such a K. Then, from (33) it holds for any t ≥ t₁ that $K\subset \partial X\backslash J_{{\theta }_0}(t)$, and B_θ(γ(t)) ≥ 0 for any $\theta \in \partial X\backslash J_{{\theta }_0}(t)$. So,

$$\begin{array}{l}{\displaystyle {\int }_{\partial X}{B}_{\theta }(\mathrm{\gamma }(t))}d\mu (\theta )={\displaystyle {\int }_{J_{{\theta }_0}(t)}{B}_{\theta }(\mathrm{\gamma }(t))}d\mu (\theta )+{\displaystyle {\int }_{\partial X\backslash J_{{\theta }_0}(t)}{B}_{\theta }(\mathrm{\gamma }(t))}d\mu (\theta )\\ \ge {\displaystyle {\int }_{J_{{\theta }_0}(t)}{B}_{\theta }(\mathrm{\gamma }(t))}d\mu (\theta )+{\displaystyle {\int }_K{B}_{\theta }(\mathrm{\gamma }(t))}d\mu (\theta ).\end{array}$$

Since K is compact, we choose a constant C > 0 satisfying

$$B_{\theta }(\mathrm{\gamma }(t_1))\ge C>0,\forall \theta \in K.$$

From (32), we have

$${\displaystyle {\int }_{\partial X}{B}_{\theta }(\mathrm{\gamma }(t))}d\mu (\theta )\ge \frac{t}{t_1}{\displaystyle {\int }_{J_{{\theta }_0}(t)}{B}_{\theta }(\mathrm{\gamma }(t_1))}d\mu (\theta )+C\frac{t}{t_1}\mu (K).$$

To estimate the first term of the RHS we choose D ≥ 0 satisfying

$$B_{\theta }(\mathrm{\gamma }(t))\ge -\sup \{|B_{\theta }(\mathrm{\gamma }(t))|;\theta \in \partial X\}=-D.$$

In fact, since ∂X is compact, B_θ(γ(t₁)), as a continuous function of θ, is bounded with respect to θ. Therefore, the above is written as

$${\displaystyle {\int }_{\partial X}{B}_{\theta }(\mathrm{\gamma }(t))d\mu (\theta )\ge \frac{t}{t_1}(-D\mu (J_{{\theta }_0}(t))+C\mu (K)).}$$

We let t → +∞ and then, from (34) we have

$$\underset{t\to \infty }{\lim }{\mathcal{B}}_{\mu }(\mathrm{\gamma }(t))=\lim {\displaystyle {\int }_{\partial X}{B}_{\theta }(\mathrm{\gamma }(t))d\mu (\theta )=+\infty }$$

from which it follows that the closed set A_C is bounded and hence is compact. Therefore, B_μ admits a minimal point x ∈ X, namely, x is a barycenter of μ. □

Proposition 6. Let (X, g) be an Hadamard manifold of bounded Ricci curvature. If (X, g) is asymptotically harmonic, then the following holds; If there exists${\mu }_0\in \mathcal{P}(M)$ such that μ-average Hessian$\nabla d{\mathcal{B}}_{{\mu }_0}$is positive definite at every point in X, then, for any μ ∈ P(M) μ-average Hessian ∇dB_μ is also positive definite at every point in X.

Proof. Let x ∈ X and u ∈ T_xX. Then, for a geodesic γ in X, γ(0) = x, $\dot{\mathrm{\gamma }}(0)=u$we have

$$\begin{array}{l}{(\nabla d{\mathrm{B}}_{{\mu }_0})}_x(u,u)=\frac{d^2}{d{t}^2}{|}_{t=0}{\mathrm{B}}_{{\mu }_0}(\mathrm{\gamma }(t))={\displaystyle {\int }_{\theta \in \partial X}\frac{{\partial }^2}{\partial t^2}{|}_{t=0}{B}_{\theta }}(\mathrm{\gamma }(t))d{\mu }_0(\theta )\\ ={\displaystyle {\int }_{\partial X}{(\nabla d{B}_{\theta })}_x(u,u)}d{\mu }_0(\theta ).\end{array}$$

Similarly for any $\mu \in \mathcal{P}(M)$, we have

$$\begin{array}{l}{(\nabla d{\mathrm{B}}_{\mu })}_x(u,u)=\frac{d^2}{d{t}^2}{|}_{t=0}{\mathrm{B}}_{\mu }(\mathrm{\gamma }(t))={\displaystyle {\int }_{\theta \in \partial X}\frac{{\partial }^2}{\partial t^2}{|}_{t=0}{B}_{\theta }}(\mathrm{\gamma }(t))d\mu (\theta )\\ ={\displaystyle {\int }_{\partial X}{(\nabla d{B}_{\theta })}_x(u,u)}d\mu (\theta ).\end{array}$$

(35)

Let € $C={min}_{\theta \in \partial X}\frac{d\mu }{d{\mu }_0}(\theta )>0$. It is concluded then from the above

$${(\nabla d{\mathrm{B}}_{\mu })}_x(u,u)\ge C{(\nabla d{\mathrm{B}}_{{\mu }_0})}_x(u,u)>0,\forall u\in T_{x}X(\ne 0),x\in X.$$

□

Theorem 13. Let (X, g) be an Hadamard manifold satisfying the above assumptions. Then, any$\mu \in \mathcal{P}(M)$ admits a unique barycenter.

In fact, Theorem 12 asserts an existence of a barycenter for any μ. From Proposition 6 a barycenter must be unique, since, if, otherwise, μ admits barycenters y₁, y₂, y₁ ≠ y₂, then f(t) := B_μ(γ(t)) along a geodesic γ : R → X joining y₁ and y₂ satisfies f′(0) = f′ (d) = 0 (d = d(y₁, y₂)) and f″ (t) > 0, t ∈ [0, d] because of the positive definiteness of μ-average Hessian (35). So, f(t) must be constant along γ. This contradicts property (ii) of μ-average Busemann function. Hence uniqueness is proved.

Proposition 7 (average Busemann cocycle formula). Let ϕ be an isometry of an Hadamard manifold (X, g). Then for any$\mu \in \mathcal{P}(\partial X)$

$${\mathrm{B}}_{\mu }({\mathrm{\Phi }}^{-1}x)={\mathrm{B}}_{(\widehat{\mathrm{\Phi }})\#\mu }(x)+{\mathrm{B}}_{\mu }({\mathrm{\Phi }}^{-1}{x}_o).$$

(36)

Proof. Integrate the Busemann cocycle formula (28)

$$B_{\theta }({\mathrm{\Phi }}^{-1}(x))=B_{\widehat{\mathrm{\Phi }}(\theta )}(x)+B_{\theta }({\mathrm{\Phi }}^{-1}{x}_o)$$

for the inverse isometry ϕ⁻¹ with respect to a measure μ. We then get (36). □

From Theorem 13 we define a map, called barycenter map

$$\text{bar}:\mathcal{P}(\partial X)\to X;\mu \mapsto y,$$

(37)

by assigning a barycenter y to μ.

Example 5. The standard measure dθ has bar(dθ) = x_o, the base point as its barycenter. In fact, we observe

$${\displaystyle {\int }_{\partial X}\langle {(\nabla B_{\theta })}_{x_o},u\rangle }d\theta =0,\forall u\in T_{x_o}X,$$

since$d\theta ={({\mathit{\beta }}_{x_o})}_{\#}{(d\theta )}_{x_o}$is the push-forward of the spherical measure${(d\theta )}_{x_o}$ of$S_{x_o}X$, where${\mathit{\beta }}_{x_o}$is a map given in (27), and one has${\mathit{\beta }}_{x_o}(-{(\nabla B_{\theta })}_{x_o})=\theta$so that${\mathit{\beta }}_{x_o}(v)=\theta$ implies$v=-{(\nabla B_{\theta })}_{x_o}$. Then the LHS of the above is written as

$${\displaystyle {\int }_{v\in S_{x_o}X}\langle {(\nabla B_{{\mathit{\beta }}_{x_o}(v)})}_{x_o},u\rangle }d{\theta }_{x_o}(v)=-{\displaystyle {\int }_{v\in S_{x_o}X}<v,u>d{\theta }_{x_o}=0.}$$

Here, the last integration is derived from a standard formula on Sⁿ⁻¹ which is described as${\displaystyle {\sum }_{i=1}^n{({\theta }^i)}^2=1}$ with respect to the standard coordinates θ = (θ¹, …, θⁿ) ∈ Rⁿ;

$${\displaystyle {\int }_{\theta \in S^{n-1}}{\theta }^i{(d\theta )}_{S^{n-1}}}=0,i=1,\dots ,n.$$

4.4. Barycenter Map

In this subsection we will verify that the barycenter map (37) enjoys a fibration over an Hadamard manifold X in terms of Fisher metric G. Before giving a detailed argument, we prepare some special probability measures on ∂X which play a crucial role in the barycenter map, that is, Poisson kernel measures. Here, Poisson kernel is a fundamental solution of Dirichlet problem at ∂X; given a C⁰-function f on ∂X, find a function u on X which satisfies the Laplace equation Δu = 0 and the boundary condition lim_x→θ u(x) = f(θ) for θ ∈ ∂X.

Definition 7 ([3,35]). A function P_θ(x) = P (x, θ) on X is called a Poisson kernel, normalized at x_o, for θ ∈ ∂X if it satisfies

• ΔP (x, θ) = 0 and P (x, θ) > 0 for any x ∈ X and θ ∈ ∂X.
• P (x_o, θ) = 1 for any θ ∈ ∂X.
• for any θ ∈ ∂X, P (x, θ) ∈ C⁰(X ∪ ∂X \ {θ}) as an extension function on X ∪ ∂X and lim_x→θ1 P (x, θ) = 0 for θ₁ ≠ θ.

The solution u = u(x) of the Dirichlet problem on ∂X is described as an integration form;

$$u(x)={\displaystyle {\int }_{\theta \in \partial X}P(X,\theta )f(\theta )d\theta }$$

Example 6. On a real hyperbolic space Hⁿ(R) of standard hyperbolic metric of Poicaré ball model, the Poisson kernel is given by

$$P(x,\theta )={\left(\frac{1-|x{|}^2}{|x-\theta {|}^2}\right)}^{n-1},\theta \in \partial X=S^{n-1}.$$

Example 7. The Poisson integral formula, well known in potential theory, is for a bounded harmonic function h = h(z), z = re^iφ ∈ {z ∈ C||z| ≤ 1}

$$h(r{e}^{i\phi })=\frac{1}{2\mathrm{\pi }}{\displaystyle {\int }_{0\le \theta \le 2\mathrm{\pi }}\frac{1-r^2}{1-2r\cos (\phi -\theta )+r^2}f(e^{i\theta })d\theta ,}$$

where f is a bounded function on S¹. The kernel function (1 − r²)/(1 − 2r cos(ϕ − θ) + r²) is just the Poisson kernel P (z, θ) = (1 − |z|²)/|z − θ|² of the hyperbolic plane H²(R). See, for example [20].

Definition 8. A Poisson kernel on an Hadamard manifold (X, g) is called Busemann-Poisson kernel, when it has the following form

$$P(x,\theta )=\exp \{-Q{B}_{\theta }(x)\},x\in X,\theta \in \partial X,$$

where Q > 0 is volume entropy of (X, g), the exponential growth rate of the volume of (X, g)

$$Q=\underset{r\to \infty }{\lim }\frac{1}{r}\log \text{vol}B(x,r)$$

for a geodesic ball B(x, r).

Remark 5. For volume entropy refer to [8] in which the following theorem, Theorem of Manning, ([7]) is cited; if Q_top denotes the topological entropy of a compact Riemannian manifold Y of non-positive curvature, then one has

• Q(Ỹ) ≤ Q_top(Y),
• Q(Ỹ) = Q_top(Y), provided the curvature of Y is negative or zero.
  Here, Ỹ is the universal covering space of Y and the topological entropy Q_top(Y) is defined by

  $$Q_{top}(Y)=\underset{R\to \infty }{\lim }\frac{1}{R}\log (\#\{\mathrm{\gamma }|\ell (\mathrm{\gamma })\le R\}),$$

  where γ denotes a periodic geodesic in Y of length ℓ (γ) and #{γ | ℓ (γ) ≤ R} denotes the number of periodic geodesics of length not greater than R.

For example Q = 0 for a Euclidean space and Q = n − 1 for a real hyperbolic space Hⁿ(R) of standard hyperbolic metric.

Remark 6. Any Damek-Ricci space admits a Busemann-Poisson kernel (refer to [4]). See also [8] for a rank one symmetric space of non-compact type which is just a member of Damek-Ricci spaces, as observed by using Iwasawa decomposition of isometry groups.

Theorem 14. Let (X, g) be an Hadamard manifold satisfying the assumptions in Theorem 12 and Proposition 6. If (X, g) admits a Busemann-Poisson kernel, then, for${\mu }_x:=P(x,\theta )d\theta \in \mathcal{P}(\partial X)$

• bar(μ_x) = x for any x ∈ X and
• at any point y ∈ X, (∇dB_μx)_y is positive definite.

The statement (i) is shown in [8]. From (i) the barycenter map bar is onto.

Definition 9. Let$\mu =p(\theta )d\theta \in \mathcal{P}(\partial X)$and x ∈ X be a barycenter of μ. We define a linear map

$$v_x^{\mu }:T_{x}X\to T_{\mu }\mathcal{P}(\partial X);u\mapsto v_x^{\mu }(u)={(d{B}_{\theta })}_x(u)\mu =\langle {(\nabla B_{\theta })}_x,u\rangle p(\theta )d\theta .$$

(38)

Notice that the map $v_x^{\mu }$ is injective.

Proof of Theorem 14. (i) Let u ∈ T_xX and x(t) a C¹-curve in X such that x(0) = x, $\dot{x}(0)=u$. Differentiate ${\displaystyle {\int }_{\partial X}P(x(t),\theta )d\theta \equiv 1}$ as

$$\begin{array}{l}0=\frac{d}{dt}{|}_{t=0}{\displaystyle \int P(x(t),\theta )d\theta }\\ ={\displaystyle \int \frac{\partial }{\partial t}{|}_{t=0}\exp \{-Q{B}_{\theta }(x(t))\}}d\theta \\ ={\displaystyle \int -Q{(d{B}_{\theta })}_x(\dot{x}(0))\exp \{-Q{B}_{\theta }(x(0))\}}d\theta \\ =-Q\frac{d}{dt}{|}_{t=0}{\displaystyle {\int }_{\partial X}{B}_{\theta }}(x(t))d{\mu }_x=-Q{(d{B}_{{\mu }_x})}_x(u).\end{array}$$

So, x = bar μ_x.

For a proof of (ii) we first show

Assertion 4. The measure μ_x satisfies

$${(\nabla d{B}_{{\mu }_x})}_x(u,v)=Q{G}_{{\mu }_x}(v_x^{{\mu }_x}(u),v_x^{{\mu }_x}(v)),u,v\in T_{x}X$$

(39)

in terms of the Fisher metric G, where $v_x^{{\mu }_x}$ is the linear map defined in (38).

It suffices to show this in case of u = v. Let γ be a geodesic in X satisfying γ(0) = x, $\dot{\mathrm{\gamma }}\left(0\right)=u$. Then we have

$$0=\frac{d^2}{d{t}^2}{|}_{t=0}{\displaystyle {\int }_{\partial X}P(\mathrm{\gamma }(t),\theta )d\theta .}$$

However, this is

$$\begin{array}{l}{\displaystyle {\int }_{\partial X}\frac{{\partial }^2}{\partial t^2}{|}_{t=0}\exp \{-Q{B}_{\theta }(\mathrm{\gamma }(t))\}d\theta }\\ =-Q{\displaystyle {\int }_{\partial X}\left\{{(\nabla d{B}_{\theta })}_x(u,u)-Q{\{{(d{B}_{\theta })}_x(u)\}}^2\right\}}\exp \{-Q{B}_{\theta }(x)\}d\theta \\ =-Q{(\nabla d{B}_{{\mu }_x})}_x(u,u)-Q{G}_{{\mu }_x}(v_x^{{\mu }_x}(u),v_x^{{\mu }_x}(u))\}\end{array}$$

showing (39).

From this assertion (ii) is proved as follows. At y ∈ X we have, since (∇dB_θ)_y(·,·) is positive semi-definite,

$$\begin{array}{c}{(\nabla d{B}_{{\mu }_x})}_y(u,u)={\displaystyle \int {(\nabla d{B}_{\theta })}_y(u,u)d{\mu }_x(\theta )=}{\displaystyle \int {(\nabla d{B}_{\theta })}_y(u,u)P(x,\theta )d\theta }\\ \ge C{\displaystyle \int {(\nabla d{B}_{\theta })}_y(u,u)P(y,\theta )d\theta =C}{(\nabla d{B}_{{\mu }_y})}_y(u,u)\end{array}$$

for any u ∈ T_yX, where C = inf_θ∈∂X P (x, θ)/P (y, θ) > 0. □

Now we will investigate the map bar : $\mathcal{P}(\partial X)\to X$.

Theorem 15. The barycenter map bar : $\mathcal{P}(\partial X)\to X$ gives a projection of a fibre space whose total space is$\mathcal{P}(\partial X)$ and base space is X with fibres bar⁻¹(x) over x ∈ X. In fact, let x ∈ X and μ ∈ bar⁻¹(x). Then

$$T_{\mu }\mathcal{P}(\partial X)=T_{\mu }{\text{bar}}^{-1}(x)\oplus \mathrm{Im}{v}_x^{\mu }(\dim \mathrm{Im}{v}_x^{\mu }=n),$$

as an orthogonal direct sum of the vertical subspace T_μbar⁻¹(x) and the horizontal subspace$\mathrm{Im}{v}_x^{\mu }$ with respect to Fisher metric G_μ.

This orthogonal decomposition indicates that N = { $N_{\mu }=\mathrm{Im}{v}_x^{\mu }$; μ ∈ bar⁻¹(x)} distributes a normal bundle to each fibre bar⁻¹(x), x ∈ X. Notice that bar⁻¹(x) is path-connected, since, for μ, μ₁ ∈ bar⁻¹(x) (1 − t)μ + tμ₁, 0 ≤ t ≤ 1, also belongs to bar⁻¹(x).

We will show that the vertical subspace T_μ bar⁻¹ (x) is orthogonal to $N_{\mu }=\mathrm{Im}{v}_x^{\mu }$. Let u ∈ T_xX and τ ∈ T_μbar⁻¹(x) and take μ(t) = μ + tτ for sufficiently small |t|. Then, μ (t) ∈ bar⁻¹ (x). So, for a sufficiently small |t|, we have

$$\begin{array}{l}0={\displaystyle \int {(d{B}_{\theta })}_x(u)d\mu (t)(\theta )}={\displaystyle \int {(d{B}_{\theta })}_x(u)d(\mu +t\tau )(\theta )}\\ ={\displaystyle \int {(d{B}_{\theta })}_x(u)d\mu (\theta )}+t{\displaystyle \int {(d{B}_{\theta })}_x(u)d\tau (\theta )}\\ =t{\displaystyle \int {(d{B}_{\theta })}_x(u)d\tau (\theta )}=t{G}_{\mu }(v_x^{\mu }(u),\tau ).\end{array}$$

This means the orthogonality of T_μbar⁻¹(x) and $N_{\mu }=\mathrm{Im}{v}_x^{\mu }$.

Since the image $N_{\mu }=\mathrm{Im}{v}_x^{\mu }$ is a finite dimensional subspace of ${\mathrm{T}}_{\mu }\mathcal{P}(\partial X)$, the direct sum decomposition is easily shown and so we skip.

4.5. Fibres bar⁻¹(x) and Geodesics

We discussed in Section 3 several properties and propositions of geodesics on a space of probability measures. In this section we will investigate under which condition a geodesic of $\mathcal{P}(\partial X)$ is contained in a fibre bar⁻¹(x).

Theorem 16. Let (X, g) be an Hadamard manifold satisfying the assumptions in Theorem 12 and Proposition 6, and admitting a Busemann-Poisson kernel.

Let μ ∈ bar⁻¹(x) and τ ∈ T_μbar⁻¹(x), |τ|_G,μ = 1. Then a geodesic μ(t) = exp_μ tτ entirely belongs to bar⁻¹(x) for any t at which μ(t) is well-defined, if and only if τ fulfills H_μ(τ, τ) = 0.

Here H is the second fundamental form of a submanifold bar⁻¹(x) of the ambient space $\mathcal{P}(\partial X)$ (see Equation (5) in Section 2).

Proof. From Theorem 10, Section 3 the geodesic μ(t) is given by

$$\mu (t)={\left(\cos \frac{t}{2}+\sin \frac{t}{2}\frac{d\tau }{d\mu }(\theta )\right)}^2\mu .$$

Then μ(t) ∈ bar⁻¹(x) for all t if and only if for any u ∈ T_xX

$$0={\displaystyle {\int }_{\theta \in \partial X}{(d{B}_{\theta })}_x(u)d\mu (t)(\theta ).}$$

However the RHS is

$${\cos }^2\frac{t}{2}{\displaystyle \int {(d{B}_{\theta })}_x(u)d\mu (\theta )+2\cos }\frac{t}{2}\sin \frac{t}{2}{\displaystyle \int {(d{B}_{\theta })}_x(u)\frac{d\tau }{d\mu }(\theta )+2}{\sin }^2\frac{t}{2}{\displaystyle \int {(d{B}_{\theta })}_x(u){\left(\frac{d\tau }{d\mu }\right)}^2(\theta )d\mu (\theta )}$$

for all t. Since μ ∈ bar⁻¹(x) and τ ∈ T_μbar⁻¹(x), this is equivalent to

$$0={\displaystyle \int {(d{B}_{\theta })}_x(u)}{\left(\frac{d\tau }{d\mu }\right)}^2(\theta )d\mu (\theta )$$

which is reduced by the aid of Levi–Civita connection formula to

$$0={\displaystyle \int {(d{B}_{\theta })}_x(u)\left(\frac{d{\nabla }_{\tau }\tau }{d\mu }\right)}(\theta )d\mu (\theta )=-2G_{\mu }(v_x^{\mu }(u),{\nabla }_{\tau }\tau )$$

which means that H(τ, τ) = 0. Conversely, if τ satisfies H(τ, τ) = 0, then it is easy to see that μ(t) = exp_μ tτ belongs to the fibre bar⁻¹(x) by following reversely the above argument.

Theorem 17. Let μ, μ* ∈ bar⁻¹(x), x ∈ X (μ ≠ μ*). Then, a geodesic μ(t) joining μ and μ* lies entirely on bar⁻¹(x) if and only if

$${\displaystyle {\int }_{\partial X}{(d{B}_{\theta })}_x(u)\sqrt{\frac{d\mu *}{d\mu }}(\theta )d\mu (\theta )=0},\forall u\in T_{x}X.$$

(40)

Proof. The geodesic μ(t) joining μ and μ* is written from Theorem 11 by exp_μ tτ of an initial vector

$$\tau =\frac{1}{\sin \frac{\mathrm{l}}{2}}\left\{\sqrt{\frac{d\mu *}{d\mu }}(\theta )-\cos \frac{\mathrm{l}}{2}\right\}\mu ,\mathrm{l}=\mathrm{l}(\mu ,\mu *)>0.$$

(41)

Then, μ(t) lies on bar⁻¹(x) if and only if the following conditions hold, that is, τ is tangent to bar⁻¹(x), namely,

$$G_{\mu }(v_x^{\mu }(u),\tau )=0$$

(42)

for any u ∈ T_xX, and that

$$H(\tau ,\tau )=-\frac{1}{2}{\displaystyle {\int }_{\partial X}{(d{B}_{\theta })}_x(u){\left(\frac{d\tau }{d\mu }\right)}^2}(\theta )d\mu (\theta )=0.$$

(43)

Equation (42) is equivalent to (40), since τ is given by (41). On the other hand, (43) is written as

$$\begin{array}{l}0={\displaystyle {\int }_{\partial X}{(d{B}_{\theta })}_x(u){\left(\sqrt{\frac{d\mu *}{d\mu }}(\theta )-\cos \frac{\mathrm{l}}{2}\right)}^{2}d\mu (\theta )}\\ =-2\cos \frac{\mathrm{l}}{2}{\displaystyle {\int }_{\partial X}{(d{B}_{\theta })}_x(u)\sqrt{\frac{d\mu *}{d\mu }}(\theta )d\mu (\theta )}\end{array}$$

for any u ∈ T_xX. This condition is also (40), so we get Theorem 17.

Example 8. Let μ = dθ. Then, bar(dθ) = x_o, as seen in Example 5. We exhibit tangent vectors τ, τ₁ at dθ satisfying H(τ; τ) = 0, whereas H(τ₁; τ₁) ≠ 0, as follows;

• Identify ∂X with $S_{x_o}X\cong S^{n-1}$ via ${\mathit{\beta }}_{x_o}$, and dθ with ${(d\theta )}_{x_o}$. Choose on Sⁿ⁻¹ a function q = q(θ) = θⁱθ^j, i ≠ j and define τ = q(θ) dθ as a measure on ∂X. Then, $\tau \in T_{d\theta }\mathcal{P}(\partial X)$. Moreover, τ ∈ T_dθ bar⁻¹ (x_o), since $G_{d\theta }(v_{x_o}^{d\theta }(u),\tau )=0$ for any $u\in T_{x_o}X$and H(τ, τ) = 0. These are directly from the integral formulae; ${\displaystyle {\int }_{S^{n-1}}{\theta }^i{\theta }^j{\theta }^k}{(d\theta )}_{x_o}=0$, ${\displaystyle {\int }_{S^{n-1}}{({\theta }^i{\theta }^j)}^2{\theta }^k}{(d\theta )}_{x_o}=0$ for any k = 1, …, n. By normalizing τ′ = τ/|τ|_G in terms of G, from Theorem 16 γ(t) = exp_dθ tτ′ gives a geodesic lying on bar⁻¹(x_o).
• Let q₁ = q₁(θ) is a function on Sⁿ⁻¹, n ≥ 3, defined by q₁(θ) = θ¹θ²θ³+θ²θ³ and set τ₁ = q₁(θ) dθ. Then $({\mathrm{\gamma }}_1)(t)={\exp }_{d\theta }t{{\tau }^{\prime }}_1$, ${{\tau }^{\prime }}_1={\tau }_1/|{\tau }_1{|}_G$ is a geodesic being not completely on the fibre bar⁻¹(x_o).

5. Barycentrically Associated Maps

Let ϕ be an isometry of an Hadamard manifold (X, g). Then, from the average Busemann cocycle formula (36).

Theorem 18 ([8]). For any isometry ϕ of (X, g), we have

$$\text{bar}({\widehat{\varphi }}_{\#}\mu )=\varphi (\text{bar}(\mu )),\mu \in \mathcal{P}(\partial X).$$

(44)

Proof. Let $y=\text{bar(}{\widehat{\varphi }}_{\#}\mu \text{)}$. Then ${(d{\mathcal{B}}_{{\widehat{\varphi }}_{\#}\mu })}_y(u)=0$ for any u ∈ T_yX, namely, due to (36) ϕ⁻¹ y turns out to be a critical point of B_μ, that is, y = ϕ(bar(μ)), so (44) is obtained. □

Definition 10. Let Φ : ∂X → ∂X be a homeomorphism of ∂X. Then, a bijective map ϕ : X → X is said to be barycentrically associated to Φ, if Φ and ϕ satisfy the relation bar ◯ Φ = ϕ ◯ bar, that is, bar(Φ(μ)) = ϕ(bar(μ)) for any$\mu \in \mathcal{P}(\partial X)$.

Now we are ready to give a proof of Theorem 6.

Proof of Theorem 6. From the statement of the theorem, diagram (6) asserts for any x ∈ X, i.e.,

$${\mathrm{\Phi }}_{\#}(v_x^{{\mu }_x}(u))=v_{\phi x}^{{\mu }_{\phi x}}({({\phi }_{*})}_x(u)),\forall u\in T_{x}X,$$

(45)

namely

$${\mathrm{\Phi }}_{\#}{(d{B}_{\theta })}_x(u){\mu }_x(\theta ))={(d{B}_{\theta })}_{\phi x}({({\phi }_{*})}_{x}u){\mu }_{\phi x}(\theta )$$

(46)

for μ_x = Σ(x), where $\mathrm{\Sigma }:X\to \mathcal{P}(\partial X)$ is a cross section whose existence is assumed in the theorem. We write (46) as

$${(d{B}_{{\mathrm{\Phi }}^{-1}\theta })}_x(u){\mathrm{\Phi }}_{\#}{\mu }_x={(d{B}_{\theta })}_{\phi x}({({\phi }_{*})}_{x}u){\mu }_{\phi x}.$$

Since another diagram (6) implies ${\mathrm{\Phi }}_{\#{\mu }_x}={\mu }_{\phi x}$, we have

$${(d{B}_{{\mathrm{\Phi }}^{-1}\theta })}_x(u){\mu }_{\phi x}={(d{B}_{\theta })}_{\phi x}({({\phi }_{*})}_{x}u){\mu }_{\phi x},$$

(47)

that is,

$${(d{B}_{{\mathrm{\Phi }}^{-1}\theta })}_x(u)={(d{B}_{\theta })}_{\phi x}({({\phi }_{*})}_{x}u),u\in T_{x}X,\forall \theta \in \partial X,$$

or

$$\langle {(\nabla B_{{\mathrm{\Phi }}^{-1}\theta })}_x,u\rangle =\langle {(\nabla B_{\theta })}_{\phi x},{({\phi }_{*})}_{x}u\rangle u\in T_{x}X,\forall \theta \in \partial X.$$

Using the formal adjoint ((φ_*)_x)* of (φ_*)_x, we write the above as

$${(\nabla B_{{\mathrm{\Phi }}^{-1}\theta })}_x={({({\phi }_{*})}_x)}^{*}{(\nabla B_{\theta })}_{\phi x}u\in T_{x}X,\forall \theta \in \partial X.$$

Let v ∈ Sφ_xX be a unit tangent vector at φx and choose θ ∈ ∂X such that (∇B_θ)_φx = v so that the above is written as ${({\phi }_{*})}_x^{*}v={(\nabla B_{{\mathrm{\Phi }}^{-1}\theta })}_x$ and thus from property (v) in Section 4.2 it is concluded that $|{({\phi }_{*})}_x^{*}v|=|{(\nabla B_{{\mathrm{\Phi }}^{-1}\theta })}_x|=1$ which implies that ${({\phi }_{*})}_x^{*}$ and consequently (φ_*)_x is a linear isometry. Since x ∈ X is arbitrary, φ turns out to be an isometry of (X, g).

To show that Φ coincides with ∂X-extension $\widehat{\phi }$ we make use of (47) together with the following

$${(d{B}_{{\widehat{\phi }}^{-1}\theta })}_x(u){\mu }_{\phi x}={(d{B}_{\theta })}_{\phi x}({({\phi }_{*})}_{x}u){\mu }_{\phi x}$$

(48)

which is derived by differentiating the Busemann cocycle formula (28) to get for any u ∈ T_xX, x ∈ X

$${(d{B}_{{\widehat{\phi }}^{-1}\theta })}_x(u)={(B_{{\mathrm{\Phi }}^{-1}\theta })}_x(u).$$

(49)

Namely, we have $d(B_{{\widehat{\phi }}^{-1}\theta }-B_{{\mathrm{\Phi }}^{-1}\theta })=0$ on X for any θ ∈ ∂X. Since X is connected, $B_{{\widehat{\phi }}^{-1}\theta }(x)-B_{{\mathrm{\Phi }}^{-1}\theta }(x)=C$ for a constant C which depends on θ. From this it follows that $\widehat{\phi }=\mathrm{\Phi }$. In fact, assume ${\widehat{\phi }}^{-1}\theta \ne {\mathrm{\Phi }}^{-1}\theta$ for some θ ∈ ∂X, otherwise, and let x → Φ⁻¹θ. Then, from the visibility axiom (see Proposition 5) $B_{{\widehat{\phi }}^{-1}\theta }(x)-B_{{\mathrm{\Phi }}^{-1}\theta }(x)\to -\infty$ contradicting that C is constant.