# CANONICAL FOLIATIONS OF STATISTICAL MANIFOLDS WITH STATISTICAL MODELS

Emmanuel Gnandi, Michel Nguiffo Boyom, Stéphane Puechmorel

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Emmanuel Gnandi, Michel Nguiffo Boyom, Stéphane Puechmorel. CANONICAL FOLIATIONS OF STATISTICAL MANIFOLDS WITH STATISTICAL MODELS. information geometry, 2026, 10.1007/s41884-026-00193-8. hal-05527121

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# CANONICAL FOLIATIONS OF STATISTICAL MANIFOLDS WITH STATISTICAL MODELS

EMMANUEL GNANDI¹, MICHEL BOYOM², AND STÉPHANE PUECHMOREL³

**ABSTRACT.** Constructing a realisation of a statistical manifold as a statistical model, i.e a manifold with dual connections with respect to a Fisher metric, is an important question in information geometry. While a positive answer to this problem was given by the work on Hong Van Lé [39], writing explicitly the probability family giving rise to the Fisher metric is generally a difficult task. In this work, starting with the sheaf of solutions $\mathcal{J}\nu$ of the Hessian equation on a gauge structure $(M, \nabla)$, a canonical representation of the group associated to the Lie algebra formed by its sections is introduced. On the foliation it defines, a characterization of compact hyperbolic leaves is then obtained. Furthermore, these leaves can be provided with an explicit statistical model structure, that is a probability density defining a Fisher metric.

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## 1. INTRODUCTION

Hyperbolicity is an important notion in geometry and can be tackled using different approaches. In its original work [34], Koszul introduces the development map $Q$ to define a hyperbolic manifold as one for which the image by $Q$ of its universal covering is an open convex domain without straight lines. In [37], a cohomological characterization is given, stating that a necessary condition for a gauge structure $(M, \nabla)$ to be hyperbolic is the existence of a closed 1-form admitting a positive-definite covariant derivative. Finally, if $(M, \nabla)$ is a gauge structure on a Hessian manifold provided with a Riemannian metric $g$, it is hyperbolic iff $[g] = 0$ in Koszul-Vinberg cohomology [11]. In [52], Shima observes that the Fisher information of all classical statistical models is Hessian metrics. As Amari did, Shima [52] also proved that any dually flat statistical manifold is a Hessian manifold. These structures arose from the works of many mathematicians (e.g., J.-L. Koszul, Y. Matsushima, A. Nijenhuis, E. B. Vinberg) in their attempt to solve the Gerstenhaber conjecture, which states that every restricted theory of deformation generates its proper theory of cohomology. Michel Boyom partly resolved this conjecture in 2006 (see [47]) and subsequently developed the homological formulation of Hessian geometry [11, 13].

The aim of this paper is to study the hyperbolicity of the leaves of canonical foliations in statistical manifolds using representations in the affine group of a finite-dimensional vector space, and to equip these leaves with a statistical model. The paper is organized as follows: Section 2 summarizes the principal notions used in the paper. In Section 3, the Hessian equation on a gauge structure is introduced along with some basic facts. In Section 4, the canonical group representation is defined, from which a vanishing condition in cohomology is given in Theorem 4.4. Finally, the case of statistical manifolds is treated in Section 5, and a characterization of hyperbolic compact leaves is given. In Section 6, we prove that the universal coverings of these leaves are convex cones. In Section 7, we show that the leaves carry a statistical model structure pertaining to the exponential family. In Section 8, an explicit construction is given, and conclusions are drawn.

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## 2. PRELIMINARIES

2.1. **Locally Flat Structure.** Most of the material presented in this section can be found, with further details, in [9, 11, 26]. We recall here only the notions required in the remainder of the paper. Throughout the paper, $X(M)$ denotes the real Lie algebra of smooth vector fields on a manifold $M$. We adopt the Einstein summation convention over repeated indices.

Definition 2.1 (Koszul Connection). A Koszul connection on a smooth manifold $M$ is a first-order differential operator

$$ \nabla : TM \otimes TM \to TM, \quad (X, Y) \mapsto \nabla_X Y, $$

satisfying, for all vector fields $X, Y \in X(M)$ and all smooth functions $f \in C^\infty(M)$, the following properties:

$$ \begin{aligned} \nabla_{fX} Y &= f \nabla_X Y, \\ \nabla_X (fY) &= X(f)Y + f \nabla_X Y. \end{aligned} $$

Given a Koszul connection $\nabla$, its torsion tensor $T^\nabla$ of $\nabla$ is defined by

$$ (2.1) \qquad T^\nabla(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y], $$

and its curvature tensor $R^\nabla$ of $\nabla$ by

$$ (2.2) \qquad R^\nabla(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z. $$

Definition 2.2 (Gauge structure). A gauge structure is a pair $(M, \nabla)$ where $M$ is a smooth manifold and $\nabla$ is a Koszul connection on $M$.

Definition 2.3. An affinely flat structure on an $m$-dimensional manifold $M$ is a complete atlas

$$ \mathcal{A} = \{(U_i, \Phi_i)\}, $$

such that the transition maps $\Phi_i^{-1} \circ \Phi_j$ belong to the group $\mathrm{Aff}(\mathbb{R}^n)$ of affine transformations on $\mathbb{R}^m$.

Remark 2.4. Some long-standing conjectures pertain to the field of affine structures, including the Chern conjecture stating that any compact affine manifold has vanishing Euler characteristic.

Definition 2.5. A locally flat manifold is a pair $(M, \nabla)$, where $\nabla$ is a Koszul connection whose torsion and curvature tensors vanish identically:

$$ (2.3) \qquad T^\nabla = 0, \quad R^\nabla = 0. $$

A Koszul connection satisfying these two conditions is called a locally flat connection, and the pair $(M, \nabla)$ is referred to as a locally flat manifold.

A manifold admits an affine structure if and only if it is locally flat. This property can easily be obtained by considering a local chart and a the associated coordinate vector fields. While in general it is not possible to get a global connection from them, it is the case with an affine structure, since the derivatives of the transitions maps are constant. This connection has obviously vanishing torsion and curvature. Conversely, if a Koszul connection on $M$ has vanishing torsion and curvature, there is a reduction of the structure group to $\mathrm{Aff}(\mathbb{R}^n)$, hence an affine structure.

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2.2. **Cohomology of Koszul–Vinberg algebras.** A Koszul–Vinberg algebra (also known as a pre-Lie algebra, left-symmetric algebra, Koszul algebra, or Vinberg algebra) was introduced independently by Koszul [34] and Vinberg [56] in their investigations of affinely flat manifolds and homogeneous cones. The subject has since given rise to a substantial body of literature; we refer in particular to the works of Vey [54], Vinberg [57], Boyom [10,47], Matsushima [43], Helmstetter [30], and Burde [15].

Jean-Louis Koszul and Albert Nijenhuis, motivated by the relationships between the co-homology of Koszul-Vinberg (KV) algebras and the deformation theory of locally flat manifolds, have extensively developed the theory of cohomology of KV algebras [48]. In [13], Boyom provides a comprehensive review of this theory, which he has significantly enriched himself. He shows that “Koszul’s last formula” gives rise to KV cohomology with trivial coefficients and extends these results to cohomology with coefficients in a non-trivial KV module. Moreover, he highlights numerous applications of the KV cohomology of the algebra $\mathcal{A}$ to the theory of statistical models from the perspective of information geometry.

**Definition 2.6.** A Koszul–Vinberg algebra is a vector space $\mathcal{A}$ endowed with a bilinear product

$$ \cdot : \mathcal{A} \times \mathcal{A} \to \mathcal{A}, \quad (a, b) \mapsto a \cdot b, $$

satisfying the identity

$$ (2.4) \qquad (a \cdot b) \cdot c - a \cdot (b \cdot c) = (b \cdot a) \cdot c - b \cdot (a \cdot c), \quad \forall a, b, c \in \mathcal{A}. $$

To a locally flat manifold $(M, \nabla)$ we associate the algebra $\mathcal{A} = (\chi(M), \nabla)$, with product defined by

$$ (2.5) \qquad X \cdot Y := \nabla_X Y, \quad X, Y \in \chi(M). $$

The algebra $\mathcal{A}$ is a KV-algebra and will be called the KV-algebra of the locally flat manifold $(M, \nabla)$. The algebra $\mathcal{A}$ associated with a locally flat manifold is naturally a two-sided KV-module over itself. The space of smooth functions $C^{\infty}(M)$ forms a left KV-module over $\mathcal{A}$ with the action:

$$ X \cdot f = df(X). $$

**Definition 2.7.** Let $(M, \nabla)$ be a locally flat manifold. The Koszul–Vinberg complex is defined as

$$ (C(\nabla), \delta) = \left( C(\nabla) = \bigoplus_{j \ge 0} C^j(\nabla), \quad \delta_j : C^j(\nabla) \to C^{j+1}(\nabla) \right), $$

where

$$ C^0(\nabla) = \{f \in C^\infty(M) \mid \nabla^2 f = 0\}, \quad C^j(\nabla) = \operatorname{Hom}_{\mathbb{R}}(\otimes^j \chi(M), C^\infty(M)), \quad j \ge 1. $$

The coboundary operator $\delta_j$ is defined by

$$ (2.6) \qquad \delta_0 f = df, \quad \forall f \in C^0(\nabla), $$

and for $j \ge 1$,

$$ \delta_j f(X_1 \otimes \cdots \otimes X_{q+1}) = \sum_{i=1}^{j} (-1)^i \left( \begin{aligned} & d(f(\dots, \hat{X}_i, \dots, X_{j+1})) (X_i) \\ & - \sum_{k \ne i} f(\dots, \hat{X}_i, \dots, \nabla_{X_i} X_k, \dots) \end{aligned} \right), \quad (2.7) $$
88 where the hat $\hat{X}_i$ means omission of $X_i$.

89 The Koszul–Vinberg cohomology is the graded vector space

$$ (2.8) \qquad H_{KV}^j(\nabla) = \ker \delta_j / \operatorname{im} \delta_{j-1}. $$

90 2.3. Developing map of a locally flat manifold. Let $(M, \nabla)$ be a locally flat manifold and
91 fix a base point $x_0 \in M$. Consider the space of differentiable paths

$$ c : [0, 1] \to M, \quad c(0) = x_0. $$

92 Homotopy with fixed endpoints is an equivalence relation on $c$ and its quotient is the
93 universal covering space, denoted by $\tilde{M}$. The projection

$$ \pi : \tilde{M} \to M, \quad \pi([c]) = c(1), $$

94 is the universal covering map. Being a local diffeomorphism, $\pi$ allows to lift the
95 connection $\nabla$ to $\tilde{M}$, yielding the connection $\tilde{M}$.

96 Now, for a path $c : [0, 1] \to M$ and $\sigma \in [0, 1]$, let

$$ \tau_{\sigma} : T_{x_0} M \to T_{c(\sigma)} M $$

97 be the parallel transport along $c$ with respect to $\nabla$.

98 **Definition 2.8 (Developing map).** The developing map of $(M, \nabla)$ is the map

$$ D : \tilde{M} \to T_{x_0} M \cong \mathbb{R}^n $$

99 defined by

$$ (2.9) \qquad D([c]) = \int_0^1 \tau_\sigma^{-1} \left( \frac{dc}{d\sigma} \right) d\sigma. $$

100 When dealing with locally flat manifolds, the developing map has the following important
101 property.

102 **Proposition 2.1.** Let $M$ be a compact locally flat manifold with fundamental group $\pi(M)$. There exists an homomorphism $\theta: \pi(M) \to \operatorname{Aff}(\mathbb{R}^n)$ such that the developing map is an equivariant immersion with respect to $\theta$.

105 Informally, the developing map is a way of "flattening" a curve on the manifold, that is
106 representing it in $\mathbb{R}^n$.

107 2.4. Hyperbolic affine manifolds. We now turn to a crucial notion in this article: the
108 hyperbolicity of locally flat manifolds. Jacques Vey extended W. Kaup’s notion of hyper-
109 bolicity, originally developed for Riemann surfaces, to the setting of locally flat manifolds.
110 Hyperbolicity, in the sense of Vey [54, 55] and Kobayashi [31–33], can be viewed as the
111 opposite of geodesic completeness.

112 Hyperbolic affine manifolds are closely related to Hessian manifolds, which in turn are
113 strongly connected to Kähler geometry. This notion has been extensively studied by several
114 authors, including Koszul [34, 36, 37], Vey [54, 57], H Shima [52] and Kobayashi [31, 33].
115 Hyperbolicity is a fundamental concept in Koszul’s geometry and can be approached from
116 multiple perspectives. In this work, we emphasize the algebraic topology viewpoint, which
117 provides a richer framework than Riemannian geometry. In particular, the theory of affine
118 representations of Lie groups plays a central role in the study of locally flat manifolds,
119 following the work of Jean-Marie Souriau [53].
144 where

$$ (3.8) \qquad \begin{aligned} \Omega_{ij}^l(X) = & \frac{\partial^2 X^l}{\partial x_i \partial x_j} + \Gamma_{ik}^l \frac{\partial X^k}{\partial x_j} + \Gamma_{jk}^l \frac{\partial X^k}{\partial x_i} - \Gamma_{ij}^k \frac{\partial X^l}{\partial x_k} \\ & + \frac{\partial \Gamma_{jk}^l}{\partial x_i} + \Gamma_{jk}^m \Gamma_{im}^l - \Gamma_{ij}^m \Gamma_{mk}^l. \end{aligned} $$

145 The geometric symbol $\sigma_{\nabla}$ of $\nabla^2$ is

$$ (3.9) \qquad \sigma_{\nabla}(X)(\partial_i, \partial_j) = \sum_l \frac{\partial^2 X^l}{\partial x_i \partial x_j} \partial_l. $$

146 The Hessian differential operator $\nabla^2$ is involutive (the Koszul-Spencer complex is acyclic).

147 Being local, the Hessian equation can be defined on $\mathcal{X}(U)$, $U$ an open set. It is clear that a restriction of a solution is still a solution, but more can be obtained, as the next proposition indicates.

148
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**Proposition 3.1.** [9, 12] Let $(M, \nabla)$ be a gauge structure. The function $J_{\nabla}$ on the category of open sets defined by:

$$ (3.10) \qquad \mathcal{J}_{\nabla}(U) := \{X \in \mathcal{X}(U) \mid \nabla^2 X = 0\}, \quad U \subseteq M, $$
152 is a sheaf.

153
154
The next proposition shows that when dealing with solutions of the Hessian equation, the pre-Lie structure on $\chi(M)$ degenerates to an associative algebra.

155
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**Proposition 3.2.** The space of sections $J_{\nabla} = \Gamma(M, \mathcal{J}_{\nabla})$ forms an associative algebra with product defined by $X \cdot Y = \nabla_X Y$.

157
**Proof.** Let $X$ and $Y$ be two vector fields on $M$. Their product is defined by:

$$ (3.11) \qquad X \cdot Y = \nabla_X Y. $$

158
Let $U$ and $V$ be elements of $J_{\nabla}$. By a straightforward computation, we have:

$$ (3.12) \qquad \begin{aligned} X \cdot (Y \cdot (U \cdot V)) - (X \cdot Y) \cdot (U \cdot V) &= X \cdot ((Y \cdot U) \cdot V) - ((X \cdot Y) \cdot U) \cdot V \\ &= (X \cdot (Y \cdot U) - (X \cdot Y) \cdot U) \cdot V \\ &= 0. \end{aligned} $$

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Now, let us show that if $U$ and $V$ are elements of $J_{\nabla}$, then $\nabla_U V$ is also an element of $J_{\nabla}$.
By a simple calculation, we obtain:

$$ (3.13) \qquad \begin{aligned} \nabla_{X,Y}^2 \nabla_U V &= X \cdot (Y \cdot (U \cdot V)) - (X \cdot Y) \cdot (U \cdot V) \\ &= (X \cdot Y) \cdot (U \cdot V) - (X \cdot Y) \cdot (U \cdot V) \\ &= 0. \end{aligned} $$
162 **Remark 3.4.** The pair $(J_\nabla, \nabla)$ is an associative algebra together with a Lie algebra
$$(J_\nabla, [\cdot, \cdot]_\nabla), \text{ where the bracket is given by}$$

163

$$
[X, Y]_\nabla = \nabla_X Y - \nabla_Y X.
$$

164 This is in fact a immediate consequence of a pre-Lie algebra being a Lie algebra [42] with bracket:
165

166

$$
(3.14) \qquad [a, b] = a \cdot b - b \cdot a
$$

166 **Proposition 3.3** ([12]). Let (M, ∇) be a gauge structure such that ∇ is torsion-free.

167 Then $(J_\nabla, [\cdot, \cdot]_\nabla)$ is a finite-dimensional Lie subalgebra of the Lie algebra of vector fields

168

$$
(X(M), [-,-]).
$$

169 The following proposition shows that $J_\nabla$ can be trivial. Building on the previous result,
170 we now examine the behavior of $J_\nabla$ in the presence of additional geometric structures. In
171 particular, we focus on the case of a Riemannian manifold endowed with the Levi-Civita
172 connection $\nabla^{LC}$.

170

171

172

173 **Proposition 3.4.** Let $(M, g)$ be a Riemannian manifold with Ricci tensor $\mathrm{Ric}$. Then

$$
J_{\nabla LC} \subseteq \ker(\mathrm{Ric}).
$$

174 In particular, if (M, g) is Einstein, $J_{\nabla LC} = \{0\}$.

175 *Proof.* Let $(X, Y)$ be a pair of vector fields on $M$, and let $U \in J_{\nabla LC}$. We have

$$
R^{\nabla LC}(X, Y)U = (\nabla^2 U)(X, Y) - (\nabla^2 U)(Y, X).
$$

176 Taking the trace, we deduce that

$$
\mathrm{Ric}(X, U) = 0,
$$

177 so that

$$
J_{\nabla LC} \subset \ker(\mathrm{Ric}).
$$

178 If $\nabla^{LC}$ is the Levi-Civita connection of an Einstein Riemannian manifold, then obviously

$$
J_{\nabla LC} = \{0\}.
$$

179

□

180 **Proposition 3.5.** Let $(M, g)$ be a compact Riemannian manifold such that the Ricci curvature is negative, i.e.,
181

$$
\mathrm{Ric} < 0.
$$

182 Then

$$
J_{\nabla LC} = \{0\}.
$$

183 *Proof.* For any vector field $U \in X(M)$, the Lie derivative of the Levi-Civita connection
184 satisfies

$$
\mathcal{L}_U \nabla^{LC} = i_U R^{\nabla LC} + (\nabla^{LC})^2 U.
$$

185 By a direct computation, one obtains

$$
R^{\nabla LC}(X, Y)U = (\nabla^2 U)(X, Y) - (\nabla^2 U)(Y, X).
$$

186 If $U \in J_{\nabla LC}$, then

$$
\mathcal{L}_U \nabla^{LC} = i_U R^{\nabla LC} U.
$$
187 Using the Bianchi identity, we deduce

$$ (\nabla^2 U)(X, \cdot) = i_X R^{\nabla LC} U. $$

188 Consequently,

$$ \mathcal{L}_U \nabla^{LC} = 0. $$

189 Since $M$ is compact, it follows from [59] that

$$ \mathcal{L}_U g = 0, $$

190 that is, $U$ is a Killing vector field.

191 Finally, under the assumption $\mathrm{Ric} < 0$, the Bochner result [8] implies

$$ U = 0. $$

192 The result follows.

□

193 Having established the algebraic structure of $J_\nabla$, it is natural to investigate its integrability
194 properties and the corresponding global geometric realizations.

195 As an immediate consequence of Lie's third theorem [23], we have:

196 **Proposition 3.6.** Up to a Lie group isomorphism, there exists a unique simply connected
197 Lie group $G_\nabla$ whose Lie algebra is isomorphic to $J_\nabla$.

198 **Definition 3.5.** [49] The algebra $J_\nabla$ is said to be completely integrable if it is the linear
199 counterpart of a locally effective differentiable action

$$ G_{\nabla} \times M \ni (\gamma, x) \mapsto \gamma \cdot x \in M. $$

200 It is important to recall that, on a compact manifold $M$, every infinitesimal action of a finite-
201 dimensional Lie group is integrable. This follows from the fact that every vector field $X$ is
202 complete, in the sense that $X$ generates a one-parameter subgroup of the diffeomorphism
203 group.

204 **4. A CANONICAL REPRESENTATION OF $G_\nabla$**

205 **Definition 4.1.** Let $W$ be a finite-dimensional real vector space. Its group of affine isomorphisms, denoted by $\mathrm{Aff}(W)$, is defined as the semi-direct product

$$ \mathrm{Aff}(W) = \mathrm{GL}(W) \ltimes W, $$

207 where, by abuse of notation, $\mathrm{Aff}(W)$ also denotes the group of translations of $W$.

208 From the definition of a locally flat connection, $\nabla$ satisfies:

$$ \begin{aligned} (4.1) \qquad \nabla_X Y - \nabla_Y X &= [X, Y], \\ [\nabla_X, \nabla_Y] &= \nabla_{[X, Y]}. \end{aligned} $$

209 for all $X, Y \in \mathcal{X}(M)$. Consequently, there is a natural affine representation of the Lie
210 algebra $J_\nabla$ on itself (viewed as a vector space):

$$ J_{\nabla} \ni X \mapsto \rho(X) = (\nabla_X, X) \in \mathrm{gl}(J_{\nabla}) \times J_{\nabla} = \mathrm{aff}(J_{\nabla}), $$

211 with affine action given by

$$ \rho(X).Y = \nabla_X Y + X, \quad \forall Y \in J_{\nabla}. $$
212 By the universal property of simply connected finite-dimensional Lie groups, there exists a
213 unique continuous affine representation

$$
\gamma \in G_{\nabla} \mapsto (f_{\nabla}(\gamma), q_{\nabla}(\gamma)) \in \mathrm{Aff}(J_{\nabla})
$$

214 satisfying

$$
(4.2) \quad \left. \frac{d}{dt} \left( (f_{\nabla}(\exp(tX)), q_{\nabla}(\exp(tX))) \right) \right|_{t=0} = (\nabla X, X).
$$

215 **Proposition 4.1.** With the above notations, $f_\nabla$ is a linear representation of $G_\nabla$ on $J_\nabla$, and
216 $q_\nabla$ is a $J_\nabla$-valued 1-cocycle with respect to $f_\nabla$.

217 *Proof.* The pair $(f_\nabla, q_\nabla)$ defines a continuous homomorphism of the Lie group $G_\nabla$ into
218 the affine group $\mathrm{Aff}(J_\nabla)$. Hence, for any $\gamma_1, \gamma_2 \in G_\nabla$, we have

$$
(4.3) \quad (f_{\nabla}(\gamma_1)f_{\nabla}(\gamma_2), f_{\nabla}(\gamma_1)q_{\nabla}(\gamma_2) + q_{\nabla}(\gamma_1)) = (f_{\nabla}(\gamma_1\gamma_2), q_{\nabla}(\gamma_1\gamma_2)).
$$

219 The cohomology of the Lie group $G_\nabla$ with coefficients in its Lie algebra $J_\nabla$ is defined via
220 the standard complex [14, 19, 21]:

$$
(4.4) \quad \dots \to C^q(G_\nabla, J_\nabla) \xrightarrow{D} C^{q+1}(G_\nabla, J_\nabla) \xrightarrow{D} C^{q+2}(G_\nabla, J_\nabla) \to \dots,
$$

221 with coboundary operator $D$ given by

$$
D\theta(\gamma_1, \dots, \gamma_{q+1}) = f_\nabla(\gamma_1) \cdot \theta(\gamma_2, \dots, \gamma_{q+1})
$$

$$
(4.5) \qquad \begin{aligned}
&+ \sum_{i=1}^{q} (-1)^i \theta(\dots, \gamma_i \gamma_{i+1}, \dots) \\
&+ (-1)^{q+1} \theta(\gamma_1, \dots, \gamma_q).
\end{aligned}
$$

222 The condition

$$
(4.6) \quad q_{\nabla}(\gamma_1 \gamma_2) = f_{\nabla}(\gamma_1) q_{\nabla}(\gamma_2) + q_{\nabla}(\gamma_1), \quad \forall \gamma_1, \gamma_2 \in G_{\nabla},
$$

223 is equivalent to $q_\nabla \in Z^1(G_\nabla, J_\nabla)$. Therefore, the following definition is natural: $\square$

224 **Definition 4.2.** The cohomology class $[q_\nabla] \in H^1(G_\nabla, J_\nabla)$ is called the radiant class of the
225 affine representation $(f_\nabla, q_\nabla)$.

226 **Remark 4.3.** The radiance obstruction is defined in [27]. The authors consider representations of a group G to the group Aff(E) of affine transformations of a vector space E.
227 Such a representation ρ can be written as ρ_g = A_g + v_g, A ∈ Aff(E), v_g ∈ E, g ∈ G. The
228 translation part compose as v_gg' = A_g'v_g + v_g', hence is a 1-cocycle with values in the
229 G-module A_G, called the radiance obstruction. An affine representation has a fixed point
230 if and only if the radiance obstruction vanishes, a property that will be used below. This
231 yields to the notion of radiant affine manifold, that is an affine manifold whose transition
232 functions can be chosen in the linear group. The classification of radiant affine 3-manifolds
233 is addressed in [22].
234

235 We are now in a position to state one of the main results of this article.

236 **Theorem 4.4.** The following statements are equivalent:

237 (1) The affine action

$$
(\gamma, X) \in G_{\nabla} \ltimes J_{\nabla} \mapsto f_{\nabla}(\gamma) \cdot X + q_{\nabla}(\gamma) \in J_{\nabla}
$$

238 has a fixed point.
239

(2) The cohomology class $[q_\nabla]$ vanishes.

240

(3) The affine representation

$$ \gamma \in G_{\nabla} \mapsto (f_{\nabla}(\gamma), q_{\nabla}(\gamma)) \in \mathrm{Aff}(J_{\nabla}) $$

241

is conjugate to the linear representation

$$ \gamma \in G_{\nabla} \mapsto f_{\nabla}(\gamma) \in \mathrm{GL}(J_{\nabla}). $$

242

243

*Proof.* We first show that (1) implies (2). Let $-Y_0$ be a fixed point of the affine action $(f_{\nabla}, q_{\nabla})$, so that

$$ f_{\nabla}(\gamma)(-Y_0) + q_{\nabla}(\gamma) = -Y_0, \quad \forall \gamma \in G_{\nabla}. $$

244

It follows that

$$ q_{\nabla}(\gamma) = f_{\nabla}(\gamma)(Y_0) - Y_0, \quad \forall \gamma \in G_{\nabla}, $$

245

hence the cocycle $q_{\nabla}$ is exact.

246

247

Next, we prove that (2) implies (3), which was obtained [27]. Consider the affine isomorphism $(e, Y_0)$, which is the translation by $Y_0$:

$$ X \mapsto X + Y_0. $$

248

Then

$$ (e, Y_0) \cdot (f_{\nabla}(\gamma), q_{\nabla}(\gamma)) \cdot (e, Y_0)^{-1} = (f_{\nabla}(\gamma), 0_{\nabla}), $$

249

where $0_{\nabla}$ denotes the zero element of $J_{\nabla}$.

250

Finally, we show that (3) implies (1). This means that there exists an affine isomorphism

$$ J_{\nabla} \ni Y \mapsto L(Y) + X_0 \in J_{\nabla} $$

251

such that

$$ (L, X_0) \cdot (f_{\nabla}(\gamma), q_{\nabla}(\gamma)) \cdot (L, X_0)^{-1} = (f_{\nabla}(\gamma), 0_{\nabla}), \quad \forall \gamma \in G_{\nabla}. $$

252

Expanding the left-hand side gives the following identities:

$$ (4.7a) \qquad L \cdot f_{\nabla}(\gamma) \cdot L^{-1} = f_{\nabla}(\gamma), $$

$$ (4.7b) \qquad L(q_{\nabla}(\gamma)) + X_0 - [L \cdot f_{\nabla}(\gamma) \cdot L^{-1}](X_0) = 0_{\nabla}. $$

253

From (b), we obtain

$$ q_{\nabla}(\gamma) = f_{\nabla}(\gamma)(L^{-1}(X_0)) - L^{-1}(X_0), \quad \forall \gamma \in G_{\nabla}. $$

254

Using (a), this simplifies to

$$ q_{\nabla}(\gamma) = f_{\nabla}(\gamma)(X_0) - X_0, \quad \forall \gamma \in G_{\nabla}, $$

255

so that the vector $-X_0$ is a fixed point of the affine representation $(f_{\nabla}, q_{\nabla})$. □

256

257

**Definition 4.5.** The affine representation $(f_{\nabla}, q_{\nabla})$ is called the canonical affine representation of the gauge structure $(M, \nabla)$.

258

259

260

**Remark 4.6.** When the infinitesimal action $J_{\nabla}$ is integrable, the proposition above is a key tool to relate the canonical affine representation of $(M, \nabla)$ to the hyperbolicity problem for the orbits of $G_{\nabla}$.
## 5. APPLICATION TO STATISTICAL MANIFOLDS

### 5.1. Introduction to Statistical Manifolds.
In this section, certain relevant notations and useful definitions related to information geometry are recalled. Information geometry is an emerging field which provides a certain correspondence between differential geometry and statistics through the Fisher information matrix (cf. [2,3,38,51]). Several authors have examined the process of constructing a proper distance between probability distributions referring to its significance in practical and theoretical statistics, information theory, signal processing, quantitative finance, and machine learning. To solve this problem, Rao [50] proposed the geodesic distance induced by the Fisher information metric, known in the literature as the Fisher-Rao distance (or Rao's distance).

Let us consider a statistical model $\mathcal{P}_d(\Xi)$, which is a set of parametric densities defined as follows:

$$ \mathcal{P}_d(\Xi) = \{p_{\theta} : \theta \in \Theta\}, $$

where $\Theta$ (the parameter space) is an open subset of $\mathbb{R}^d$ and $\Xi$ is a sample space, with parameter $\theta = (\theta^1, \dots, \theta^d)$. Assuming that $p_{\theta}(x) > 0$ for all relevant $x$, the log-likelihood function is

$$ \ell_{\theta}(x) = \log p_{\theta}(x). $$

The Fisher information tensor is given by

$$ (5.1) \qquad g^F(\theta)_{ij} = \mathbb{E}_{\theta} \left[ \frac{\partial \ell_{\theta}(X)}{\partial \theta_i} \frac{\partial \ell_{\theta}(X)}{\partial \theta_j} \right] = - \mathbb{E}_{\theta} \left[ \frac{\partial^2 \ell_{\theta}(X)}{\partial \theta_i \partial \theta_j} \right], $$

for all $\partial/\partial\theta^i, \partial/\partial\theta^j \in T_{\theta}\Theta \simeq \mathbb{R}^d$. A direct computation gives the alternative representation

$$ (5.2) \qquad g^F(\theta)_{ij} = 4 \int_{\Xi} \frac{\partial \sqrt{p_{\theta}(x)}}{\partial \theta_i} \frac{\partial \sqrt{p_{\theta}(x)}}{\partial \theta_j} dx. $$

The obtained geometry is called the information geometry of the parametric family $\mathcal{P}_d(\Xi)$. The Fisher information matrix measures the amount of information present in the data about a parameter $\theta$. Rao [50] showed that, under mild regularity conditions, $\mathcal{P}_d(\Xi)$ (or $\Theta$) is a Riemannian manifold endowed with the Fisher-information metric (also called the Fisher-Rao metric). Cencov [20] proved a uniqueness result for the Fisher metric on finite sample spaces; this has been extended to infinite sample spaces (see [4]). Another extension of Cencov's result was provided by [17,41], who proved that the Fisher-Rao metric is the unique invariant metric tensor under Markov embeddings, up to a multiplicative constant. The Fisher metric is invariant under reparametrizations of the sample space $\Xi$ and covariant under reparametrizations of the parameter space $\Theta$ (see [16]). Hence the Fisher-Rao distance is designed to be invariant under diffeomorphisms of both $\Xi$ and $\Theta$.

Amari [1,3] and Chentsov [20] introduced a family of $\alpha$-connections $\nabla^{(\alpha)}$, $\alpha \in \mathbb{R}$, which are symmetric but generally non-metric on the statistical model $\mathcal{P}_d(\Xi)$, satisfying

$$ \nabla^{(\alpha)} g^F = \alpha T^{\alpha-C}, $$

where $T^{\alpha-C}$ denotes the Amari-Chentsov tensor. This is a totally symmetric 3-tensor defined by

$$ (5.3) \qquad T_{ijk}^{\alpha-C} = \mathbb{E}_{p_{\theta}}[\partial_i \log p_{\theta}(X) \partial_j \log p_{\theta}(X) \partial_k \log p_{\theta}(X)], \quad i, j, k = 1, \dots, d. $$

The Christoffel symbols of the $\alpha$-connection are commonly written as
$$ \Gamma_{ij,k}^{(\alpha)}(\theta) = \mathbb{E}_{p_{\theta}} \left[ \frac{\partial^2 \ell_{\theta}(X)}{\partial \theta_i \partial \theta_j} \frac{\partial \ell_{\theta}(X)}{\partial \theta_k} \right] + \frac{1-\alpha}{2} \mathbb{E}_{p_{\theta}} \left[ \frac{\partial \ell_{\theta}(X)}{\partial \theta_i} \frac{\partial \ell_{\theta}(X)}{\partial \theta_j} \frac{\partial \ell_{\theta}(X)}{\partial \theta_k} \right]. $$

295 For $\alpha = 0$ one recovers the Levi-Civita connection of the Fisher metric:

$$ \Gamma_{ij,k}^{\mathrm{lc}}(\theta) = \mathbb{E}_{p_{\theta}} \left[ \frac{\partial^2 \ell_{\theta}(X)}{\partial \theta_i \partial \theta_j} \frac{\partial \ell_{\theta}(X)}{\partial \theta_k} \right] + \frac{1}{2} \mathbb{E}_{p_{\theta}} \left[ \frac{\partial \ell_{\theta}(X)}{\partial \theta_i} \frac{\partial \ell_{\theta}(X)}{\partial \theta_j} \frac{\partial \ell_{\theta}(X)}{\partial \theta_k} \right]. $$

296 In a statistical model $\mathcal{P}_d(\Xi)$ we have the duality relation

$$ \partial_{\theta_k} g^F(\partial_{\theta_i}, \partial_{\theta_j}) = g^F(\nabla_{\partial_{\theta_k}}^{(\alpha)} \partial_{\theta_i}, \partial_{\theta_j}) + g^F(\partial_{\theta_i}, \nabla_{\partial_{\theta_k}}^{(-\alpha)} \partial_{\theta_j}), $$

297 where $g^F$ is the Fisher information metric. Lauritzen [38] was inspired by this construction and by the Amari-Chentsov tensor to formalize the notion of a statistical manifold.
298

299 **Definition 5.1.** A statistical manifold is a quadruple $(M, g, \nabla, \nabla^*)$ where:

$$ \begin{array}{l} \text{300} \quad (1) \quad \text{For all } X, Y, Z \in \mathcal{X}(M), \\ \phantom{\text{300} \quad (1) \quad \text{}} X(g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X^* Z). \end{array} $$

301 (2) Both $\nabla$ and $\nabla^*$ are torsion-free affine connections.

302 Equivalently, a statistical manifold may be defined as a Riemannian manifold $(M, g)$
303 equipped with a totally symmetric 3-tensor $B$ (often $B = \nabla g$).

304 For a statistical manifold $(M, g, \nabla, \nabla^*)$, the triple $(g, \nabla, \nabla^*)$ is called a statistical structure, and the pair $(\nabla, \nabla^*)$ are referred to as statistical connections, and the Levi-Civita connection $\nabla^{\mathrm{lc}}$ satisfies

$$ 2 \nabla^{\mathrm{lc}} = \nabla + \nabla^*. $$

307 Alternatively, one can construct dual connections $(\nabla^D, \nabla^{*D})$ from a smooth function $D(\theta_1, \theta_2)$ (usually called a divergence or contrast function) which is $C^3$ and satisfies suitable conditions (see [24, 25, 44]):

$$ \begin{array}{l} \text{308} \quad \Gamma_{ijk}^{D}(\theta) = - \partial_{\theta_1}^{i} \partial_{\theta_1}^{j} \partial_{\theta_2}^{k} D(\theta_1, \theta_2) \Big|_{\theta_1=\theta_2=\theta}, \\ \text{309} \quad \Gamma_{ijk}^{*D}(\theta) = - \partial_{\theta_2}^{i} \partial_{\theta_2}^{j} \partial_{\theta_1}^{k} D(\theta_1, \theta_2) \Big|_{\theta_1=\theta_2=\theta}. \end{array} $$

310 where $\Gamma_{ijk}^D(\theta)$ and $\Gamma_{ijk}^{*D}(\theta)$ are the Christoffel symbols of $\nabla^D$ and $\nabla^{*D}$ respectively.

311 **Example 1:** Any Riemannian manifold $(M, g)$ with $\nabla^{\mathrm{lc}}$ the Levi-Civita connection is a statistical manifold (trivial case), since $\nabla^{\mathrm{lc}}$ is self-dual.

313 **Example 2:** Given a statistical manifold $(M, g, \nabla, \nabla^*)$, the family $(M, g, \nabla^{(\alpha)}, \nabla^{(-\alpha)})_{\alpha \in \mathbb{R}}$ with $\nabla^{(\alpha)} = \frac{1+\alpha}{2} \nabla + \frac{1-\alpha}{2} \nabla^*$ is again a statistical manifold for every $\alpha \in \mathbb{R}$.

315 **Example 3:** Let $(M, g)$ be a Riemannian manifold with Levi-Civita connection $\nabla^{\mathrm{lc}}$, and let $\xi \in TM \setminus \{0\}$. For all $X, Y \in \mathcal{X}(M)$, define

$$ \begin{aligned} \nabla_X Y &= \nabla_X^{\mathrm{lc}} Y + g(X, \xi) g(Y, \xi) \xi, \\ \nabla_X^* Y &= \nabla_X^{\mathrm{lc}} Y - g(X, \xi) g(Y, \xi) \xi. \end{aligned} $$
317 Then $(M, g, \nabla, \nabla^*)$ is a statistical manifold.

318 A statistical manifold $(M, g, \nabla, \nabla^*)$ naturally arises from a statistical model, that is, a
319 parameterized family of probability distributions {$p_{\theta} | \theta \in \Theta$}. In this framework, the
320 Riemannian metric $g$ is provided by the Fisher information metric:

$$
g_{ij} = \mathbb{E}[\partial_{\theta_i} l_{\theta} \partial_{\theta_j} l_{\theta}], \quad l_{\theta} = \log p_{\theta}.
$$

321 Motivated by this construction, Lauritzen and Amari raised the following question: Can
322 every statistical manifold be realized as a finite-dimensional statistical model?

323 This question received a positive answer from Hong Van Le [40], who demonstrated, using
324 Nash’s embedding theorem [45, 46] for the statistical metric and Gromov’s immersion
325 theorem [29] for the Amari–Chentsov tensor, that such a realization is indeed possible.
326 More precisely, Hong Van Le [40] showed that there exists an isostatistical embedding of
327 any given compact statistical manifold into the space of probability measures over a finite
328 set.

329 **Theorem 5.2** ( [40]). Every smooth compact statistical manifold $(M^n, g, T)$ of finite dimension admits, for a sufficiently large integer $N$, an isostatistical embedding (i.e., an embedding that preserves both the tensor $T$ and the metric $g$) into $\text{Cap}_+^N$, the space of all positive probability distributions defined on a discrete sample space $\Omega^N$ consisting of $N$-elementary events, where

$$
\mathrm{Cap}_+^N := \left\{ (p_1, \dots, p_n) \mid \begin{array}{l} p_i > 0 \text{ for } i = 1, \dots, n \text{ and } \displaystyle\sum_{i=1}^n p_i = 1 \end{array} \right\}.
$$

334 This result shows that compact statistical manifolds of finite dimension can indeed be
335 realized as statistical models, although it does not provide an explicit form for such an
336 embedding.

337 In our approach, which is based on global analysis, we explicitly construct a statistical model
338 from a canonical foliation defined on the statistical manifold under consideration.

339
340
341
342
343

**5.2. Hessian manifolds.** Hessian manifolds exhibit remarkable topology and geometry
and are closely related to Kähler manifolds. Their influence encompasses both the theoreti-
cal foundations of information geometry, Kähler geometry, and complex analysis, as well as
numerous practical applications in optimization, mathematical statistics, and quantitative
finance.

344 **Definition 5.3.** A Riemannian metric $g$ on a locally flat manifold $(M, \nabla)$ is called a Hessian metric if $g$ can be locally expressed as

$$
(5.5) \qquad g = \nabla^2 h,
$$

that is:

$$
(5.6) \qquad g_{ij} = \frac{\partial^2 h}{\partial x_i \partial x_j},
$$

347 where $(x_1, \dots, x_n)$ is a system of affine coordinates with respect to $\nabla$.

348 A locally flat manifold $(M, \nabla)$ equipped with a Hessian metric is termed a Hessian manifold.

349
350

**Definition 5.4.** A foliation $\mathcal{F}$ of a statistical manifold $(M, g, \nabla, \nabla^*)$ is said to be Hessian if the restriction of the statistical structure to its leaves defines Hessian manifolds.
### 5.3. Canonical Hessian Foliations in Statistical Manifolds.
Let $(M, g, \nabla, \nabla^*)$ be a compact statistical manifold, and let $\nabla^{LC}$ denote its Levi-Civita connection. We consider three gauge structures $(M, \nabla)$, $(M, \nabla^*)$, and $(M, \nabla^{LC})$. Since $M$ is compact, the infinitesimal algebras $J_\nabla$, $J_{\nabla^*}$, and $J_{\nabla^{LC}}$ correspond to the following locally effective differentiable dynamical systems:

$$ (5.7) \qquad \begin{aligned} G_{\nabla} \times M &\to M, \\ G_{\nabla^*} \times M &\to M, \\ G_{\nabla^{LC}} \times M &\to M. \end{aligned} $$

**Remark 5.5.** From the Riemannian geometry viewpoint, the orbits of $G_{\nabla^{LC}}$ are flat Riemannian manifolds. Up to isometry and finite covering, each $n$-dimensional orbit is a flat cylinder over a Euclidean torus [58], that is,

$$ \left( \frac{\mathbb{T}^k}{\Gamma} \times \mathbb{R}^{n-k}, g_0 \right), $$

where $g_0$ is the flat metric induced by the ambient Euclidean metric of $\mathbb{R}^n$, $k$ is the first Betti number of the orbit, and $\Gamma$ is a finite group of isometries.

Let $\mathcal{F}_\nabla$ (resp. $\mathcal{F}_{\nabla^*}$) denote the foliation defined by the orbits of the Lie group $G_\nabla$ (resp. $G_{\nabla^*}$), and let $g_\nabla$ (resp. $g_{\nabla^*}$) be the restriction of $g$ to $\mathcal{F}_\nabla$ (resp. $\mathcal{F}_{\nabla^*}$).

**Theorem 5.6.** Let $(M, g, \nabla, \nabla^*)$ be a compact statistical manifold. Then the foliations $\mathcal{F}_\nabla$ and $\mathcal{F}_{\nabla^*}$ are singular Hessian foliations in $(M, g, \nabla, \nabla^*)$.

*Proof.* Since $M$ is assumed to be compact, it follows that $\mathcal{F}_\nabla$ defines a singular foliation on $M$. Clearly, the leaves of $\mathcal{F}_\nabla$ are statistical sub-manifolds obtained by restricting the statistical structure on $M$. Moreover, since $R^\nabla(\cdot, \cdot)Z = 0$ for any $Z \in J_\nabla$, it follows that $\mathcal{F}_\nabla$ is a Hessian foliation on $(M, g, \nabla, \nabla^*)$. Applying the same reasoning to $\mathcal{F}_{\nabla^*}$, the theorem follows. $\square$

**Remark 5.7.** Let $S_\nabla$ be a leaf of $\mathcal{F}_\nabla$, and denote by $g_\nabla$ the restriction of $g$ to $S_\nabla$. By the previous theorem, $(S_\nabla, g_\nabla, \nabla, \nabla^*)$ is a Hessian manifold. We can therefore associate two KV-algebras, defined by

$$ \mathcal{A}_{\nabla} = (\mathcal{X}(S_{\nabla}), \nabla), \quad \mathcal{A}_{\nabla^*} = (\mathcal{X}(S_{\nabla}), \nabla^*). $$

By a direct computation, one obtains

$$ \delta g_{\nabla}(X, Y, Z) = g_{\nabla}(T^{\nabla}(X, Y), Z), $$

and

$$ \delta g_{\nabla}(X, Y, Z) = g_{\nabla}(T^{\nabla^*}(X, Y), Z), $$

for all vector fields $X, Y, Z \in \mathcal{X}(S_\nabla)$. From these identities, we deduce that the metric $g_\nabla$ defines two cohomology classes, which will be referred to as Hessian classes in the following text:

$$ [g_\nabla] \in H_{KV}^2(\mathcal{A}_\nabla, C^\infty(M)), \quad [g_{\nabla^*}] \in H_{KV}^2(\mathcal{A}_{\nabla^*}, C^\infty(M)). $$

**Theorem 5.8.** In a compact statistical manifold $(M, g, \nabla, \nabla^*)$, the following assertions are equivalent:

$$ (1) \quad [g_\nabla] = 0 \in H_{KV}^2(\mathcal{A}_\nabla, C^\infty(M)), $$

$$ (2) \quad [q_{\nabla^*}] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*}), $$
382 (3) The pair $(f_{\nabla^*}, q_{\nabla^*})$ has a fixed point,

383 (4) The pair $(f_{\nabla^*}, q_{\nabla^*})$ is affinely conjugate to its linear component $f_{\nabla^*}$.

384 *Proof.* According to Theorem 4.4, assertions (2), (3) and (4) are equivalent. Therefore, it is sufficient to prove that assertions (1) and (2) are equivalent.
Let us demonstrate first that (1) implies (2)

386 Since the class $[g_{\nabla}] \in H_{KV}^2(\mathcal{A}_{\nabla}, C^{\infty}(M))$ vanishes, there exists a de Rham closed differential 1-form $\theta$ such that:

$$g_{\nabla}(X, Y) = X\theta(Y) - \theta(\nabla_X Y), \quad \forall (X, Y).$$

388 By the defining property of statistical manifolds, it comes:

$$Xg_{\nabla}(Y, Z) = g_{\nabla}(\nabla_X^*Y, Z) + g_{\nabla}(Y, \nabla_X Z), \quad \forall (X, Y, Z).$$

389 Let $H$ be the unique vector field satisfying:

$$\theta(X) = g_{\nabla}(H, X), \quad \forall X.$$

390 Using once again the defining property of statistical manifolds, we get:

$$Xg_{\nabla}(H, Y) - g_{\nabla}(\nabla_X^*H, Y) - g_{\nabla}(H, \nabla_X Y) = 0.$$

391 Since left hand member is $C^{\infty}(M)$-multilinear we can assume that $H \in J_{\nabla}$ and we get the following identity

$$g_{\nabla}(\nabla_X^*H, Y) = Xg_{\nabla}(H, Y) - g_{\nabla}(H, \nabla_X Y) = g_{\nabla}(X, Y), \quad \forall (X, Y) \subset \mathcal{X}(M).$$

393 Thus one has:

$$\nabla_X^*(H) - X = 0_{\nabla}, \quad \forall X \in \mathcal{X}(M).$$

394 So the $H$ is a fixed point of $(f_{\nabla^*}, q_{\nabla^*})$.
Let us demonstrate now that (2) implies (1).

395 Let us assume that $(f_{\nabla^*}, q_{\nabla^*})$ has a fixed point $Y_0 \in J_{\nabla^*}$. Then:

$$f_{\nabla^*}(\gamma)(Y_0) + q_{\nabla^*}(\gamma) = Y_0, \quad \forall \gamma \in G_{\nabla^*}.$$

396 To every $X \in J_{\nabla^*}$, we assign the one parameter subgroup

$$\{\mathrm{Exp}(tX), t \in \mathbb{R}\} \subset G_{\nabla^*}.$$

397 We have:

$$f_{\nabla^*}(\mathrm{Exp}(tX))(Y_0) + q_{\nabla^*}(\mathrm{Exp}(tX)) = Y_0, \quad \forall t \in \mathbb{R}.$$

398 Calculating the derivative at $t=0$, one obtains:

$$\nabla_X^* Y_0 + X = 0.$$

399 Finally:

$$Xg_{\nabla}(Y_0, Y) = g_{\nabla}(\nabla_X^*Y_0, Y) + g_{\nabla}(Y_0, \nabla_X Y)$$

400 Using $\nabla_X^* Y_0 = -X, \forall X$, one obtains the following identity:

$$Xg_{\nabla}(Y_0, Y) = -g_{\nabla}(X, Y) + g_{\nabla}(Y_0, \nabla_X Y).$$

401 By putting $\theta(Y) = -g_{\nabla}(Y_0, Y)$, it comes:

$$g_{\nabla}(X, Y) = X\theta(Y) - \theta(\nabla_X Y), \quad \forall (X, Y)$$

402 Then one has $[g_{\nabla}] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla}, C^{\infty}(M))$, concluding the proof. $\square$
**Corollary 5.1.** Let $(M, g, \nabla, \nabla^*)$ be a compact statistical manifold such that one of the assertions of Theorem 5.8 is satisfied. Then any compact leaf $(S_\nabla, \nabla)$ is a hyperbolic manifold.

The nullity of the Hessian class of a Hessian metric is equivalent to that of the radiant class of the dual affine representation $(f_{\nabla^*}, q_{\nabla^*})$; thus, the hyperbolicity of a locally flat manifold can be expressed as the nullity of the radiant class of the affine dynamics of the dual connection.

**Theorem 5.9.** For a compact Hessian manifold $(M, g, \nabla, \nabla^*)$, the following statements are equivalent:

(1) $(M, \nabla)$ is hyperbolic;

(2) $[g] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla}, C^{\infty}(M))$;

(3) $[q_{\nabla^*}] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*})$;

(4) The pair $(f_{\nabla^*}, q_{\nabla^*})$ has a fixed point;

(5) The pair $(f_{\nabla^*}, q_{\nabla^*})$ is affinely conjugate to its linear component $f_{\nabla^*}$.

**Proof.** This follows directly by combining Theorem 5.8 with Koszul's Theorem 2.11. □

Koszul showed that the hyperbolicity of $(M, \nabla)$ is equivalent to the existence of a closed 1-form $\omega$ whose covariant differential $\nabla\omega$ is positive definite, i.e., a Riemannian metric. This Riemannian metric

$$ (5.8) \qquad g = \nabla\omega $$

is a Hessian metric. Consequently, the hyperbolic connection $\nabla$ admits a locally flat dual connection, denoted $\nabla^*$.

Hence, in what follows, a hyperbolic manifold $(M, \nabla)$ can be understood as a Hessian manifold $(M, g, \nabla, \nabla^*)$ where $\nabla$ is a hyperbolic connection.

## 6. WHEN THE ORBITS ARE QUOTIENTS OF CONVEX CONES

Convex cones are fundamental examples of bounded domains. The study of convex cones has long been, and continues to be, a topic of great interest in both geometry and analysis. The pioneering works were carried out by Élie Cartan, but there is a wealth of subsequent research; see, for instance, [18], [34], [57], among many others. Nowadays, the analysis of convex cones plays an important role in information geometry [5]. In [55], Jacques Vey established a geometric condition under which a convex domain in $\mathbb{R}^n$ is necessarily a cone. His theorem can be stated as follows:

**Theorem 6.1.** Let $\Omega \subset \mathbb{R}^n$ be a convex domain containing no lines. If there exists a properly discontinuous group $D$ of affine transformations acting on $\Omega$ such that the quotient $\Omega/D$ is compact, then $\Omega$ is a cone.

It is noteworthy that Koszul obtained a related result in [35], showing that if the group of affine automorphisms of $\Omega$ acts transitively on $\Omega$, then $\Omega$ is also a cone.
Theorem 6.2. Let $(M, \nabla)$ be a compact hyperbolic manifold. If the universal coverings of $M$ are isomorphic to a cone. Then the cohomology classes

$$ (6.1) \qquad [q_{\nabla^*}] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*}) \quad \text{and} \quad [q_{\nabla}] = 0 \in H^1(G_{\nabla}, J_{\nabla}). $$

Proof. By assumption, the universal covering space $\tilde{M}$ is diffeomorphic to a convex cone in $\mathbb{R}^n$ that contains no straight line. Let us denote this cone by $\Omega$. Define an affine map $q: \Omega \to M$ by

$$ (6.2) \qquad q = \pi \circ D^{-1}, $$

where $\pi: \tilde{M} \to M$ is the covering projection and $D: \tilde{M} \to \Omega$ is the developing map.

Let $H' \in \mathcal{X}(\mathbb{R}^n)$ be the vector field defined by

$$ (6.3) \qquad H'(x) = \sum_{i=1}^{n} x^i \frac{\partial}{\partial x^i}, $$

that is, the homothetic vector field (i.e. the infinitesimal generator of the one-parameter group of homotheties with center 0 and ratio $e^t$). Then, for any $X' \in \mathcal{X}(\mathbb{R}^n)$, we have

$$ (6.4) \qquad \nabla_{X'}^0 H' = X', $$

where $\nabla^0$ denotes the canonical flat connection on $\mathbb{R}^n$.

The vector field $H'$ is $GL(\mathbb{R}^n)$-invariant. Since $\Omega$ containing no lines, every affine automorphism of $\Omega$ is the restriction of an element of $GL(\mathbb{R}^n)$. Hence, the restriction of $H'$ to $\Omega$ is invariant under the affine automorphisms of $\Omega$.

Therefore, there exists a unique vector field $H$ on $M$ such that $dq(H) = H'$. Because $q$ is locally an isomorphism of locally flat manifolds (that is, $dq(\nabla^0) = \nabla$), we have

$$ (6.5) \qquad \nabla_X H = \nabla_{X'}^0 H' = X. $$

Consequently,

$$ (6.6) \qquad [q_{\nabla}] = 0 \in H^1(G_{\nabla}, J_{\nabla}). $$

By assumption, $(M, \nabla)$ is a hyperbolic manifold. Then, by Theorem 5.9, we deduce that

$$ (6.7) \qquad [q_{\nabla^*}] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*}). $$

□

Corollary 6.1. Let $(M, g, \nabla, \nabla^*)$ be a compact hyperbolic manifold. The following cohomology classes vanish:

$$ (1) \qquad [g] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla}, C^{\infty}(M)), $$

$$ (2) \qquad [q] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*}), $$

$$ (3) \qquad [q_{\nabla}] = 0 \in H^1(G_{\nabla}, J_{\nabla}), $$

$$ (4) \qquad [g] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla^*}, C^{\infty}(M)). $$

Proof. By definition of hyperbolicity in the sense of Koszul (Theorem 2.11), we immediately have

$$ (6.8) \qquad [g] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla}, C^{\infty}(M)). $$

According to Theorem 5.9, the first statement implies the second.
We now show that (2) implies (3). Suppose that

$$ (6.9) \qquad [q_{\nabla^*}] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*}). $$

Then there exists a vector field $H$ such that

$$ (6.10) \qquad \nabla_X^*(H) = X, \quad \forall X \in \mathcal{X}(M). $$

Hence, $(M, \nabla^*, H)$ is a radiant manifold. By Goldman [28], the image of the universal cover $\tilde{M}$ under the development map is an open cone in $\mathbb{R}''$. Therefore, by Theorem 6.2, we obtain

$$ (6.11) \qquad [q_{\nabla}] = 0 \in H^1(G_{\nabla}, J_{\nabla}). $$

Finally, we show that (3) implies (4). Assume (3) holds. Then there exists a vector field $Y_0$ such that

$$ (6.12) \qquad \nabla_X Y_0 = X, \quad \forall X. $$

Using the statistical property, we have

$$ (6.13) \qquad X g(Y_0, Y) = g(\nabla_X Y_0, Y) + g(Y_0, \nabla_X^* Y). $$

Substituting $\nabla_X Y_0 = X$, it follows that

$$ (6.14) \qquad X g(Y_0, Y) = g(X, Y) + g(Y_0, \nabla_X^* Y). $$

Defining

$$ (6.15) \qquad \theta(Y) = g(Y_0, Y), $$

we obtain

$$ (6.16) \qquad g(X, Y) = X\theta(Y) - \theta(\nabla_X^* Y), \quad \forall X, Y, $$

which implies

$$ (6.17) \qquad [g] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla^*}, C^\infty(M)). $$

This completes the proof. □

**Corollary 6.2.** Let $(M, g, \nabla, \nabla^*)$ be a compact Hessian manifold. The following conditions are equivalent:

(1) $(M, \nabla)$ is a hyperbolic manifold;

(2) $(M, \nabla^*)$ is a hyperbolic manifold.

*Proof.* (1) implies (2) follows from the previous corollary. Now let us check (2) implies (1). By assumption $(M, \nabla^*)$ is hyperbolic manifold, then

$$ (6.18) \qquad [g] = 0 \in H_{KV}^2(\mathcal{A}_{\nabla^*}, C^\infty(M)). $$

Then there exists a 1-form $\theta$ on $M$ such that, for all vector fields $X, Y$,

$$ (6.19) \qquad g(X, Y) = X\theta(Y) - \theta(\nabla_X^* Y). $$

Define a vector field $Y_0$ by

$$ (6.20) \qquad \theta(Y) = g(Y_0, Y), \quad \forall Y. $$
486 Substituting into the previous relation and using the statistical property of $(g, \nabla, \nabla^*)$, we get

487

$$ (6.21) \qquad g(X, Y) = g(\nabla_X Y_0, Y) + g(Y_0, \nabla_X^* Y), \quad \forall X, Y. $$

488

Comparing terms, this implies

$$ (6.22) \qquad \nabla_X Y_0 = X, \quad \forall X. $$

489

490

491

Hence, $Y_0$ is a radiant vector field for $(M, \nabla)$. By Goldman [28], the image of the universal cover $\tilde{M}$ under the development map is an open cone in $\mathbb{R}^n$. Denote this cone by $\Omega$. It is well known [52, 57] that there exists a real-analytic, closed 1-form

$$ (6.23) \qquad \alpha = d \log \psi $$

492

493

494

495

on $\Omega$, where $\psi$ is the Koszul–Vinberg characteristic function of the cone. This 1-form, called the Koszul 1-form, is invariant under the affine automorphisms of $\Omega$, and its covariant derivative $\nabla\alpha$ defines an affine-invariant Riemannian metric on $\Omega$. Since $M$ is compact, then from Theorem 2.11, it follows that $(M, \nabla)$ is a hyperbolic manifold. $\square$

496

497

498

499

**Theorem 6.3.** Let $(M, g, \nabla, \nabla^*)$ be a compact statistical manifold, and let $S_\nabla$ be a compact leaf of the Hessian foliation $\mathcal{F}_\nabla$. If one of the equivalent conditions of Theorem 5.8 holds, then $S_\nabla$ is diffeomorphic to the quotient of a sharp open convex cone by a discrete Lie subgroup of $GL(\mathbb{R}^n)$.

500

*Proof.* The action

$$ (6.24) \qquad G_\nabla \times S_\nabla \to S_\nabla $$

501

induces an action

$$ (6.25) \qquad G_\nabla \times \tilde{S}_\nabla \to \tilde{S}_\nabla. $$

502

503

504

505

506

507

508

The manifold $(S_\nabla, \nabla)$ is homogeneous under the action of the Lie group $G_\nabla$. Therefore, both $\pi_1(S_\nabla)$ and $G_\nabla$ can be viewed as subgroups of the affine automorphism group $\text{Aff}(\tilde{S}_\nabla)$. It is straightforward to verify that $G_\nabla$ is contained in the normalizer of $\pi_1(S_\nabla)$. Since $S_\nabla$ is transitively acted upon by $G_\nabla$, every orbit of $G_\nabla$ in $\tilde{S}_\nabla$ is an open sub-manifold of $\tilde{S}_\nabla$. Because $\tilde{S}_\nabla$ is connected, the action of $G_\nabla$ on $\tilde{S}_\nabla$ is transitive. Hence, by [35], we conclude that $S_\nabla$ is diffeomorphic to the quotient of a sharp open convex cone by a discrete subgroup of $GL(\mathbb{R}^n)$. $\square$

509

510

**Corollary 6.3.** Let $(M, g, \nabla, \nabla^*)$ be a compact statistical manifold. Assume that a compact leaf $S_\nabla$ of the foliation $\mathcal{F}_\nabla$ is hyperbolic. Then the following cohomology classes vanish:

511

$$ (1) \quad [g_\nabla] = 0 \in H_{KV}^2(\mathcal{A}_\nabla, C^\infty(M)), $$

512

$$ (2) \quad [q_{\nabla^*}] = 0 \in H^1(G_{\nabla^*}, J_{\nabla^*}), $$

513

$$ (3) \quad [q_\nabla] = 0 \in H^1(G_\nabla, J_\nabla), $$

514

$$ (4) \quad [g_\nabla] = 0 \in H_{KV}^2(\mathcal{A}_\nabla, C^\infty(M)). $$

515

*Proof.* This result follows directly from Corollary 6.1. $\square$
## 7. STATISTICAL MODELS ON THE LEAVES OF $\mathcal{F}_V$

### 7.1. Characteristic Function and Barbaresco Formalism.
The locally flat structure of $(M, \nabla)$ lifts to its universal covering $\tilde{M}$ as a locally flat structure $(\tilde{M}, \tilde{\nabla})$, for which the projection

is an affine map. The locally flat structure on $\tilde{M}$ is defined by a local affine diffeomorphism

$\mathcal{D} : (\tilde{M}, \tilde{\nabla}) \to (\mathbb{R}^n, \nabla^0)$,
where $\nabla^0$ is the standard flat connection defined by

$$ \nabla^0 \frac{\partial}{\partial x_i} = 0, \quad \forall i = 1, 2, \dots, n. $$

The development map induces a representation $H(\nabla)$ of the fundamental group $\pi_1(M)$ of $M$ in $\text{Aff}(\mathbb{R}^n)$, called the holonomy representation of $M$, defined by the following commutative diagram:

$$ \begin{tikzcd} (\tilde{M}, \tilde{\nabla}) \arrow[r, "\pi"] & (M, \nabla) \arrow[d, "\mathcal{D}"] \arrow[d, "\mathcal{H}(\nabla)"'] \\ (\mathbb{R}^n, \nabla^0) \arrow[r, "\mathcal{H}(\nabla)"'] & (\mathbb{R}^n, \nabla^0) \arrow[d, "\mathcal{D}"] \arrow[d, "\mathcal{D}"] \end{tikzcd} $$

Let $\Omega \subset \mathbb{R}^n$ be a sharp open convex cone, and let $\bar{\Omega}$ denote its closure. The dual cone of $\Omega$ is defined by

$$ (7.2) \qquad \Omega^* = \{ \Psi \in (\mathbb{R}^n)^* \mid \langle \Psi, x \rangle > 0, \forall x \in \bar{\Omega} \setminus \{0\} \}. $$

The holonomy $H(\nabla)$ acts on $\Omega^*$ via the dual representation:

$$ (7.3) \qquad H(\nabla) : \Omega^* \ni \xi \mapsto \tilde{\gamma} \cdot \xi = \xi \circ \gamma^{-1} \in \Omega^*, \quad \gamma \in H(\nabla). $$

J.-L. Koszul and E. B. Vinberg introduced the following integral associated with a sharp convex cone $\Omega$:

$$ (7.4) \qquad \rho_{\Omega}(\theta) = \int_{\Omega^*} e^{-\langle \theta, \xi \rangle} d\xi, \quad \forall \theta \in \Omega, $$

where $d\xi$ is the Lebesgue measure on $\mathbb{R}^n$. The function $\rho_{\Omega}$ is analytic on $\Omega$, strictly positive $\rho_{\Omega}(\theta) \in (0, +\infty)$, and is called the *Koszul-Vinberg characteristic function* of the cone $\Omega$. It satisfies the following properties [55]:

534 (1) **Logarithmic Convexity:** $\rho_{\Omega}$ is logarithmically strictly convex, and $\Phi_{\Omega}(\theta) = \log \rho_{\Omega}(\theta)$ is strictly convex.

536 (2) **Holonomy Invariance:** For $\gamma \in H(\nabla)$,

$$ \rho_{\Omega}(\gamma\theta) = \det(\gamma)^{-1} \rho_{\Omega}(\theta). $$

537 (3) **Koszul 1-form:** The differential 1-form

$$ \alpha = d \log \rho_{\Omega} = \frac{d\rho_{\Omega}}{\rho_{\Omega}} $$
is invariant under $H(\nabla)$. For $\theta \in \Omega$ and $u \in \mathbb{R}^n$,

$$ \langle \alpha_{\theta}, u \rangle = - \int_{\Omega^{*}} \langle \xi, u \rangle e^{-\langle \xi, \theta \rangle} d\xi, \quad \text{and} \quad \alpha_{\theta} \in -\Omega^{*}. $$

(4) **Koszul 2-form:** The symmetric differential 2-form

$$ \beta = \nabla \alpha = d^2 \log \rho_{\Omega} $$

is a positive definite symmetric bilinear form on $\mathbb{R}^n$, invariant under $H(\nabla)$.

(5) **Koszul-Vinberg Metric:** The Riemannian metric

$$ g = d^2 \log \rho_{\Omega} $$

is invariant under $H(\nabla)$.

In [7], [6], Frédéric Barbaresco introduced a probability density called the Koszul density, defined as follows. For $\theta \in \Omega$ and $\xi \in \Omega^*$,

$$ (7.5) \qquad P(\theta, \xi) = \frac{e^{-\langle \theta, \xi \rangle}}{\rho_{\Omega}(\theta)} = \exp \{ -\langle \theta, \xi \rangle - \log \rho_{\Omega}(\theta) \}. $$

Then the family

$$ \{P(\theta, \xi) | \xi \in \Omega^*\} $$

forms an exponential family of probability distributions on $\Omega^*$ parameterized by $\theta \in \Omega$.

7.2. **Construction of a Probability Density on the Leaves of $\mathcal{F}_{\nabla}$**. Our goal is to endow the leaves of the foliation $\mathcal{F}_{\nabla}$ with a statistical model structure when the holonomy group $H(\nabla) \subset GL(n, \mathbb{R})$ is amenable. This means that there exists a mean

$$ \mu \in \text{Hom}(L^{\infty}(H(\nabla)), \mathbb{R}) $$

such that

$$ (7.6) \qquad \mu(f) = \int_{H(\nabla)} f(\gamma) d\mu(\gamma), $$

and satisfying the following properties:

(1) If $f > 0$, then $\mu(f) > 0$.

(2) $\mu(f \circ \gamma) = \mu(f)$ for all $f$ and all $\gamma \in H(\nabla)$.

(3) $\mu(1) = 1$.

Let us define, for $\theta \in \Omega$ and $\xi \in \Omega^*$, the function

$$ (7.7) \qquad f_{\theta, \xi} : \gamma \in H(\nabla) \mapsto P(\theta\gamma, \xi) = P(\theta, \gamma'\xi) \in \mathbb{R}. $$

Since $P(\theta\gamma, \xi) \le 1$, we have

$$ (7.8) \qquad f_{\theta, \xi} \in L^{\infty}(H(\nabla)). $$

**Proposition 7.1.** The function

$$ (7.9) \qquad Q(\theta, \xi) = \mu(P_{\theta, \xi}) = \int_{H(\nabla)} P(\theta\gamma, \xi) d\mu(\gamma) $$

is $H(\nabla)$-invariant, i.e.

$$ (7.10) \qquad Q(\gamma\theta, \xi) = Q(\theta, \xi). $$
560 **Proposition 7.2.** Let

$$Q : \frac{\Omega}{H(\nabla)} \times \Omega^* \to \mathbb{R}.$$

561 Then

$$ (7.11) \qquad \int_{\Omega^*} Q(\theta, \xi) \, d\xi = 1. $$

562 **Proof.** We have

$$ (7.12) \qquad \int_{\Omega^*} Q(\theta, \xi) \, d\xi = \int_{\Omega^*} \left[ \int_{H(\nabla)} P(\theta\gamma, \xi) \, d\mu(\gamma) \right] d\xi. $$

563 By the Fubini–Tonelli theorem,

$$ (7.13) \qquad \int_{\Omega^*} Q(\theta, \xi) \, d\xi = \int_{\Omega^* \times H(\nabla)} P(\theta\gamma, \xi) \, d(\mu \otimes \xi) $$

$$ (7.14) \qquad = \int_{H(\nabla)} \left[ \int_{\Omega^*} P(\theta\gamma, \xi) \, d\xi \right] d\mu(\gamma). $$

564 Since for all $\gamma \in H(\nabla)$ we have $\int_{\Omega^*} P(\theta\gamma, \xi) \, d\xi = 1$, it follows that

$$ (7.15) \qquad \int_{\Omega^*} Q(\theta, \xi) \, d\xi = \int_{H(\nabla)} 1 \, d\mu(\gamma) = 1. $$

565

□

566 **Theorem 7.1.** Let $(M, g, \nabla)$ be a compact statistical manifold, and let $\mathcal{F}_\nabla$ denote its canonical Hessian foliation. Let $S_\nabla$ be a compact leaf of $\mathcal{F}_\nabla$. If one of the equivalent conditions of Theorem 5.8 holds on $S_\nabla$, then $S_\nabla$ can be identified, up to an affine diffeomorphism, with a parameter space of statistical models on the measurable space $(\Omega^*, \mathcal{B}(\Omega^*))$, with model

$$ \langle S_\nabla, Q \rangle, $$

570 where $Q$ is given by

$$ (7.16) \qquad Q(\Theta, \xi) = \int_{H(\nabla)} \frac{e^{-\langle \gamma\Theta, \xi \rangle}}{\int_{\Omega^*} e^{-\langle \gamma\Theta, \xi \rangle} \, d\xi} \, d\mu^{H(\nabla)}(\gamma), $$

571 with $H(\nabla) \subset GL(\mathbb{R}^n)$.

572

8. A WORKED EXAMPLE

573 We present an example of a compact Hessian manifold whose orbits are quotients of
574 statistical models by a discrete subgroup.

575 Let $\mathbb{R}^2$ be a 2-dimensional real affine space equipped with the natural flat affine connection $\mathcal{D}$, and let $\{x, y\}$ be an affine coordinate system on $\mathbb{R}^2$.

577 Consider the domain

$$ (8.1) \qquad \Omega = \{(x, y) \in \mathbb{R}^2 \mid x > 0, y > 0\}. $$

578 Define on $\Omega$ the Riemannian metric

$$ (8.2) \qquad g = \frac{1}{x^2} dx^2 + \frac{1}{y^2} dy^2. $$

579 Then $(D, g)$ defines a Hessian structure on $\Omega$.
580 Let $\Sigma$ and $\chi$ be linear transformations on $\Omega$ given by
$$ (8.3) \qquad \Sigma : (x, y) \mapsto (2x, y), \quad \chi : (x, y) \mapsto (2x, 3y). $$

581 These transformations preserve the Hessian structure $(D, g)$.

582 Denote by $\Gamma$ the group generated by $\{\Sigma, \chi\}$, which can also be written as

$$ (8.4) \qquad \Gamma = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}. $$

583 The group $\Gamma$ acts properly discontinuously on $\Omega$, and the quotient $\Omega/\Gamma$ is a compact Hessian manifold diffeomorphic to a torus.

585 Let $\pi : \Omega \to \Omega/\Gamma$ be the natural projection, and denote by $(D, g)$ the induced Hessian structure on $\Omega/\Gamma$. The space of $D$-parallel and $\Gamma$-invariant 1-forms on $\Omega$ is spanned by $dx$ and $dy$. Thus, the space of $D$-parallel 1-forms on $\Omega/\Gamma$ is spanned by $\omega$ and $\Phi$ such that

$$ (8.5) \qquad dx = \pi^*\omega, \quad dy = \pi^*\Phi. $$

589 Define vector fields $X$ and $\tilde{X}$ on $\Omega/\Gamma$ by

$$ (8.6) \qquad \omega(Y) = g(X, Y), \quad \Phi(Y) = g(\tilde{X}, Y) \quad \text{for all vector fields } Y \text{ on } \Omega/\Gamma. $$

590 Then

$$ (8.7) \qquad X = \pi_* \left( \frac{\partial}{\partial x} \right), \quad \tilde{X} = \pi_* \left( \frac{\partial}{\partial y} \right), $$

591 and the space $J_\nabla$ of Hessian vector fields on $\Omega/\Gamma$ is spanned by $\{X, \tilde{X}\}$. Since $T^2$ is compact, $J_\nabla$ is integrable.

593 Let $\exp(tX)$ and $\exp(tY)$ denote the one-parameter groups generated by $X$ and $\tilde{X}$. Define

$$ (8.8) \qquad G_{\nabla} = \langle \exp(tX), \exp(tY) \rangle. $$

595 For $a > 0$ and $b > 0$, the compact orbit

$$ (8.9) \qquad G_{\nabla} \cdot \pi(a, b) = \{ \gamma \cdot \pi(a, b) \mid \gamma \in G_{\nabla} \} $$

596 is a circle. Hence, $(S^1, D, g_{S^1})$ is a Hessian manifold satisfying

$$ (8.10) \qquad D \frac{\partial}{\partial \theta} = 0, \quad g_{S^1} = \frac{1}{\theta^2} d\theta \otimes d\theta. $$

597 By conjugation, we have

$$ (8.11) \qquad D^* \frac{\partial}{\partial \theta} = -\frac{2}{\theta} \frac{\partial}{\partial \theta}. $$

598 Let

$$ (8.12) \qquad H^* = h(\theta) \frac{\partial}{\partial \theta} $$

599 be a vector field on $S^1$. Then $H^*$ is a homothetic vector field of $D^*$ if $h$ is a solution of the differential equation

$$ (8.13) \qquad y' - \frac{2}{\theta^2} y = 1. $$
601 We obtain

$$ (8.14) \qquad h(\theta) = Ae^{-2/\theta} + 2e^{-2/\theta} \int_{-2/\theta}^{+\infty} \frac{e^{-s}}{s} ds + \theta, $$

602 where A is a constant. Since $D^*H^* = \text{Id}$, $(S^1, D)$ is hyperbolic.

603 Finally, let $p: \mathbb{R} \to S^1$ be the universal covering map. The map

$$ (8.15) \qquad (\mathbb{R}, \tilde{D}) \to (S^1, D) $$

604 shows that the compact orbits $G_\nu \cdot \pi(a, b)$ are quotients

$$ (8.16) \qquad \frac{\mathbb{R}^{>0}}{\Gamma}, $$

605 where $\Gamma$ is a discrete subgroup of $\text{GL}(\mathbb{R})$.

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708 INSA DE TOULOUSE, DÉPARTEMENT DE GÉNIE MATHÉMATIQUE, UNIVERSITÉ DE TOULOUSE, FRANCE

709 Email address: kpanteemmanuel@gmail.com, gnandi@insa-toulouse.fr

710 INSTITUT ALEXANDRE GROTHENDIECK, IMAG-UMR CNRS 5149-C.C.051, UNIVERSITÉ DE MONTPELLIER

711 Email address: boyom@math.univ-montp2.fr

712 ECOLE NATIONALE DE L'AVIATION CIVILE, UNIVERSITÉ DE TOULOUSE

713 Email address: stephane.puechmorel@enac.fr