On a constant curvature statistical manifold

Abstract

We will show that a statistical manifold $(M, g, \nabla )$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $\nabla$ is projectively flat and the curvatures satisfies $R=R^*$, where $R^*$ is the curvature of the dual connection $\nabla ^*$. Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $-1$ statistical structures and vice versa.

Appendix A: A sketch of the proof of Theorem 3.1

As explained in Sect. 3, property convex $\mathbb {R} \mathbb {P}^{n}$-structures on a compact manifold M have been characterized by Condition (E), and we have rephrased it in terms of constant curvature $-1$ statistical manifold structure on M, and we have obtained Theorem 3.1. In this appendix, we will briefly give a proof of this theorem along the proof of Theorem 3.2.1 in [12].

Let us prove the necessary part. Let $\widetilde{M}$ be the universal cover of M and consider the bundle $T \widetilde{M} \oplus L$, where $L = {\mathbb R} \times \widetilde{M}$. Then it is evident that by the flatness of $\nabla ^g$ (which is equivalent to the constant curvature $-1$ statistical structure), $T \widetilde{M} \oplus L$ is isomorphic to the trivial bundle ${\mathbb R}^{n+1} \times \widetilde{M}$. Let us take the projection p from $T \widetilde{M} \oplus L \cong \mathbb R^{n+1} \times \widetilde{M}$ to $\mathbb R^{n+1}$. Moreover, let us take the canonical section of $T \widetilde{M} \oplus L$ by $u_0 : m \rightarrow (0, 1)$, and define $\phi = p \circ u_0$. Then by the construction, $\phi$ is a $\rho$-equivariant mapping from $\widetilde{M}$ to $\mathbb R^{n+1}$, where $\rho$ is the holonomy representation of the flat connection $\nabla ^g$. Then Labourie has proved that $\phi$ is a proper immersion, and the image $\phi (M)$ is a locally convex proper hypersurface in a sequence of Propositions, 3.2.2 (immersion), 3.2.3 (strictly locally convex and radial) and 3.2.4 (proper) in [12], respectively. Moreover, the geodesic $\gamma (t)$ with respect to $\nabla$ gives a sub-bundle

$$\begin{aligned} P = \mathbb R (\dot{\gamma }) \oplus \mathbb R \subset TM \oplus L, \end{aligned}$$

which is parallel along $\gamma (t)$. Then $\phi (\gamma (t))$ is the projective line defined by P, and $\nabla$ and $\nabla ^g$ define the same projective flat structure. Therefore the connection $\nabla$ gives a properly convex $\mathbb {R} \mathbb {P}^{n}$-structure on M.

Let us prove the sufficient part. An important step has been proved by Vinberg [21] and see Lemma 3.1.1 in [12]: For a given properly convex $\mathbb {R} \mathbb {P}^{n}$-structure on a manifold M induced by the pair $(f, \rho )$, there exists a proper $\rho$-equivariant immersion $\tilde{f}$ from a universal cover $\widetilde{M}$ into $\mathbb R^{n+1}$ such that the image is strictly convex and radial and $\pi \circ \tilde{f} = f$, where $\pi$ is the projection $\mathbb R^{n+1}\setminus \{0\}$ to $\mathbb {R} \mathbb {P}^{n}$. Note that a hypersurface is strictly convex means that it does not contain any segment and a hypersurface is radial means that if the vector pointing from the origin points inward. Let $\Sigma = \tilde{f} (\widetilde{M})$ be the locally strictly convex hypersurface in $\mathbb R^{n+1}$. Since $\Sigma$ is radial, thus

$$\begin{aligned} T\mathbb R^{n+1}|_{\Sigma }= T \Sigma \oplus \mathbb R N. \end{aligned}$$

The standard flat connection $\nabla ^0$ on $\mathbb R^{n+1}$ then induces a volume preserving connection $\nabla$ given as

$$\begin{aligned} \nabla _X^0 (Z + \lambda N) = \nabla _X Z + \lambda \nabla _X^0 N + (L_X \lambda ). N + g(Z, X).N \end{aligned}$$

for vector fields X, Z on $\Sigma$. Since $\Sigma$ is strictly locally convex and radial, $\nabla _X^0 N = X$ and $g(X, X)>0$. Now the pair $(\nabla , g)$ clearly gives a constant curvature $-1$ statistical manifold structure on M.