## 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.

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## Acknowledgements

We would like to thank Prof. Jun-ichi Inoguchi and Porf. Hitoshi Furuhata for comments on the manuscripts and letting us know several related references. Moreover, we would like to thank the four anonymous reviewers for their suggestions, comments and additional references.

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The first named author is partially supported by Kakenhi 18K03265.

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The first named author is partially supported by Kakenhi 18K03265.

## Appendix A: A sketch of the proof of Theorem 3.1

### 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

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

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

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*.

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Kobayashi, S., Ohno, Y. On a constant curvature statistical manifold.
*Info. Geo.* **5**, 31–46 (2022). https://doi.org/10.1007/s41884-022-00065-x

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DOI: https://doi.org/10.1007/s41884-022-00065-x

### Keywords

- Statistical manifolds
- constant curvatures
- Conjugate symmetries
- Projective flatness
- Properly convex structures

### Mathematics Subject Classification

- Primary 53B12
- 53C15