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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.

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References

  1. Amari, S.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics, vol. 28. Springer, Berlin (1985)

    Book  Google Scholar 

  2. Amari, S., Nagaoka, H.: Method of Information Geometry. Amer. Math. Soc. Oxford Univ. Press, Oxford (2000)

    MATH  Google Scholar 

  3. Bokan, N., Nomizu, K., Simon, U.: Affine hypersurfaces with parallel cubic forms. Tohoku Math. J. (2) 42(1), 101–108 (1990)

    MathSciNet  Article  Google Scholar 

  4. Calin, O., Udrişte, C.: Geometric Modeling in Probability and Statistics. Springer, Cham (2014)

    Book  Google Scholar 

  5. Eisenhart, L.P.: Non-Riemannian Geometry, Amer. Math. Soc. Colloq. Publ. 8 (1927)

  6. Furuhata, H., Inoguchi, J., Kobayashi, S.-P.: A characterization of the alpha-connections on the statistical manifold of normal distributions. Inf. Geom. 4(1), 1–12 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Globke, W., Quiroga-Barranco, R.: Information geometry and asymptotic geodesics on the space of normal distributions. Inf. Geom. 4(1), 131–153 (2021)

    MathSciNet  Article  Google Scholar 

  8. Goldman, W.M.: Convex real projective structures on compact surfaces. J. Differ. Geom. 31(3), 791–845 (1990)

    MathSciNet  Article  Google Scholar 

  9. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry II, Interscience Tracts in Pure and Applied Math, vol. 15. Interscience Publishers, Geneva (1969)

    Google Scholar 

  10. Kobayashi, S.P., Ohno, Y.: On a constant curvature statistical manifold (2020). PreprintarXiv:2008.13394

  11. Kurose, T.: Dual connections and affine geometry. Math. Z. 203(1), 115–121 (1990)

    MathSciNet  Article  Google Scholar 

  12. Labourie, F.: Flat projective structures on surfaces and cubic holomorphic differentials. Pure Appl. Math. Q. 3, 1057–1099 (2007)

    MathSciNet  Article  Google Scholar 

  13. Lauritzen, S.L.: Statistical manifolds. In: Differential Geometry in Statistical Inference, IMS Lecture Notes: Monograph Series, vol. 10. Institute of Mathematical Statistics, Hayward, California, pp. 163–216 (1987)

  14. Loftin, J.C.: Affine spheres and convex \(\mathbb{RP}^n\)-manifolds. Am. J. Math. 3(2), 255–274 (2001)

  15. Mikeš, J., Stepanova, E.: A five-dimensional Riemannian manifold with an irreducible \({\rm SO}(3)\)-structure as a model of abstract statistical manifold. Ann. Glob. Anal. Geom. 45(2), 111–128 (2014)

  16. Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge Univ, Oxford (1994)

    MATH  Google Scholar 

  17. Opozda, B.: Bochner’s technique for statistical structures. Ann. Glob. Anal. Geom. 48, 357–395 (2015)

    MathSciNet  Article  Google Scholar 

  18. Opozda, B.: A sectional curvature for statistical structures. Linear Algebra Appl. 497, 134–161 (2016)

    MathSciNet  Article  Google Scholar 

  19. Rylov, A.: Constant curvature connections on statistical models. In:Information Geometry and its Applications, Springer Proc. Math. Stat., vol. 252. Springer, Cham, pp. 349–361 (2018)

  20. Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the affine differential geometry of hypersurfaces, Lecture Notes of the Science University of Tokyo, (1991)

  21. Vinberg, E.B.:The theory of convex homogeneous cones, Translations Moscow Math. Soc., 340–403 (1963)

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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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Correspondence to Shimpei Kobayashi.

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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

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

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 XZ 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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Keywords

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

Mathematics Subject Classification

  • Primary 53B12
  • 53C15