## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 303 - 321

### Statistical manifolds and affine differential geometry

#### Abstract

In this paper, we give a summary of geometry of statistical manifolds, and discuss relations between information geometry and affine differential geometry. Dually flat spaces and canonical divergence functions are important objects in information geometry. We show that such objects can be generalized in the framework of affine differential geometry.

In addition, we give a brief summary of geometry of statistical manifolds admitting torsion, which is regarded as a quantum version of statistical manifolds. We discuss relations between statistical manifolds admitting torsion and geometry of affine distributions.

#### Article information

**Source***Probabilistic Approach to Geometry*, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 303-321

**Dates**

Received: 20 January 2009

Revised: 28 February 2009

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543086326

**Digital Object Identifier**

doi:10.2969/aspm/05710303

**Mathematical Reviews number (MathSciNet)**

MR2648267

**Zentralblatt MATH identifier**

1201.53011

**Subjects**

Primary: 53A15: Affine differential geometry 53A30: Conformal differential geometry 53B05: Linear and affine connections 62B10: Information-theoretic topics [See also 94A17] 81Q70: Differential-geometric methods, including holonomy, Berry and Hannay phases, etc.

**Keywords**

Statistical manifold affine differential geometry information geometry semi-Weyl manifold affine distribution

#### Citation

Matsuzoe, Hiroshi. Statistical manifolds and affine differential geometry. Probabilistic Approach to Geometry, 303--321, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710303. https://projecteuclid.org/euclid.aspm/1543086326