Advanced Studies in Pure Mathematics

Statistical manifolds and affine differential geometry

Hiroshi Matsuzoe

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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


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