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A remark on statistical manifolds with torsion

김화정
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2023.06|한국수학논문집 (Korean Journal of Mathematics)|제31권 제2호|pp. 133-137

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키워드

torsionmetric connectiondual connectionsstatistical manifold
주제분류수학

초록

Consider a Riemannian manifold $M$ equipped with a metric $g$. In this article, we study a notion for statistical manifolds $(M,g,\nabla)$, which can have a non-zero torsion, abbreviated to SMT. Then it turns out that the tensor fields $\nabla g$ and $\tilde{\nabla} g$ obtained from two different SMT-connections are different.

Abstract

Consider a Riemannian manifold $M$ equipped with a metric $g$. In this article, we study a notion for statistical manifolds $(M,g,\nabla)$, which can have a non-zero torsion, abbreviated to SMT. Then it turns out that the tensor fields $\nabla g$ and $\tilde{\nabla} g$ obtained from two different SMT-connections are different.

참고문헌 (11)

  1. S. Amari, "Information Geometry and Its Applications, Applied Mathematical Sciences" (2016), Springer Japan
  2. A. M. Blaga, "Conformal-projective transformations on statistical and semi-Weyl manifolds with torsion", arXiv.2209.00689vl, ~ (2022)
  3. O. Calin, "Geometric Modeling in Probability and Statistics" (2013), Springer
  4. E. Cartan, "Sur les varietes a connexion affine et la theorie de la relativite generalisee", Ann. Ec. Norm. Sup, 42, pp.17~88 (1925)
  5. 김화정, "A note on statistical manifolds with torsion", 대한수학회논문집, 38(2), pp.621~628 (2023), 대한수학회[DOI]
  6. T. Kurose, "Geometry and Something" (2007)
  7. S. L. Lauritzen, "Statistical manifolds", Differential Geometry in statistical inference, 10, pp.163~216 (1987)
  8. H. Matsuzoe, "Statistical manifolds and affine differential geometry", Advanced Studies in Pure Mathematics, 57, pp.303~321 (2010)
  9. Differential geometry of smooth families of probability distributions, "H. Nagaoka", Dept. of Math. Eng. and Instr. Phys., Univ. of Tokyo, ~, pp.Technical Report (METR) 82-7 (1982)
  10. A. P. Norden, "On pairs of conjugate parallel displacements in multidimensional spaces", Doklady Akademii nauk SSSR, 49, pp.1345~1347 (1945)
  11. U. Simon, "Handbook of Differential Geometry, vol. I", pp.905~961 (2000), Elsevier Science

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