Information geometry with (Para-)Kähler structures

Jun Zhang, Teng Fei

Research output: Chapter in Book/Report/Conference proceedingChapter

2 Scopus citations

Abstract

We investigate conditions under which a statistical manifold M (with a Riemannian metric g and a pair of torsion-free conjugate connections ∇, ∇ ) can be enhanced to a (para-)Kähler structure. Assuming there exists an almost (para-)complex structure L compatible with g on a statistical manifold M (of even dimension), then we show (M, g, L, ∇ ) is (para-)Kähler if ∇ and L are Codazzi coupled. Other equivalent characterizations involve a symplectic form ω≡ g(L·, · ). In terms of the compatible triple (g, ω, L), we show that (i) each object in the triple induces a conjugate transformation on ∇ and becomes an element of an (Abelian) Klein group; (ii) the compatibility of any two objects in the triple with ∇ leads to the compatible quadruple (g, ω, L, ∇ ) in which any pair of objects are mutually compatible. This is what we call Codazzi-(para-)Kähler manifold [8] which admits the family of torsion-free α -connections (convex mixture of ∇, ∇ ) compatible with (g, ω, L). Finally, we discuss the properties of divergence functions on M× M that lead to Kähler (when L= J, J2= - id ) and para-Kähler (when L= K, K2= id ) structures.

Original languageEnglish (US)
Title of host publicationSpringer Proceedings in Mathematics and Statistics
PublisherSpringer New York LLC
Pages297-321
Number of pages25
DOIs
StatePublished - 2018
Externally publishedYes

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume252
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Codazzi coupling
  • Codazzi-(para-)Kähler
  • Compatible quadruple
  • Compatible triple
  • Conjugation of connection
  • Kähler structure
  • Para-Kähler structure
  • Statistical manifold
  • Torsion

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