Transformations and Coupling Relations for Affine Connections - James Tao, Jun Zhang
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last edited by Geo-Sci-InfoAuthor(s): James Tao, Jun Zhang
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_36
Video: http://www.youtube.com/watch?v=ebmReSVXZ1E
Slides: Tao_Transformation coupling relations.pdf
Presentation: https://www.see.asso.fr/node/14290
Creative Commons Attribution-ShareAlike 4.0 InternationalAbstract:
The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇ g is totally symmetric, forming, by definition, a “Codazzi pair” { ∇ , g}. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.
