Transformations and Coupling Relations for Affine Connections

Abstract

AbstractThe statistical structure on a manifold \(\mathfrak {M}\) is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection \(\nabla \) on the \(T\mathfrak {M}\), such that \(\nabla g\) is totally symmetric, forming, by definition, a “Codazzi pair” \(\{\nabla , g\}\). 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 \(T\mathfrak {M}\)), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of \(\nabla \) 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.KeywordsConformal TransformationGeneral TransformationProjective TransformationInvertible OperatorLeibniz RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.