Transformations and Coupling Relations for Affine Connections

Abstract

The 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.