Transformations and coupling relations for affine connections

Abstract

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 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 bilinear 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 framework allows affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. In addition to reproducing some known results regarding Codazzi transformations, conformal-projective transformations, etc., we extend many of these geometric relations, and hence obtain new geometric insights, for the general case of a non-degenerate bilinear form h (not required to be a symmetric form g) in relation to an affine connection with possibly non-vanishing torsion. In particular, we provide a generalization to the conformal-projective transformation of {∇,g} which preserves their Codazzi coupling. Our systematic approach establishes a general setting for the study of information geometry based on transformations and coupling relations between affine connections.