Dual Connections and Holonomy

TL;DR

This paper extends the concept of dual affine connections from Riemannian manifolds to general vector bundles with inner products, providing a geometric interpretation of the Amari tensor through Cartan decompositions.

Contribution

It introduces a generalized notion of dual connections on vector bundles, linking them to Cartan decompositions and offering a new geometric perspective on the Amari tensor.

Findings

Generalization of dual connections to vector bundles with inner products

Geometric interpretation of the Amari tensor as a connection form term

Connection to Cartan decompositions of Lie algebras

Abstract

Dual affine connections on Riemannian manifolds have played a central role in the field of information geometry since their introduction by Amari. Here I would like to extend the notion of dual connections to general vector bundles with an inner product, in the same way as a unitary connection generalizes a metric affine connection, using Cartan decompositions of Lie algebras. This gives a natural geometric interpretation for the Amari tensor, as a "connection form term" which generates dilations, and which is reversed for the dual connections.