What is the best way to understand the Bianchi Identities?

Abstract: The Bianchi identities are relations involving the curvature and torsion that will always be satisfied for a smooth manifold with an affine connection. These identities are equivalent to the Jacobi identity with respect to the skew-symmetrization of the tensor product in the free tensor algebra over the smooth vector fields. To get this equivalence we must include the connection to make the tensor algebra into what is called a D-algebra. It is also known that the Bianchi identities can be derived from the canonical principal connection and the canonical solder form of the frame bundle. 

In this presentation we will be in two parts:

1. First we look at the frame bundle. What structure is induced on the frame bundle from the connection on the underlying manifold? The goal is to get a good understanding of the curvature, torsion and the Bianchi identities on the frame bundle.
    
2. Secondly we will see what the D-algebra is and how the Bianchi identities can be represented here. We might speculate on the relation between the frame bundle and the D-algebra.

This talk is based on early stage of a potential research project.

Keywords: Bianchi identities, connection, frame bundle, D-algebra

Photo:

deepai.org. "Jonatan working on the Bianchi identities"