What exactly is the connection between the Jacobi and Bianchi identities?

I) The proofs of both the first (algebraic) Bianchi identity and the second (differential) Bianchi identity crucially use that the connection $\nabla$ is torsionfree, so they are not entirely consequences of the Jacobi identity. Proofs of the Bianchi identities are e.g. given in Ref. 1.

II) The second Bianchi identity may be formulated not only for a tangent bundle connection but also for vector bundle connections.

III) The Lie bracket in the pertinent Jacobi identities is the commutator bracket $[A,B]:=A\circ B -B\circ A$. The Jacobi identity follows because operator composition "$\circ$" is associative.

IV) In the context of Yang-Mills theory and EM, the second Bianchi identity follows because the gauge potential $A_{\mu}$ and the field strength $F_{\mu\nu}$ may be viewed as (part of) a covariant derivative and corresponding curvature tensor, respectively.


  1. M. Nakahara, Geometry, Topology and Physics, Section 7.4.