I am going to review the powerful Cartan calculus of differential forms applied to differential geometry. In particular, I will derive the structure equations and the Bianchi identities. Yes!
Firstly, in a 2-dim manifold, we and introduce the Cartan 1-forms
The connection form reads
Now, we can introduce the so-called curvature and the curvature 2-form, since from , we will get
The generalization to -dimensional manifolds is quite straightforward. The torsion 1-forms are defined through the canonical 1-forms via
With matrices and , being antisymmetric , we can derive the structure equations:
The connection forms satisfy
The gauging of the connection and curvature forms provide
since , and , as matrices. Note, as well, the characteristic classes
is satisfied, with
Now, we also have the Bianchi identities
Check follows easily:
From these equations:
sinde . By the other hand, we also deduce the 2nd Bianchi identity
Note that . Then,
Remember: gives the 1st structure equation, gives the 2nd structure equation, gives the first Bianchi identity, and provides the 2nd Bianchi identity.