# LOG#244. Cartan calculus.

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

(1)

The connection form reads

(2)

Now, we can introduce the so-called curvature and the curvature 2-form, since from , we will get

(3)

The generalization to -dimensional manifolds is quite straightforward. The torsion 1-forms are defined through the canonical 1-forms via

(4)

such as

(5)

With matrices and , being antisymmetric , we can derive the structure equations:

(6)

(7)

Note that

(8)

The connection forms satisfy

(9)

The gauging of the connection and curvature forms provide

(10)

since , and , as matrices. Note, as well, the characteristic classes

(11)

is satisfied, with

(12)

Now, we also have the Bianchi identities

(13)

(14)

Check follows easily:

(15)

From these equations:

and then

sinde . By the other hand, we also deduce the 2nd Bianchi identity

Note that . Then,

and thus

Remember: gives the 1st structure equation, gives the 2nd structure equation, gives the first Bianchi identity, and provides the 2nd Bianchi identity.

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