LOG#244. Cartan calculus.

Posted on by July 22, 2020

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)   \begin{align*} d\theta^1+\omega^1_{\;\;2}\wedge \theta^2=0\\ d\theta^2+\omega^2_{\;\;1}\wedge \theta^1=0 \end{align*}

The connection form reads

(2)   \begin{equation*} \omega=\begin{pmatrix} \omega^1_{\;\; 1} & \omega^1_{\;\; 2}\\ \omega^2_{\;\; 1} & \omega^2_{\;\; 2} \end{pmatrix} \end{equation*}

Now, we can introduce the so-called curvature k=\Omega^1_{\;\; 2}(e_1,e_2) and the curvature 2-form, since from d\omega^1_{\;\;2}=k\theta^1\wedge\theta^2, we will get

(3)   \begin{equation*} \Omega=\begin{pmatrix} \Omega^1_{\;\; 1} & \Omega^1_{\;\; 2}\\ \Omega^2_{\;\; 1} & \Omega^2_{\;\; 2}\end{pmatrix} \end{equation*}

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

(4)   \begin{equation*} \theta=\begin{pmatrix}\theta^1 \\ \vdots \\ \theta^n\end{pmatrix} \end{equation*}

such as

(5)   \begin{equation*} \Theta=\begin{pmatrix} \Theta^1 \\ \vdots \\ \Theta^n\end{pmatrix} \end{equation*}

With matrices \omega=\omega^i_{\;\; j} and \Omega^i_{\;\; j}, being antisymmetric n\times n, we can derive the structure equations:

(6)   \begin{equation*} \tcboxmath{\Theta=d\theta+\omega\wedge \theta} \end{equation*}

(7)   \begin{equation*} \tcboxmath{\Omega=d\omega+\omega\wedge\omega} \end{equation*}

Note that

(8)   \begin{equation*} \Theta^k=T^k_{ij}\theta^i\wedge\theta^j \end{equation*}

The connection forms satisfy

(9)   \begin{align*} \nabla_X e=e\omega(X)\\ \nabla e=e\omega \end{align*}

The gauging of the connection and curvature forms provide

(10)   \begin{align*} \overline{\omega}=a^{-1}\omega a+a^{-1}da\\ \overlin{\Omega}=a^{-1}\Omega a \end{align*}

since \overline{e}=ea, and e=\overline{e}a^{-1}, as matrices. Note, as well, the characteristic classes

(11)   \begin{equation*} \int_M e(M)=\int_M \mbox{Pf}\left(\dfrac{\Omega}{2\pi}\right)=\chi(M) \end{equation*}

is satisfied, with

(12)   \begin{equation*} \mbox{det}\left(I+\dfrac{i\Omega}{2\pi}\right)=1+c_1(E)+\cdots+c_k(E) \end{equation*}

Now, we also have the Bianchi identities

(13)   \begin{equation*} \tcboxmath{d\Theta=\Omega\wedge\theta-\omega\wedge\Theta} \end{equation*}

(14)   \begin{equation*} \tcboxmath{d\Omega=\Omega\wedge\omega-\omega\wedge\Omega} \end{equation*}

Check follows easily:

(15)   \begin{align*} d\theta=\Theta-\omega\wedge\theta\\ d\omega=\Omega-\omega\wedge\omega\\ d\Theta=\Omega\wedge\theta-\omega\wedge\Theta\\ d\Omega=\Omega\wedge\omega-\omega\wedge\Omega\\ d(\Omega^k)=\Omega^k\wedge\omega-\omega\wedge\Omega^k \end{align*}

From these equations:

    \[d\Theta=d(d\theta)+d\omega\wedge\theta-\omega\wedge d\theta\]

    \[d\Theta=(\Omega-\omega\wedge\omega)\wedge\omega-\omega\wedge(\Theta-\omega\wedge\theta)\]

and then

    \[d\Theta=\Omega\wedge\omega-\omega\wedge\Theta\;\;\; Q.E.D.\]

sinde \omega\wedge\omega\wedge\theta=0. By the other hand, we also deduce the 2nd Bianchi identity

    \[d\Omega=d^2\omega+d\omega\wedge\omega-\omega\wedge d\omega\]

Note that d(d\omega)=d^2\omega=0. Then,

    \[d\Omega=d\omega\wedge\omega-\omega\wedge d\omega=(\Omega-\omega\wedge\omega)\wedge \omega-\omega\wedge(\Omega-\omega\wedge\omega)\]

and thus

    \[d\Omega=\Omega\wedge\omega-\omega\wedge\Omega\;\; Q.E.D.\]

Remember: d\theta gives the 1st structure equation, d\omega gives the 2nd structure equation, d\Theta gives the first Bianchi identity, and d\Omega provides the 2nd Bianchi identity.

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