June 2022
Here we first introduce and prove Cartan’s structural equations. Note that these are Problem 4-5 and Problem 7-2 of (Lee 1997).
Let \(\nabla\) be a linear connection on a Riemannian manifold \(M\). Let \(U \subset M\) be an open subset with a local frame \(\{e_i\}\) and \(\{\omega^i\}\) the dual coframe, that is, \(\omega^i(e_j) = \delta^i_j\). Notice that for a vector field \(X = X^ke_k\), we have that
\[ \nabla_Xe_i = \nabla_{X^ke_k}e_i = X^k\nabla_{e_k}e_i = X^k\Gamma^j_{ki}e_j \]
Then we can define the unique connection 1-forms associated with this connection and this local frame by defining \({\omega_i}^j\) by \({\omega_i}^j(X) = X^k\Gamma^j_{ki}\). In other words, we define
\[ {\omega_i}^j = \Gamma^j_{ki}\omega^k \]
It is easy to check that these connection 1-forms are linear and tensorial and thus are indeed 1-forms.
In addition, given a linear connection \(\nabla\) on \(M\), we can define a map \(\tau:\mathscr{T}(M)\times\mathscr{T}(M)\to\mathscr{T}(M)\) by
\[ \tau(X,Y) = \nabla_XY - \nabla_YX - [X,Y] \]
called the torsion tensor of \(\nabla\).
We now show the following formulas, which is known as Cartan’s first structural equations
Proposition 1 (Cartan’s first structural equations). For each \(\omega^j\) in the dual coframe, we have \[ d\omega^j = \omega^i \wedge {\omega_i}^j + \tau^j \]
where \(\{\tau^1,\dots,\tau^n\}\) are the torsion 2-forms, defined in terms of the torsion tensor \(\tau\) and the frame \(\{e_i\}\) by
\[ \tau(X,Y) = \tau^j(X,Y)e_j. \]
Proof. Let \(e_p, e_q\) be elements in the local frame. Then since \(\omega^j\) is a 1-form, we have that \[ \begin{aligned} d\omega^j(e_p,e_q) &= e_p(\omega^j(e_q)) - e_q(\omega^j(e_p)) - \omega^j([e_p,e_q]) \\ &= e_p(\delta^j_q) - e_q(\delta^j_p) - \omega^j\left(\nabla_{e_p}e_q - \nabla_{e_q}e_p - \tau(e_p,e_q)\right) \\ &= -\omega^j\left(\Gamma^k_{pq}e_k - \Gamma^k_{qp}e_k - \tau^k(e_p,e_q)e_k\right), \end{aligned} \]
where the second equality is by the definition of the torsion tensor of \(\nabla\) and the last equality is because \(e_p(\delta^j_q) - e_q(\delta^j_p) = 0 - 0 = 0\) if \(p \neq q\) or \(e_p(\delta^j_q) - e_q(\delta^j_p) = 1 - 1 = 0\) if \(p = q\).
Then for the terms with the Christoffel symbols we use the definition of the connection 1-forms
\[ {\omega_q}^k(e_p) = \Gamma^k_{sq}\omega^s(e_p) = \Gamma^k_{sq}\delta^s_p = \Gamma^k_{pq}. \]
So continuing, we have
\[ \begin{aligned} d\omega^j(e_p,e_q) &= -\omega^j\left(\Gamma^k_{pq}e_k - \Gamma^k_{qp}e_k - \tau^k(e_p,e_q)e_k\right) \\ &= -\omega^j\left({\omega_q}^k(e_p)e_k - {\omega_p}^k(e_q)e_k - \tau^k(e_p,e_q)e_k\right) \\ &= {\omega_p}^k(e_q)\omega^j(e_k) - {\omega_q}^k(e_p)\omega^j(e_k) + \tau^k(e_p,e_q)\omega^j(e_k) \\ &= {\omega_p}^k(e_q)\delta^j_k - {\omega_q}^k(e_p)\delta^j_k + \tau^k(e_p,e_q)\delta^j_k \\ &= {\omega_p}^j(e_q) - {\omega_q}^j(e_p) + \tau^j(e_p,e_q). \end{aligned} \]
Observe that
\[ (\omega^i\otimes{\omega_i}^j)(e_p,e_q) = \omega^i(e_p){\omega_i}^j(e_q) = \delta^i_p{\omega_i}^j(e_q) = {\omega_p}^j(e_q), \]
and similarly \((\omega^i\otimes{\omega_i}^j)(e_q,e_p) = {\omega_q}^j(e_p)\). So continuing, we have
\[ \begin{aligned} d\omega^j(e_p,e-q) &= {\omega_p}^j(e_q) - {\omega_q}^j(e_p) + \tau^j(e_p,e_q) \\ &= (\omega^i\otimes{\omega_i}^j)(e_p,e_q) - (\omega^i\otimes{\omega_i}^j)(e_q,e_p) + \tau^j(e_p,e_q) \\ &= (\omega^i\wedge{\omega_i}^j)(e_p,e_q) + \tau^j(e_p,e_q). \end{aligned} \]
This implies that
\[ d\omega^j(X,Y) = X^pY^qd\omega^j(e_p,e_q) = X^pY^q(\omega^i\wedge{\omega_i}^j+\tau^j)(e_p,e_q) = (\omega^i\wedge{\omega_i}^j+\tau^j)(X,Y). \]
In other words, that
\[ d\omega^j = \omega^i\wedge{\omega_i}^j + \tau, \]
as required.
Similarly, we define the connection 2-forms \({\Omega_i}^j\) in the following way. Let \(\nabla\) now be the unique Riemann/Levi-Civita connection on \((M,g)\) and \(R\) be the Riemann curvature endomorphism. Recall that \(R\) is given by
\[ R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, \]
for smooth vector fields \(X,Y,Z\) on \(M\). Then we define \({\Omega_i}^j\)’s by the following relation
\[ {\Omega_i}^j = \frac{1}{2}{R_{k\ell i}}^j\omega^k\wedge\omega^\ell. \]
And along with this definition of the connection 2-forms we have Cartan’s second structural equations.
Proposition 2 (Cartan’s second structural equations). For the connection 2-forms \({\Omega_i}^j\), we have
\[ {\Omega_i}^j = d{\omega_i}^j - {\omega_i}^k\wedge{\omega_k}^j. \]
Proof. We have that
\[ {R_{k\ell i}}^je_j = R(e_k,e_\ell)e_i = \nabla_k\nabla_\ell e_i - \nabla_\ell\nabla_k e_i - \nabla_{[e_k,e_\ell]}e_i, \](2.1)
where \(\nabla_i\) is shorthand for \(\nabla_{e_i}\).
Similarly to above, we have that \(\nabla_\ell e_i = {\omega_i}^p(e_\ell)e_p\), and since \(\nabla\) is torsion free, after some calculation we also find that \(\nabla_{[e_k,e_\ell]}e_i = {\omega_i}^p([e_k,e_\ell])e_p\). So inserting these into (2.1) above, we have that
\[\begin{aligned} R(e_k,e_\ell)e_i &= \nabla_k({\omega_i}^p(e_\ell)e_p) - \nabla_\ell({\omega_i}^p(e_k)e_p) - {\omega_i}^p([e_k,e_\ell])e_p \\ &= (e_k{\omega_i}^p(e_\ell))e_p + {\omega_i}^p(e_\ell)\nabla_ke_p - (e_\ell{\omega_i}^p(e_k))e_p - {\omega_i}^p(e_k)\nabla_\ell e_p - {\omega_i}^p([e_k,e_\ell])e_p \\ &= e_k({\omega_i}^p(e_\ell))e_p + {\omega_i}^p(e_\ell){\omega_p}^j(e_k)e_j - e_\ell({\omega_i}^p(e_k))e_p - {\omega_i}^p(e_k){\omega_p}^j(e_\ell)e_j - {\omega_i}^p([e_k,e_\ell])e_p. \end{aligned}\](2.2)
Since \({\omega_i}^p\) is a 1-form we have that
\[ d{\omega_i}^p(e_k,e_\ell) = e_k({\omega_i}^p(e_\ell)) - e_\ell({\omega_i}^p(e_k)) - {\omega_i}^p([e_k,e_\ell]) \]
So (2.2) above becomes
\[\begin{aligned} R(e_k,e_\ell)e_i &= e_k({\omega_i}^p(e_\ell))e_p + {\omega_i}^p(e_\ell){\omega_p}^j(e_k)e_j - e_\ell({\omega_i}^p(e_k))e_p - {\omega_i}^p(e_k){\omega_p}^j(e_\ell)e_j - {\omega_i}^p([e_k,e_\ell])e_p \\ &= d{\omega_i}^p(e_k,e_\ell)e_p + {\omega_i}^p(e_\ell){\omega_p}^j(e_k)e_j - {\omega_i}^p(e_k){\omega_p}^j(e_\ell)e_j. \end{aligned}\](2.3)
Now notice that
\[\begin{aligned} {\omega_i}^p\wedge{\omega_p}^j(e_\ell,e_k) &= {\omega_i}^p\otimes{\omega_p}^j(e_\ell,e_k) - {\omega_i}^p\otimes{\omega_p}^j(e_k,e_\ell) \\ &= {\omega_i}^p(e_\ell){\omega_p}^j(e_k) - {\omega_i}^p(e_k){\omega_p}^j(e_\ell). \end{aligned}\]
Here we are using the determinant convention for the wedge product. So (2.3) above becomes
\[\begin{aligned} R(e_k,e_\ell)e_i &= d{\omega_i}^p(e_k,e_\ell)e_p + {\omega_i}^p(e_\ell){\omega_p}^j(e_k)e_j - {\omega_i}^p(e_k){\omega_p}^j(e_\ell)e_j \\ &= d{\omega_i}^j(e_k,e_\ell)e_j - ({\omega_i}^p\wedge\omega{_p^j})(e_k,e_\ell)e_j, \end{aligned}\]
where in the last line we have renamed the dummy variable from \(p\) to \(j\) for the first term and for the second term we have swapped the order of the arguments going into the wedge product (hence the change in sign).
So we have that
\[ {R_{k\ell i}}^j = d{\omega_i}^j - {\omega_i}^p\wedge{\omega_p}^j. \]
So finally we have that
\[\begin{aligned} {\Omega_i}^j(e_p,e_q) &= \frac{1}{2}{R_{k\ell i}}^j\omega^k\wedge\omega^\ell(e_p,e_q) \\ &= \frac{1}{2}{R_{k\ell i}}^j(\delta^k_p\delta^\ell_q - \delta^k_q\delta^\ell_p) \\ &= \frac{1}{2}\left({R_{k\ell i}}^j - {R_{\ell ki}}^j\right) \\ &= {R_{k\ell i}}^j = (d{\omega_i}^j - {\omega_i}^k\wedge{\omega_k}^j)(e_p,e_q), \end{aligned}\]
as required.