Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by $\Omega_i^j=\frac{1}{2}R_{klij}\phi^k \wedge \phi^l$.

We have the following identities also known as Cartan's first and second structure equations:

i) $d\phi^j=\phi^i \wedge \omega_i^j + \tau^j$ where $\tau^1,...,\tau^n$ are the torsion 2 forms.

ii) $\Omega_i^j=d\omega_i^j-\omega_i^k \wedge \omega_k^j$

I have two questions:

1)Is there a geometric meaning attached to these equations?

2) Why are these equations important and what are they useful for?