Question

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?

Answer 1

The $1$-forms $\omega^i_j$ define an affine connection on the tangent bundle, and the first structure equation gives the formula for the torsion tensor. It is equivalent to the equation $$ \nabla_X Y - \nabla_Y X - [X,Y] = \tau(X,Y), $$ where the connection $\nabla$ is defined using the $1$-forms $\omega^i_j$. I think of these equations as describing what happens when you parallel transport a vector along a 1-parameter family of curves with respect to the connection.

The second structure equation is equivalent to $$ [\nabla_X, \nabla_Y]Z - \nabla_{[X,Y]}Z = R(X,Y)Z $$ and defines the curvature tensor. I personally find these equations to be inscrutable, but if you study how families of geodesics vary, then the curvature tensor arises naturally as part of the Jacobi equation. This gives it a nice geometric interpretation.

Answer 2

I think you may find an answer in Differential Geometry by Sharpe.