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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?

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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.

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    $\begingroup$ And for the torsion, the following MO question might be useful: mathoverflow.net/questions/20493/… $\endgroup$ – José Figueroa-O'Farrill Apr 16 '11 at 1:16
  • $\begingroup$ Of course, this is 100% correct, but I think there is also another point of view, perhaps closer to Cartan's methods, and is recursive. You start with with an adapted co-frame, apply d, then express the result in terms of the co-frame + something new, but with values in the Lie algebra. Now we have two new things to apply d to, namely the $\omega_{ij}$ and the $\tau_j$. We repeat expressing them in terms of "older" things. I am not sure how to formalize it, but it probably has to do with EDS and prolongation. $\endgroup$ – Malkoun Aug 8 '17 at 13:05
  • $\begingroup$ Malkoun, you're correct, and the process is in fact known as prolongation. It is in fact, how I originally learned how to derive the structure equations. But to me that's a formal algebraic description and not geometric. $\endgroup$ – Deane Yang Aug 8 '17 at 18:00
  • $\begingroup$ Let's say you have a Riemannian manifold $(M,g)$, and let $(\theta^i)$, for $1 \leq i \leq m$, with $m=\dim M$, be a smooth local orthonormal coframe. Applying $d$ to the coframe gives our first set of "invariants" (or perhaps I should write $O(m)$-invariants). That the first set of "invariants" is nothing but the Levi-Civita connection is the meaning of the first structure equations. $\endgroup$ – Malkoun Aug 22 '17 at 22:41
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    $\begingroup$ Malkoun, your understanding looks correct to me. $\endgroup$ – Deane Yang Aug 22 '17 at 23:26
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I think you may find an answer in Differential Geometry by Sharpe.

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