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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
    Commented Apr 16, 2011 at 1:16
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    $\begingroup$ 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
    Commented Aug 22, 2017 at 22:41
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    $\begingroup$ We then apply $d$ a second time, and get a second set of "invariants". That this second set of invariants can be broken in 2 parts, the first quadratic in the Levi-Civita connection and the second one nothing but the curvature of $g$, is the content of Cartan's second structure equation. $\endgroup$
    – Malkoun
    Commented Aug 22, 2017 at 22:41
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    $\begingroup$ Malkoun, your understanding looks correct to me. $\endgroup$
    – Deane Yang
    Commented Aug 22, 2017 at 23:26
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    $\begingroup$ Why is the second equation ${\textrm{hor d}}\omega=d\omega - \frac{1}{2}[\omega\wedge\omega]$ equivalent to $[\nabla_X, \nabla_Y]Z - \nabla_{[X,Y]}Z = R(X,Y)Z$? Is there a direct geometrical understanding? $\endgroup$
    – Alex
    Commented Nov 25, 2022 at 12:04
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I think you may find an answer in Differential Geometry by Sharpe.

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  • $\begingroup$ Also Richard Sharpe's pdf "An introduction to Cartan Geometries" $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2021 at 14:14
  • $\begingroup$ Could you kindly summarise it in a few sentences? The book is quite long. $\endgroup$
    – Alex
    Commented Nov 25, 2022 at 11:33

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