Cartan structural equations

To deduce the Cartan structural equations in a coordinated frame we are going to use the definition of the Christoffel symbols (connection coefficients) and where we always are going to use the Einstein sum convention:

\abla_{\\partial_{i}}\\partial_{j}={\\Gamma^{s}}_{ij}\\partial_{s}

and the curvature tensor

R(X,Y)Z=\abla_{X}\abla_{Y}Z-\abla_{Y}\abla_{X}Z-\abla_{[X,Y]}Z

where $X, Y, Z$ are any three vector fields in a riemannian manifold $\\cal{M}$ with the Levi-Civita connection $\abla$ .

First, we define through the relation $\abla_{X}\\partial_{i}={\\omega^{s}}_{i}(X)\\partial_{s}$ a set of scalar function ${\\omega^{s}}_{i}$ which are easily to see that they actually are 1-forms. We observe that ${\\omega^{s}}_{i}(\\partial_{j})={\\Gamma^{s}}_{ij}$ .

They satisfy skew-symmetry rule: $\\omega_{si}=-\\omega_{is}$ ,which arises from the covariant constancy of the metric tensor $g_{kl}$ i.e.

$\\displaystyle 0$	$\\displaystyle=$	$\\displaystyle\abla_{X}g_{kl}$
	$\\displaystyle=$	$\\displaystyle\abla_{X}\\langle\\partial_{k},\\partial_{l}\\rangle$
	$\\displaystyle=$	$\\displaystyle\\langle\abla_{X}\\partial_{k},\\partial_{l}\\rangle+\\langle\\partial%_{k},\abla_{X}\\partial_{l}\\rangle$
	$\\displaystyle=$	$\\displaystyle\\langle{\\omega^{s}}_{k}(X)\\partial_{s},\\partial_{l}\\rangle+%\\langle\\partial_{k},{\\omega^{s}}_{l}(X)\\partial_{s}\\rangle$
	$\\displaystyle=$	$\\displaystyle{\\omega^{s}}_{k}(X)g_{sl}+{\\omega^{s}}_{l}(X)g_{ks}$
$\\displaystyle 0$	$\\displaystyle=$	$\\displaystyle\\omega_{lk}(X)+\\omega_{kl}(X)$

that last equation is valid for each vector field $X$ , then $\\omega_{lk}=-\\omega_{kl}$ .

Next we define through the relation

R(X,Y)\\partial_{i}={\\Omega^{s}}_{i}(X,Y)\\partial_{s}

the scalars ${\\Omega^{s}}_{i}(X,Y)$ which are the so called connection 2-forms.That they are really 2-forms is an easy caligraphic exercise.

Now by the use of the Riemann curvature tensor above we see

$\\displaystyle R(X,Y)\\partial_{i}$	$\\displaystyle=$	$\\displaystyle\abla_{X}\abla_{Y}\\partial_{i}-\abla_{Y}\abla_{X}\\partial_{i}%-\abla_{[X,Y]}\\partial_{i}$
	$\\displaystyle=$	$\\displaystyle\abla_{X}({\\omega^{s}}_{i}(Y)\\partial_{s})-\abla_{Y}({\\omega^{s%}}_{i}(X)\\partial_{s})-{\\omega^{s}}_{i}[X,Y]\\partial_{s}$
	$\\displaystyle=$	$\\displaystyle X({\\omega^{s}}_{i}(Y))\\partial_{s}+{\\omega^{s}}_{i}(Y)\abla_{X}%\\partial_{s}-Y({\\omega^{s}}_{i}(X)\\partial_{s}-{\\omega^{s}}_{i}(X)\abla_{Y}%\\partial_{s}-{\\omega^{s}}_{i}[X,Y]\\partial_{s}$
	$\\displaystyle=$	$\\displaystyle X({\\omega^{s}}_{i}(Y))\\partial_{s}+{\\omega^{s}}_{i}(Y){\\omega^{t%}}_{s}(X)\\partial_{t}-Y({\\omega^{s}}_{i}(X)\\partial_{s}-{\\omega^{s}}_{i}(X){%\\omega^{t}}_{s}(Y)\\partial_{t}-{\\omega^{s}}_{i}[X,Y]\\partial_{s}$
	$\\displaystyle=$	$\\displaystyle[X({\\omega^{s}}_{i}(Y))+{\\omega^{t}}_{i}(Y){\\omega^{s}}_{t}(X)-Y(%{\\omega^{s}}_{i}(X))-{\\omega^{t}}_{i}(X){\\omega^{s}}_{t}(Y)-{\\omega^{s}}_{i}[X%,Y]]\\partial_{s}$
$\\displaystyle{\\Omega^{s}}_{i}(X,Y)\\partial_{s}$	$\\displaystyle=$	$\\displaystyle[X({\\omega^{s}}_{i}(Y))-Y({\\omega^{s}}_{i}(X))-{\\omega^{s}}_{i}[X%,Y]+{\\omega^{s}}_{t}(X){\\omega^{t}}_{i}(Y)-{\\omega^{s}}_{t}(Y){\\omega^{t}}_{i}%(X)]\\partial_{s}$

In this last relation we recognize -in the first three terms- the exterior derivative of ${\\omega^{s}}_{i}$ evaluated at $(X,Y)$ i.e.

d{\\omega^{s}}_{i}(X,Y)=X({\\omega^{s}}_{i}(Y))-Y({\\omega^{s}}_{i}(X))-{\\omega^{%s}}_{i}[X,Y]

and in the last two terms its wedge product

{\\omega^{s}}_{t}\\wedge{\\omega^{t}}_{i}(X,Y)={\\omega^{s}}_{t}(X){\\omega^{t}}_{i%}(Y)-{\\omega^{s}}_{t}(Y){\\omega^{t}}_{i}(X)

all these for any two fields $X, Y$ . Hence

{\\Omega^{s}}_{i}=d{\\omega^{s}}_{i}+{\\omega^{s}}_{t}\\wedge{\\omega^{t}}_{i}

which is called the second Cartan structural equation for the coordinated frame field $\\partial_{i}$ .

More interesting things happen in an an-holonomic basis.