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:

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

and the curvature tensor

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

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

First, we define through the relation $\nabla_{X}\partial_{i}={\omega^{s}}_{i}(X)\partial_{s}$ 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.

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

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

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

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.

Title	Cartan structural equations
Canonical name	CartanStructuralEquations
Date of creation	2013-03-22 17:35:46
Last modified on	2013-03-22 17:35:46
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	12
Author	juanman (12619)
Entry type	Result
Classification	msc 53A45
Classification	msc 58A12
Classification	msc 58A10