I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors.
I need to compute $\nabla_{\dot\gamma}{\dot\gamma}$. Apparently, the Christoffel symbols in that case are computed with :
$\omega^i{}_{k\ell}=\frac{1}{2}g^{im} \left( g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m} + c_{mk\ell}+c_{m\ell k} - c_{k\ell m} \right)\,$
where $c_{k\ell m}=g_{mp} {c_{k\ell}}^p\ $ are the commutation coefficients of the basis; that is, $[\mathbf{u}_k,\mathbf{u}_\ell] = c_{k\ell}{}^m \mathbf{u}_m\,\ $
(wiki)
I have several questions regarding the commutation coefficients:
- I read that the commutator is defined by $[X,Y]=\nabla_X Y - \nabla_Y X$ : isn't it a catch 22? how is it possible to compute that commutator, knowing that the connection $\nabla$ is obtained by the Christoffel symbols that uses the commutator that uses the connection that uses [...] ?
- For one of my applications, my tangent space is made of symmetric matrices (and the manifold has a fancy metric $g$) : in that case, is the commutator only defined by $[X,Y]=X*Y-Y*X$ with $*$ the standard matrix multiplication ?
- For another application, my tangent space are vectors in $R^N$ and my space is Euclidean. Of course, in that case there are probably easier ways to handle that, but I'd still like to understand what would the commutator be in that case.
- In the definition $[\mathbf{u}_k,\mathbf{u}_\ell] = c_{k\ell}{}^m \mathbf{u}_m\,\ $ , is-it necessary to solve a linear system to get the coefficients $c_{k\ell}{}^m$ or can I just project $[\mathbf{u}_k,\mathbf{u}_\ell]$ on $\mathbf{u}_m$ using the metric $g$ (my basis is not orthogonal, so I guess the linear system is needed?).
Thanks!