Here is a short question which has been bugging me for a long time: In many textbooks, the components of the Riemann curvature in local coordinates/abstract index notation are defined as follows: If $\partial_1, \dots, \partial_n$ is a basis of some tangent space with dual basis $dx^1, \dots, dx^n$, then we set $$R^i{}_{jkl} = dx^i(R(\partial_k, \partial_l)\partial_j).$$ I never managed to find a good reason for this. Why do they place the $j$-index after the $i$-index?
The version which seems more logical to me, is the following. Set $$R_{klj}{}^i = dx^i(R(\partial_k, \partial_l)\partial_j).$$ In particular we then have $$R(\partial_k, \partial_l) \partial_j = \sum_{i=1}^n R_{klj}{}^i \partial_i$$ and $$\langle R(\partial_k, \partial_l) \partial_j, \partial_i\rangle = \sum_{m=1}^n R_{klj}{}^m g_{mi} = R_{klji},$$ so everything makes sense and is easy to remember.
What would move anyone to do things differently from this? Is there a good explanation for the other indexing?
Thanks.
Ok, the answer is "because of Cartan's moving frames". In terms of a moving (dual) frame $\theta^1, \dots, \theta^n$, we have connection one-forms $\omega^i{}_j$ determined by $$d\theta^i = \sum_j\omega^i{}_j\wedge \theta^j$$ and the curvature two-form $\Omega^i{}_j$ determined by $$\Omega^i{}_j = d\omega^i{}_j + \sum_m \omega^i{}_m\wedge \omega^m{}_j.$$ In terms of $\Omega^i{}_j$, we then have $$\sum_\ell R^\ell{}_{kij}E_\ell = R(E_i,E_j)E_k = \sum_{\ell,t}\Omega^\ell{}_t(E_i, E_j) \theta^t(E_k)E_\ell = \sum_\ell \Omega^\ell{}_k(E_i,E_j)E_\ell.$$ So $\Omega^\ell{}_k = \sum_{i<j} R^\ell{}_{kij} \theta^i \wedge \theta^j$ makes perfect sense from the moving frames point of view.