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I am trying to solve this exercise from Lee's Riemannian Manifolds, but am getting stuck. The problem is: let $\{E_i\}$ be a local frame with dual coframe $\{ \varphi_i\}$. Show that there exists a unique matrix of $1$-forms $\omega_i^j$ such that $\nabla_X (E_i)=\omega^j_i(X)E_j$

I began by writing $X=X^j E_j\Rightarrow \nabla_X E_i=X^j\nabla_{E_j} E_i=X^j \Gamma_{ji}^k E_k$ and then letting $\omega_i^j(X)=X^j \Gamma_{ji}^k$. However, I don't think that this is fully answering the question...maybe it is, but I'm a bit confused because I did not use the dual coframe at all, and for the next problem I will have to show that $d\varphi^j = \varphi^i\wedge\omega_i^j + \tau^j$ where $\tau$ is the torsion $2$-form defined by the torsion tensor. I would think that my answer for this question should help me in answering the next one.

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  • $\begingroup$ You'll use the dual coframe if you write $\omega_i^j$ and not $\omega_i^j(X)$. $\endgroup$ – Ted Shifrin Jan 23 '17 at 7:59

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