If $(M,\langle\cdot,\cdot\rangle)$ is a pseudo-Riemannian manifold and $\nabla$ denotes the Levi-Civita connection of the metric, we can define the connection $1$-forms and curvature $2$-forms relative to any frame $(E_1,\ldots,E_n)$, not necessarily orthonormal, by $$\nabla_XE_j = \sum_i \omega^i_{\;j}(X)E_i \quad \mbox{and}\quad R(X,Y)E_j = \sum_i \Omega^i_{\;j}(X,Y)E_i,$$ok. Then, if $(\theta^1,\ldots,\theta^n)$ denotes the dual coframe, one proves the structure equations $${\rm d}\theta^i = \sum_{j} \theta^j \wedge \omega^i_{\;j}\quad\mbox{and}\quad\Omega^i_{\;j} = {\rm d}\omega^i_{\;j}+\sum_k \omega^i_{\;k}\wedge \omega^k_{\;j}.$$I recall seeing somewhere that these structure equations actually characterize the connection and curvature forms, but I don't recall where. So I'd like a reference or proof for this result. More precisely:
If $\widetilde{\omega}^i_{\;j}$ and $\widetilde{\Omega}^i_{\;j}$ satisfy$${\rm d}\theta^i = \sum_{j} \theta^j \wedge \widetilde{\omega}^i_{\;j}\quad\mbox{and}\quad\widetilde{\Omega}^i_{\;j} = {\rm d}\widetilde{\omega}^i_{\;j}+\sum_k \widetilde{\omega}^i_{\;k}\wedge \widetilde{\omega}^k_{\;j},$$then $\widetilde{\omega}^i_{\;j} = \omega^i_{\;j}$ and $\widetilde{\Omega}^i_{\;j} = \Omega^i_{\;j}$?
Thanks.
Edit: inspired by the comments... do we get the desired characterization adding the assumption that
$${\rm d}g_{ij}=\sum_k (g_{ik}\omega^k_{\;j}+g_{jk}\omega^k_{\;i})$$
?
I use the conventions
- $R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z$;
- $\alpha \wedge \beta = \frac{(k+\ell)!}{k!\ell!}{\rm Alt}(\alpha\otimes\beta)$, for $\alpha \in \Omega^k(M)$ and $\beta \in \Omega^\ell(M)$;
- ${\rm Alt}\gamma= \frac{1}{r!}\sum_{\sigma \in S_r}(-1)^{|\sigma|} \gamma^\sigma$, where $\gamma \in \Omega^r(M)$.