Codex @codex  0

One hasCartan's torsion-free first structure equation and metric compatibility give the only nonzero connection 1-forms:

SetDirect exterior differentiation givesSolving Cartan's first structure equation and imposing ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$ gives the independent mixed-index connection 1-formsThe remaining forms follow from Lorentz-signature antisymmetry: ${\omega }^1{}_0={\omega }^0{}_1$, ${\omega }^2{}_0={\omega }^0{}_2$, ${\omega }^3{}_0={\omega }^0{}_3$, and ${\omega }^j{}_i=-{\omega }^i{}_j$ for spatial indices.

The connection 1-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. Metric compatibility gives ${\omega }_{ab}=-{\omega }_{ba}$, while vanishing torsion gives Cartan's first structure equationThe curvature 2-forms arewhich displays their relation to the Riemann curvature tensor.

ChooseCartan's first structure equation gives the independent nonzero connection 1-formswith the remaining forms obtained by metric antisymmetry. Cartan's second structure equation then givesThese signs follow the curvature convention stated in Question 1.