= Connection 1-form
{title2=$\omega^a{}_b$}
{wiki}
A <connection 1-form> records how a chosen <local frame> changes under a <covariant derivative>. If $e_b$ is the <local frame> and $\theta^a$ its dual <coframe>, then
$$
\nabla e_b=\omega^a{}_b\otimes e_a.
$$
Thus evaluating $\omega^a{}_b$ on a <tangent vector> gives the mixing of vectors of the <local frame> under differentiation in that direction. The coefficients depend on the chosen <local frame>; their <curvature 2-form> describes the geometrical obstruction to locally consistent <parallel transport> around a loop.
For the <Levi-Civita connection>, the <torsion tensor> vanishes, giving <Cartan's first structure equation>
$$
d\theta^a+\omega^a{}_b\wedge\theta^b=0.
$$
In an <orthonormal frame>, <metric compatibility> also gives $\omega_{ab}=-\omega_{ba}$ after lowering the first index with the <metric tensor> in that <local frame>. In a general <local frame> the metric components need not be constant, so this simple antisymmetry is not an unrestricted statement about every connection matrix.
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