= Parallelism of the Riemannian volume form
{title2=$\nabla\omega_g=0$}
The <Levi-Civita connection> of an oriented <Riemannian manifold> preserves its <Riemannian volume form>. In a positively oriented orthonormal frame its connection matrix is skew-symmetric. Differentiating the wedge of the dual coframe gives minus the trace of that matrix times the <volume form>, which is zero. Equivalently <parallel transport> preserves the metric and orientation, so it preserves the normalized <volume form>.
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