Metric connection

A connection on a Riemannian vector bundle is metric-compatible whenEquivalently, its parallel transport maps are isometries.

The unique torsion-free metric connection satisfiesThis formula both proves uniqueness of the Levi-Civita connection and computes it from the metric and Lie bracket of vector fields.

A connection on a complex vector bundle with Hermitian metric $h$ is unitary whenIn a unitary local frame its connection matrix is skew-Hermitian, and so is its curvature matrix.