Metric connection
= Metric connection
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A connection on a Riemannian vector bundle is metric-compatible when
$$
d\langle s,t\rangle
=\langle\nabla s,t\rangle+\langle s,\nabla t\rangle.
$$
Equivalently, its parallel transport maps are isometries.
= Metric-compatible connection
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= Orthogonal connection
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