= Parallel transport preserves a fibre metric
{title2=$\langle P_\gamma a,P_\gamma b\rangle=\langle a,b\rangle$}
For a <metric connection> on a <vector bundle>, the <covariant derivative along a curve> satisfies $(d/dt)\langle u,v\rangle=\langle D_tu,v\rangle+\langle u,D_tv\rangle$. If both sections are parallel, the right side vanishes. Thus <parallel transport> preserves the fibre <inner product>, not merely vector lengths, and is a linear isometry between the endpoint fibres.
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