Parallel section along a curve

A section $V$ along a smooth curve is parallel if $D_{t}V=0$. By Local frame formula for covariant differentiation along a curve, in a pulled-back frame this means $v^{\mathrm{\prime }}(t)=-{\omega }_{\gamma (t)}(\dot{\gamma }(t))v(t)\mathrm{.}$ For a piecewise smooth curve, parallel means continuous on the full parameter interval and parallel on each smooth piece. One-sided derivatives are used at the endpoints of a piece; no equality of the two derivatives at a corner is required. On a singleton, every prescribed fibre vector is a parallel section by convention; on an empty interval the empty section is parallel.

The zero section is parallel, and along a constant curve the coefficients in a constant fibre frame are constant precisely for parallel sections. The definition does not assert that all local coefficients in arbitrary moving frames are constant.