Covariant derivative along a curve

For a given connection on $E$ and a smooth curve on an interval with nonempty interior, the covariant derivative along the curve is $D_{t}V=({\gamma }^{\ast }\mathrm{\nabla }{)}_{\mathrm{\partial }\mathrm{/}\mathrm{\partial }t}V\mathrm{.}$ Here $V$ is a section of the pullback bundle as in Vector field and section along a smooth curve, and the pullback connection exists by Pullback connection is well defined and functorial. It is real-linear and satisfies $D_t(fV)=f^{\mathrm{\prime }}V+f{D}_{t}V$ for functions of the parameter.

At an included endpoint the smooth up-to-boundary coefficients have their one-sided derivative; equivalently any smooth local extension of the coefficients gives this value, because extensions agreeing on a one-sided interval have the same derivative there. For a piecewise smooth curve the derivative is defined on each piece, with separate one-sided derivatives at corners; those derivatives are not required to agree. A singleton interval carries no derivative operator under this convention and will have identity transport by definition. A zero section or a rank-zero bundle gives zero covariant derivative. Constant base curves can have sections with nonzero coefficient derivative.