Vector field along a curve ($V\in \Gamma ({\gamma }^{\ast }TM)$)

A vector field along a curve $\gamma :I\to M$ is a smooth section of a vector bundle with $V(t)\in T_{\gamma (t)}M$. Its values may differ at distinct times mapping to the same base point. The covariant derivative along a curve differentiates its time-dependent coefficients using the pulled-back connection on a vector bundle; it does not require extending it to one ambient vector field.