 Local extension of geodesic velocity

The velocity of a geodesic can be extended to one smooth ambient vector field on a sufficiently short neighbourhood of any parameter value. If its initial velocity is zero, uniqueness of the geodesic equation makes it locally constant, and the zero field works. Otherwise one coordinate has nonzero velocity; the inverse function theorem supplies coordinates straightening the small embedded arc to $(t,0,\dots ,0)$. The coordinate field $\partial /\partial t$ is the required extension. Shrinking the interval is essential: arbitrary fields along self-intersecting curves need not have one ambient extension.