= 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 $(t,0,\ldots,0)$. 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.
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