A curve on a manifold \(M\) is a continuous map \(\gamma\colon I\to M\), where \(I\subset \mathbb{R}\) is an interval.
In the case \(I\) is compact, we call \(\gamma\) curve segment. It is smooth if \(M\) is smooth and \(\gamma\) has a smooth extension on a neighbourhood at each endpoint of \(I\).