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I read in several books (Do Carmo, Riemannian Geometry or John M. Lee, Smooth manifolds) that a vector field $X$ on a smooth manifold $M$ is a mapping which associates to each point $p \in M$ a tangent vector $X(p) \in \mathrm{T}_{p}M$. With a more general point of view, a vector field is a section of the tangent bundle $TM$.

Given a differentiable curve $c : I \subset \mathbb{R} \to M$, the derivative $\dot{c}$ of the curve is sometimes called velocity field. However, I fail to see how $\dot{c}$ is a vector field of $M$. What am I missing ?

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    $\begingroup$ It's not a vector fields on $M$. It's a vector field along a curve. See the answer here $\endgroup$ – user99914 Jul 20 '16 at 18:11
  • $\begingroup$ There is a nice expository answer that might help you here. $\endgroup$ – ITA Jul 20 '16 at 18:17
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The velocity field $\dot{c}$ is not a vector field on the manifold, but it is a section of the pullback of the tangent bundle by $c$, usually denoted by $c^*TM$. This basically means that it is a "vector field defined on the image of $c$" (and then, not really, as it could have more than one value at a point of the image, if $c$ crosses itself).

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