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Let $γ : I→R^3$ be a regular parametrization of a curve C. If asked what a vector field on C is I would perhaps answer like this:

1) "a smooth function $v$ associating to any point $γ(t)$ of C an element of the tangent space of $R^3$ at $γ(t)$".

An alternative answer: 2) "a smooth function $f$ associating to any t $∈$ I an element of the tangent space of $R^3$ at $γ(t)$".

My question: is the second answer technically wrong? Thanks.

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1 Answer 1

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In fact, I'd prefer the second definition - it's what Lee uses in his Riemannian Manifolds book:

enter image description here

This allows a curve to intersect itself at a point in different directions while being able to record this fact in the velocity vector. I think if one takes seriously the idea that "curve" refers to the map $\gamma:I\to M$, and not its image $\gamma(I)\subset M$, then this is the right definition.

However, I'd say your first definition is appropriate if you're talking about vector fields on the subset $\gamma(I)$, having forgotten about $\gamma$. From Lee's Smooth Manifolds book:

enter image description here

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    $\begingroup$ The point about self intersections is a nice one! $\endgroup$ May 22, 2015 at 16:04
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    $\begingroup$ Good answer. It really depends on whether you're thinking of "a curve" as a parametrized curve (i.e. a function from an interval into some space) or as a set of points (typically the image of some parametrized curve). To work with "vector fields along curves," it's usually most useful to work with the former interpretation, in which case the OP's second definition is the right one. $\endgroup$
    – Jack Lee
    May 22, 2015 at 16:25
  • $\begingroup$ @Jack Lee : And we use the differential structure of the tangent bundle of the manifold to determine smoothness of the assignment? $\endgroup$
    – MSIS
    Feb 1 at 0:14
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    $\begingroup$ @MSIS: Yes, that's right. $\endgroup$
    – Jack Lee
    Feb 1 at 0:17
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    $\begingroup$ @MSIS: Yes, you should ask this as a separate question. $\endgroup$
    – Jack Lee
    Feb 1 at 0:29

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