Does a curve (differentiable manifold of dim 1) always have a parametrization?

Question

I don't know much about this field, so this is a basic question. I think there are 2 similar basic concepts referring to curve:

• A $C^k$ differential curve, can be defined as a differentiable $C^k$ manifold of dimension 1. This is what one can read here: https://en.wikipedia.org/wiki/Curve#Differentiable_curve .

• A $C^k$-parametric curve is roughly a $C^k$ function $f$ from an interval $I$ of $\mathbb{R}$, to a normed vector space $E$. This concept is studied here https://en.wikipedia.org/wiki/Differentiable_curve, surprisingly I haven't seen this page mentioning the first definition despite its title.

Now I think it is obvious that the trajectory of a $C^k$-parametric curve (ie $f(I)$) is a $C^k$ differential curve, because one of the equivalent definitions of manifold says that, each point of it needs to have some local parametrization, so we can use the global parametrization provided by the parametric curve for it.

It naturally raises the converse question: If i am given a set of points that is a manifold of dimension 1, is it the trajectory of at least one parametric arc?

A more accurate question: if I have a $C^k$ differential curve, is it the trajectory of at least one $C^k$ parametric arc?

I thought that maybe one can use the local parametrizations provided by the manifold and somehow stick them together as a chain, but I feel like there is no guarantee that I can cover the whole curve, because maybe the paramatrizations are getting smaller and smaller...

Answer 1

When you say

Now I think it is obvious that the trajectory of a $C^k$-parametric curve (ie $f(I)$) is a $C^k$ differential curve...

this is in fact not correct. Consider for example a lemniscate.

The converse is not true either. Consider a manifold of dimension one with two connected components.