I'm working through a Differential Geometry text. The author makes a statement I'm having a hard time understanding the validity of.

He defines a curve in $\mathbb{R}^3$ as a diffentiable function $\alpha: I \rightarrow \mathbb{R}^3$ from an open interval $I$ into $\mathbb{R}^3$.

Then he says "since a curve $\alpha: I \rightarrow \mathbb{R}^3$ is a function, it makes sense to say that $\alpha$ is one-to-one."

I'm struggling to see why this is the case...are self-intersecting curves non-differentiable and therefore ruled out? Or am I just over-thinking this? Under-thinking?

  • $\begingroup$ Where, in what text? $\endgroup$ – Jonas Meyer Jan 29 '15 at 7:56
  • $\begingroup$ I'm reading this straight out of O'neill's Elementary Differential Geometry. On page 16 he defines a curve and on page 21 he makes that statement. I'm starting to think this is a property of all such differentiable curves. $\endgroup$ – Dmitri Valentine Jan 29 '15 at 7:59

Jonas Meyer's reading is right: O'Neill is talking about injectivity as a property that a curve could possibly possess, but may not.

Admittedly, the wording is not the clearest, but I think the fuller context helps:

"Since a curve $\alpha \colon I \to \mathbb{R}^3$ is a function, it makes sense to say that $\alpha$ is one-to-one; that is, $\alpha(t) = \alpha(t_1)$ only if $t = t_1$. Another special property of curves is periodicity: A curve $\alpha \colon \mathbb{R} \to \mathbb{R}^3$ is periodic if there is a number $p > 0$ such that $\alpha(t+p) = \alpha(t)$ for all $t$ -- and the smallest such number $p$ is then called the period of $\alpha$."

Is O'Neill really saying that all curves have the "special property" of periodicity, or simply that it's a special property that some curves might have? (He obviously means the latter.)

(And as an aside, by the way, periodic curves are never injective.)

  • $\begingroup$ Nor are closed curves parametrized by $[a,b]$ with $\alpha(a)=\alpha(b)$. :) $\endgroup$ – Ted Shifrin Jan 29 '15 at 14:11

It probably just means that it is a property the function might have, or might not have, but at least the curve, being a function, lies in the category of things for which it could make sense to determine one way or the other whether they're injective. It is to make sure you have clear that it is a function, as opposed to a set of points.

If $n$ is a natural number, it makes sense to say that $n$ is odd, but if $n=16$, it is also false to say it. It is not like saying $\pi$ is odd which is just nonsense instead of wrong. Likewise, you can say a curve is injective (or not), but for the set of points traced out by the curve (i.e. the range) it would not make sense to say it is injective.

Added: From a preview in Google Books I got more context. Right after that, in the same paragraph, it says "Another special property of curves is periodicity...". This reinforces that it is just pointing out special properties a curve might or might not have.


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