Let's suppose that $M$ is a *connected* $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely $\mathbb R$ and $S^1$. These are clearly both orientable, and one "high-powered" way to show this it is to show that the top exterior power $\Lambda^n (T^*M)$ of its cotangent bundle, which is the cotangent bundle itself since $n=1$, is trivial, because each of these manifolds is a Lie group. Of course, if you already know that the only ones are $\mathbb R^1$ and $S^1$, you can also directly show from the "oriented cover" definition of orientability that they are indeed both orientable.

I am teaching a third-year curves and surfaces course, and what I am looking for is the following. Suppose you *don't* know what all the connected $1$-dimensional smooth manifolds are. (In my course we can suppose they are embedded submanifolds of $\mathbb R^n$, but that's not very important.) How can one show, using elementary ideas, that any connected curve has to be orientable?

I believe that one way to do this is the following: show that any connected curve can be expressed as the image of a *single* regular parametrized curve. Once we have this, we're done. It is clear that the argument should use the fact that we can always "reparametrize by arc-length", and by measuring the arc-length from a fixed point in a fixed initial direction, one either gets a diffeomorphism with $\mathbb R$ or with $S^1$, depending on whether or not the curve is closed. Is there an easy way to justify this to students in a curves and surfaces class? Is there an easier argument that I haven't noticed?

The reason I am thinking about this is because all the undergraduate curves and surfaces texts spend a lot of time explaining why surfaces need not be orientable, but never discuss why orientability is never an issue for curves. They also spend a lot of time talking about "covering a surface with multiple coordinate charts" but never discuss (except for $S^1$, sometimes) the need to cover a curve by more than one chart. It would be nice to be able to give my students an easy (but rigorous) explanation of the orientability of any connected curve.