Among the smooth 1-manifolds (with or without boundary) which embed into $\mathbb{R}^2$, which ones can be represented by a single parametrization $z = (x,y) = f(t)$, for $t \in I$, where $I$ is an interval (not necessarily open or closed), and $f$ is smooth (i.e. infinitely differentiable)?
The reason I ask is that in my reading, I've come across two definitions for "line integral" (which I'm sure turn out to be equivalent):
The first is the standard definition given in most multivariable calculus and complex analysis classes, which relies on defining a "curve" as a (sufficiently nice) function (or image set) $z = \gamma(t)$. That is: $\int_\gamma f(z) dz = \int_a^b f(g(t))g'(t) dt$.
The second is the more high-powered definition involving 1-manifolds and differential 1-forms.
So my question is not really about whether or not these two definitions are equivalent per se, but rather about how much generality is lost by looking at only the "special" 1-manifolds which admit representations as $z = \gamma(t)$.
(My own thoughts: (1) Do these 1-manifolds end up being the 1-manifolds with an atlas consisting of one chart? I'm thinking not, because the circle cannot be given an atlas with one chart, yet can still be parametrized by a single function. (2) Does the Implicit Function Theorem have any role to play?)