I saw papers saying something like "let $\gamma:S^1 \times [0,T] \to \mathbb{R}^2$ parametrise a curve. The second interval above just makes it time dependent, but why parametrise (for fixed time) the curve on S^1, the unit circle? I think it's to make it closed but what is confusing is that in papers they write $\gamma$ a function of a real variable $u$ and $t$, so as $\gamma(u, t)$ so how can the domain of the first variable $u$ be the unit circle? Should it not be $\gamma(u, v, t)$ for $u^2 + v^2 = 1$?
Thanks