Cfrgtkky

According to the discussion in this thread, even though the spherical coordinate $theta$ on $S^1$ is a local function, $dtheta$ is a global form on $S^1$. How to prove this rigorously? The discussion also points out it is the pullback of $frac{-ydx+xdy}{x^2+y^2}$. Why?

My attempt:

For the first question: as hinted I probably want to show that two local form $dtheta$, and $dtheta+pi$ agrees on their overlaps. The point is that I want to evaluate $d(theta+pi)(frac{partial }{partialtheta})$. To do that take a function $f: mathbb R to mathbb R$, $d(theta+pi)(frac{partial }{partialtheta})(f) = frac{partial}{partialtheta}(f circ (theta+pi))$, where $fcirc(theta + pi)(e^{itheta}) = f(theta+pi)$. So $d(theta+pi)(frac{partial }{partialtheta})(f) = f'|_{theta+pi}$ as desired. Then $d(theta+pi)$ and $dtheta$ coincides on their overlap, thus we can extend $dtheta$ to a global form on $S^1$.

For the second question: Let $i: s^1 to mathbb R^2$ be the embedding. Then $i^{ast}(frac{-ydx+xdy}{x^2+y^2}) = frac{-ycirc i(dx circ i)+xcirc i(dy circ i)}{(xcirc i)^2+(ycirc i)^2}$. The key observation is that in spherical coordinate $xcirc i(theta) = costheta, y circ i(theta) = sintheta$.

Am I right in my reasonings?