Exactness of $dx,dy$

Question

Let $dx,dy$ denote differential $1$-forms. It is easy to verify that they are closed. My question is:

Does there exist a space such that either $dx$ or $dy$ or both are exact?

(A counterexample in one dimension is $S^1$: here, $d\theta$ is not exact. It is not clear to me though whether in this case $dx$ is exact or not)

Answer 1

What's wrong with $\mathbb{R}^n$? The notation $dx$ isn't a coincidence; $dx$ is the differential of the coordinate function (zero-form) $x$ which takes a point $p=(p_1, p_2,...,p_n)$ to the number $p_1$.

$dx$ is exact on $S^1$ because the restriction to a subspace of an exact form is exact: If $\omega=df$ on $X$, than $\omega|_S = d(f|_S)$ and is exact on $S\subset X$.

Answer 2

If $M$ is a manifold and $(U, \varphi)$ is a coordinate chart with $\varphi = (x^1, \dots, x^n)$ then $dx^i$ is a one-form on $U$ and is exact as $dx^i = d(x^i)$; note, this one-form is only defined on $U$, not $M$. In $M$ has a global coordinate chart, which occurs if and only if $M$ is diffeomorphic to $\mathbb{R}^n$, then $dx^1, \dots, dx^n$ are exact one-forms on $M$.

The highlighted paragraphs address the cases where $dx^i$ is a global form on a manifold but isn't exact. It is slightly technical; if you don't really understand it, go to the final paragraph where I address the $S^1$ case.

Suppose now that $G$ is a group of diffeomorphisms of $M$ which act freely and properly discontinuously, then $M/G$ is a manifold. If $\omega$ is a form on $M$ and $g^*\omega = \omega$ for every $g \in G$, then $\omega$ descends to a form on $M/G$ which is often denoted by $\omega$ as well.

If $M = \mathbb{R}^n$ and $G = \mathbb{Z}^n$ (acting by translations), the quotient is an $n$-dimensional torus. As $dx^i$ is translation invariant, $g^*dx^i = dx^i$ so we obtain a one-form, which we again call $dx^i$, on the torus. Despite the fact that $dx^i$ appears to be written as the exterior derivative of a function, note that $dx^i$ is not exact because the coordinate function $x^i$ is not defined on the entire torus.

The simplest case of the above is when $n = 1$. Instead of using the coordinate $x$ on $\mathbb{R}$, I will use $\theta$ and then $S^1 = \mathbb{R}/2\pi\mathbb{Z}$; any point on $S^1$ is given by an angle $\theta$ up to an integer multiple of $2\pi$. The one form $d\theta$ on $\mathbb{R}$ descends to $S^1$ (where it is again called $d\theta$) but it isn't exact because the $\theta$ coordinate on $S^1$ given by angle cannot be defined on all of $S^1$ simultaneously (if you try to 'connect it up' you will have a point defined by both the angle $\theta_0$ and $\theta_0 + 2\pi$).