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Let $M$ be a smooth manifold. We know that the $C^\infty (M)$-module $\Omega^1 (M)$ is finitely generated, i.e. there exists $1$-forms $\{\alpha_1, \ldots, \alpha_k \}$ such that for any $1$-form $\omega$, we can write $\omega = \sum_{i=1}^k f_i \alpha_i$ for some $f_i \in C^\infty (M)$.

I'm wondering if the $\alpha_i$ can be chosen to be closed, or, furthermore, exact.

I'm guessing there must exist a counterexample, as I haven't seen this result in any of the standard textbooks or online sets of notes, and it might make computations a little too easy. I've been toying around with this idea for a while but haven't gotten any leads in either direction, except that this is trivially true in $\mathbb{R}^n$.

Have any of you seen this result or know a counterexample?

Paul Cusson
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1 Answers1

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Unless I'm mistaken, it is always possible to find a generating set of exact 1-forms. Here's a rough sketch of one possible construction:

  • Show that $M$ is covered by finitely many coordinte charts $(U_1,\varphi_1),\cdots,(U_N,\varphi_N)$ (see here, for instance).
  • Show there are open subsets $U_1',\cdots,U_N'$ with $\overline{U_i'}\subseteq U_i$ and $M=\bigcup_iU_i'$.
  • Show that there are smooth functions $\psi_i:M\to\mathbb{R}$ with $\psi_i|_{U_i'}=1$ and $\operatorname{supp}(\psi_i)\subset U_i$.
  • Show that $\Omega^1M$ is generated by $d(\psi_ix_i^j)$ where $x_i^j:U_i\to\mathbb{R}$ are the coordinate functions of $\varphi_i$.
Kajelad
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