Let's take the so-called Liouville one-form $\theta = \sum_i p_i dq^i$ (AKA Canonical one-form, Tautological one-form, Symplectic potential, etc.), defined on cotangent bundles.

Some contributions (e.g. this discussion) cite a difference between its "local" and "global" meaning: what's the difference between the two, and what is meant by "local" and "global"?

  • $\begingroup$ Theres a difference between its local and global description - is this what you mean? For example, the description that you have of the Liouville 1-form in your question is local as it uses coordinates and coordinates are a local description. $\endgroup$ – Mozibur Ullah Dec 20 '17 at 17:16
  • $\begingroup$ So @MoziburUllah the "global" description means "coordinates-free"? $\endgroup$ – Lo Scrondo Dec 20 '17 at 17:31
  • $\begingroup$ Usually, yes; I was questioning what you meant by 'meaning'? $\endgroup$ – Mozibur Ullah Dec 20 '17 at 17:46
  • $\begingroup$ About "meaning", I just used the term from here: link. $\endgroup$ – Lo Scrondo Dec 20 '17 at 18:29
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    $\begingroup$ "Local" is description in a local chart. "Global" is a description as a differential 1-form on the whole manifold. In your case, $\theta$ is a differential 1-form on a cotangent bundle $T^*Q$. By denoting $\pi:T^*Q\to Q$ the bundle projection, a global description (or definition) of $\theta$ would be $\theta|_p(v) := p(\pi_*(v))$ for $p\in T^*Q$, $v\in T_p T^*Q$. A "global" meaning of this 1-form, the tautological aspect, is that for $\alpha\in \Omega^1(Q)$ we have $\alpha^*\theta = \alpha$. $\endgroup$ – Noé AC Dec 31 '17 at 22:36

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