Recovering a manifold from its Lie algebra of vector fields

Question

Associated to every manifold is a Lie Algebra of vector fields, where we take the collection of vector fields and equip it with the Lie Bracket.

Is it true that given the Lie Algebra of vector fields, we can (roughly) recover the manifold? I know that there is a correspondence between Lie Algebras and Lie Groups, but recovering the manifold given the Lie Algebra of vector fields feels like a different construction. Naively, it seems like the Lie Algebra of vector fields is 'not enough information' to recover the manifold (but it is enough to recover a Lie Group roughly because a Lie Group has a group operation that sort of 'makes all the points look locally similar'?)

Your idea of why the Lie algebra recovers the group is quite off, really. You should browse a proof of Lie's theorem! — Mariano Suárez-Álvarez, Feb 17 '18 at 21:31

score 4 · Accepted Answer · answered Feb 17 '18 at 21:34

It is a theorem of Janusz Grabowski that the algebraic structure of the Lie algebra of smooth vector fields on a manifold is enough to recover the manifold. The paper is

Grabowski, Janusz. (1978). Isomorphisms and ideals of the Lie algebras of vector fields. Inventiones Mathematicae. 50. 13-33. 10.1007/BF01406466.

and googling should find the full text for you.

Hm. Looking back on the paper it appears that I misremembered who originally proved this, but Grabowski's paper has the references, so that should not be too bad.

Recovering a manifold from its Lie algebra of vector fields

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