Let $M\_1,M\_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M\_1)\cong \mathfrak{X}(M\_2)$ as Lie algebras.



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> **Question.** Are $M\_1$ and $M\_2$ diffeomorphic?
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This seems like a basic question but I did not find anything on Google.



Does the Lie algebra of vector fields $\mathfrak{X}(M)$ determine the diffeomorphism class of a manifold $M$?
The answer is *yes*, and the assumptions "simply connected" and "compact" are actually unnecessary.


In fact, it is possible to reconstruct any smooth manifold $M$, up to diffeomorphisms, by using the subalgebra $\mathfrak{X}\_0(M)$ of vector fields with compact support. See


*Shanks, M. E.; Pursell, Lyle E.*, [**The Lie algebra of a smooth manifold**](http://dx.doi.org/10.2307/2031961), Proc. Am. Math. Soc. 5, 468-472 (1954). [ZBL0055.42105](https://zbmath.org/?q=an:0055.42105), [JSTOR](https://www.jstor.org/stable/2031961#metadata_info_tab_contents). 


