I have seen it mentioned in various places that the Lie algebra of the diffeomorphism group of a manifold M is identifiable with the Lie algebra of all vector fields on M, but I have not found a demonstration of this. I can show that the map(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \rho: Lie(Diff(M)) \to Vect(M), ~~~ \rho(X)_p = \frac{d}{dt}\vert_{t=0} \exp(tX) \cdot p[/tex]

is surjective but am having difficulty showing injectivity. Here Vect(M) is vector fields on M, p is in M and . denotes the action of Diff(M) on M. Is this even the right way to see the correspondence between Lie(Diff(M)) and Vect(M)?

Thanks.

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# Lie algebra of the diffeomorphism group of a manifold.

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