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Under the "The Lie algebra associated with a Lie group" section in the article Lie_group of Wikipedia, it mentions "If G is any group acting smoothly on the manifold M, then it acts on the vector fields" but how does G act on vector fields?

Based on the Lie_group–Lie_algebra_correspondence article, I think for tangent vector field the definition is:

$G$ is Lie group that acts smoothly on smooth manifold $M$,$m$ is point in $M$, $X\in\Gamma (TM)$ then $G$ act on $\Gamma (TM)$ by:

$$ (g\cdot X)_m=dg_{g^{-1}\cdot m}(X_{g^{-1}\cdot m}) $$

where $dg_{g^{-1}\cdot m}$ is the differential of the map $m \to g\cdot m$ at $g^{-1}\cdot m$

Is that correct? Can it be defined for general vector fields?

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  • $\begingroup$ yes and yes, diffeomorphisms act on vector fields $\endgroup$ Oct 24, 2021 at 12:57
  • $\begingroup$ the space of vector fields* $\endgroup$ Oct 24, 2021 at 14:42
  • $\begingroup$ @Timkinsella Is there some reference about how is it defined for general vector fields? The way for tangent vector field seems not easy to be extended to general vector field. $\endgroup$
    – jw_
    Oct 25, 2021 at 12:52
  • $\begingroup$ youve given the correct def for vector fields. maybe youre taking about the restriction of a vector field to a submanifold, so that vectors do not necessarily lie in the tangent space to the submanifold. in that case a diffeomorphism of the submanifold probably wont act of the space of "vector fields" in a sensible way $\endgroup$ Oct 25, 2021 at 14:11
  • $\begingroup$ in the wiki article when they say "vector field" they mean what you mean when you say "tangent vector field" $\endgroup$ Oct 25, 2021 at 14:15

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