# The definition of lie group action on vector field

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?

• yes and yes, diffeomorphisms act on vector fields Oct 24, 2021 at 12:57
• the space of vector fields* Oct 24, 2021 at 14:42
• @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.
– jw_
Oct 25, 2021 at 12:52
• 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 Oct 25, 2021 at 14:11
• in the wiki article when they say "vector field" they mean what you mean when you say "tangent vector field" Oct 25, 2021 at 14:15