The Lie Bracket of Vector Fields
We know that any vector field can act as an endomorphism on the space of smooth functions on . What happens if we act by one vector field followed by another? To really make things explicit, let’s say that is a coordinate patch, so we can write
where is a coefficient function and measures how fast is changing as we increase the th coordinate function through . Now if we hit this with another vector field we find
At each point the field gives a vector , which acts as a derivation on the ring of smooth functions at . That is
Now, this is obviously an endomorphism on since it’s the composite of two endomorphisms. But it is not a vector field, since at a given point we don’t get a derivation of the ring of smooth functions at . Indeed, what happens if we give it the product of two functions?
We’ve got a bunch of terms left over at the end! But one thing is nice about it: the leftover terms are symmetric between and :
So what would happen if instead of using the regular composition product of these endomorphisms, we used the associated Lie bracket? We’d find
That is, the Lie bracket of and is another vector field! Indeed, let’s see what it looks like in coordinates:
where we can cancel off the two second partial derivatives because we’re assuming that is “smooth”, which in this case entails “has mixed second partial derivatives which commute” in any local coordinate system.
And so we might appropriately write
Of course, even where we don’t have local coordinates we can still write or and get a vector field. We may also find it useful to write down the value of this field at a point: . Indeed we can check that this behaves like a vector at :
And so the space of smooth vector fields on forms a Lie subalgebra of the Lie algebra of endomorphisms of the vector space .