To put it another way, is the vector specifies for the point . On the other hand, is the image of the vector specifies for the point . If these two vectors are the same for every , then we say that “intertwines” the two vector fields, or that and are “-related”. The latter term is a bit awkward, which is why I prefer the former, especially since it does have that same commutative-diagram feel as intertwinors between representations.
Anyway, in the case that is a diffeomorphism we can actually use this to transfer vector fields from one manifold to the other. Given a point , which point should it be compared to? the inverse image , of course. This point gets the vector , which then gets sent to . That is, if we define , then is guaranteed to intertwine and .
Since is a linear map on each stalk it’s clear that if intertwines and , as well as and , then intertwines and . But we’ve just seen that vector fields form a Lie algebra, and it would be nice if we could say the same for and . The catch is that we don’t just compute these point-by-point.
Let’s pick a text function and a point . We first check that
Now we can calculate
So intertwines and , as we asserted. In the case where is a diffeomorphism, this means that the construction above gives us a homomorphism of Lie algebras from to .