This is something I always forget exists and has a name, so I end up reproving it. Since this sequence of posts is a hodge-podge of things to help me take a differential geometry test, hopefully this will lodge the result in my brain and save me time if it comes up.

I’m not sure whether to call it a lemma or not, but the setup is you have a smooth map and a vector field on , say and a vector field on say such that and are -related. Define and to be the image of flowing for time and let and be the flows of and respectively. Then the lemma says for all we have and on .

This is a “naturality” condition because all it really says is that the following diagram commutes:

Proof: Let , then is a curve that satisfies the property . Since , and integral curves are unique, we get that at least on the domain of .

Thus if then , or equivalently . But we just wrote that where defined, which is just a different form of the equation .

We get a nice corollary out of this. If our function was actually a diffeo, then take the pushforward, and we get that the flow of the pushforward is and the flow domain is actually equal .

In algebraic geometry we care a lot about families of things. In the differentiable world, the nicest case of this would be when you have a smooth submersion: , where is compact and both are connected. Then since all values are regular, is smooth embedded submanifold. If were say (of course, couldn’t be compact in this case), then we would have a nice 1-dimensional family of manifolds that are parametrized in a nice way.

It turns out to be quite easy to prove that in the above circumstance all fibers are diffeomorphic. In AG we often call this an “iso-trivial” family, although I’m not sure that is the best analogy. The proof basically comes down to the naturality of flows. Given any vector field on , we can lift it to a vector field on that is -related. I won’t do the details, but it can be done clearly in nice choice of coordinates and then just patch together with a partition of unity.

Let be the notation for . Fix an , then by the above naturality lemma is well-defined and hence a diffeomorphism since it has smooth inverse . Let . Then as long as there is a vector field on which flows to , then we’ve shown that , so since , were arbitrary, all fibers are diffeomorphic. But there is such a vector field, since is connected.