I've been trying to perform some calculations that would benefit from being phrased in a geometric manner. A lot of the time I have something like:
$$\frac{\partial}{\partial t}h \circ g_t = \frac{\partial h}{\partial g_t}\dot{g}_t$$
Where $h,g_t$ are diffeomorphisms and $\frac{\partial h}{\partial g_t}$ denotes the Jacobian. I want to say this is simply just the pushforward of the map $h$ acting on the tangent vector $\dot{g}_t$, i.e. $h_* \dot{g}_t$ and this fits with my intuition that the pushforward is essentially the Jacobian when it comes to flat space. But when I go to verify my intuition with what the definition of pushforward is this doesn't seem correct. For example, according to:
Pushforward of a Vector Field by a Diffeomorphism
Show that the chain rule $(G\circ F)_* = G_* \circ F_*$ holds.
Identity about composition of the push forward of diffeomorphisms
I should be saying that it's not a pushforward unless I had something like $\frac{\partial h}{\partial g_t}\dot{g}_t h^{-1}$. How come? I find it very disconcerting that such a natural operating such as the chain rule doesn't translate to the pushforward map. Is there any way I can make it more "geometric"?
Best Answer
The concept I was looking for was the notion of the tangent map. This is multiplying by the Jacobian matrix, without the change of coordinates that is part of the pushforward. My confusion arises from the fact that the terminology is mixed often. What Wikipedia calls the "pushforward differential" is the tangent map, and is an inequivalent operation to the pushforward discussed in the linked questions.
The book of Marsden and Ratiu on Tensor Analysis covers this.