Pushforward of Smooth Vector Fields

[Zag]

Hello everyone, my question is: what are the criteria that must be satisfied for the pushforward of a smooth vector field to be a smooth vector field on its own right?

Consider a smooth map $\phi : M \longrightarrow N$ between the smooth manifolds $M$ and $N$. The pushforward associated with this map is a map $\phi_{\ast} : TM \longrightarrow TN$ between the respective tangent bundles associated with $M$ and $N$. (For simplicity I am omitting here the point-wise nature of the pushforward definition).

Smooth vector fields on $M$ are smooth sections $\sigma : TM \longrightarrow M$ of the tangent bundle $TM$. My question is: what are the criteria that must be satisfied for the pushforward of a smooth vector field $\sigma$ on $M$ to be a smooth vector field on its own right on the target manifold $N$? In other words, what would be the conditions which guarantee $\phi_{\ast}\sigma$ to be a smooth section of $TN$?

It seems to me that a basic necessary requirement would be that $\phi$ must be bijective, so that $\phi_{\ast}$ would define a unique vector at every point of $N$. However, I am not sure what would be a set of sufficient conditions.

Any thoughts would be greatly appreciated.

Thank you for your help,
Zag

 

[WWGD]

Maybe you can think of either what happens to the standard basis under a map, or use coordinates , so the Jacobian describes what happens. When can you invert a Jacobian? Or think about what happens when pushing forward a tangent bundle.