Graduate 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.