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A Pushforward of Smooth Vector Fields

  1. Jan 12, 2018 at 11:09 AM #1

    Zag

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

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

    It seems to me that a basic necessary requirement would be that [itex]\phi[/itex] must be bijective, so that [itex]\phi_{\ast}[/itex] would define a unique vector at every point of [itex]N[/itex]. 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
     
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