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A Pushforward of Smooth Vector Fields

  1. Jan 12, 2018 #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
     
  2. jcsd
  3. Jan 17, 2018 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Jan 18, 2018 #3

    WWGD

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    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.
     
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