Skip to main content
MathOverflow

Return to Question

take out the topological manifold confusion, update the title
Source Link
Ryan Budney
Ryan Budney
  • 44.4k
  • 2
  • 139
  • 245

vector fields on transformation properties of divergence (of a topological manifoldvector field)

Hi,

If I have a divergence free vector field defined on a topologicalsmooth manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3 dimensions and we contract the disk to a line. Perhaps the question could be answered by giving first only the most general transformation that could be applied to the vector field (probably general linear transformation) followed by others that may depend increasingly on the metric (which I may want to ignore since I am more interested in diffeomorphisms rather than smooth maps) and yet further depending on the properties of the vector field.

thanks

vector fields on a topological manifold

Hi,

If I have a divergence free vector field defined on a topological manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3 dimensions and we contract the disk to a line. Perhaps the question could be answered by giving first only the most general transformation that could be applied to the vector field (probably general linear transformation) followed by others that may depend increasingly on the metric (which I may want to ignore since I am more interested in diffeomorphisms rather than smooth maps) and yet further depending on the properties of the vector field.

thanks

transformation properties of divergence (of a vector field)

Hi,

If I have a divergence free vector field defined on a smooth manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3 dimensions and we contract the disk to a line. Perhaps the question could be answered by giving first only the most general transformation that could be applied to the vector field (probably general linear transformation) followed by others that may depend increasingly on the metric (which I may want to ignore since I am more interested in diffeomorphisms rather than smooth maps) and yet further depending on the properties of the vector field.

thanks

Source Link
Ben Sprott
Ben Sprott
  • 1.3k
  • 14
  • 23

vector fields on a topological manifold

Hi,

If I have a divergence free vector field defined on a topological manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3 dimensions and we contract the disk to a line. Perhaps the question could be answered by giving first only the most general transformation that could be applied to the vector field (probably general linear transformation) followed by others that may depend increasingly on the metric (which I may want to ignore since I am more interested in diffeomorphisms rather than smooth maps) and yet further depending on the properties of the vector field.

thanks