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    $\begingroup$ The tangent bundle is an associated bundle to the frame bundle, so any tensor on the base can be 'lifted' to a function on the frame bundle with values in a representation that is equivariant under the action of the group. I'm pretty sure you can find this point of view in Sternberg's book Lectures on Differential Geometry, amongst others. Probably in Kobayashi-Nomizu I too. $\endgroup$ Commented Mar 1, 2015 at 13:22
  • $\begingroup$ If you have some time, could you give an explicit example of the correspondence between tensors and equivariant functions on the frame bundle? For example for a vector field. I am checking the references that you mentioned but there is nothing too explicit. Thanks Paul. $\endgroup$ Commented Mar 1, 2015 at 15:33
  • $\begingroup$ I guess there is now no need, after Peter's post. $\endgroup$ Commented Mar 1, 2015 at 19:09