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In Gauge Fields, Knots, and Gravity, exercise 18 is the following: Show that if $\phi:M \to N$ we can push forward a vector field $v$ on $M$ to obtain a vector field $\phi_*$ on $N$ satisfying $(\phi_* v)_q = \phi_*(v_p)$, whenever $\phi(p)=q$.

I don't understand the question. I don't see how $(\phi_* v)_q$ is defined, given that we always apply the pushforward to tangent vectors, and not vector fields. If someone could explain the question, that would be good.

Edit: $M$ and $N$ are smooth manifolds, $\phi$ is a diffeomorphism.

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You can apply the push forward pointwise. In fact, we define $(\phi_*v)_q$ to be $\phi_*(v_p)$, where $\phi(p) = q$. What you're asked to show is that the function $q\mapsto (\phi_*v)_q$

  • is well-defined; and

  • defines a legitimate vector field on $N$ (i.e. is smooth).

For the first part, you'll need to use the fact that $\phi$ is one-to-one.

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    $\begingroup$ Before showing that $q\mapsto(\phi_*v)_q$ is smooth, show that it's a function. Here you'll need that $\phi$ is one-to-one, an important hypothesis that was missing in the statement of the exercise but added in the edit. $\endgroup$ – Andreas Blass Mar 3 '13 at 22:34
  • $\begingroup$ @AndreasBlass I've updated my answer to reflect your comment. Thanks! $\endgroup$ – Avi Steiner Mar 4 '13 at 1:55

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