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I just want to check that my definition of the push forward of a vector field is correct. So I have a vector field $X$ on $M$ and a diffeomorphism $\phi:M\to N$.

Then $\phi_*X$ is defined as follows

$$(\phi_*X)(x)(f) = (\phi_{*\phi^{-1}(x)}X_{\phi^{-1}(x)})(f) = X_{\phi^{-1}(x)}(f\circ\phi)$$

where $x\in N$ and $f\in C^{\infty}(N)$

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    $\begingroup$ I don't understand the middle term in that equality, because I don't know what $X_\star$ means. The last one, however, looks OK. $\endgroup$ – Giuseppe Negro May 29 '18 at 8:58
  • $\begingroup$ my bad. It should be $X_{\phi^{-1}(x)}$ $\endgroup$ – tomak May 29 '18 at 9:07
  • $\begingroup$ Yeah, that's right. $\endgroup$ – Praneet Srivastava May 29 '18 at 9:32
  • $\begingroup$ Possible duplicate of push forward of vector field $\endgroup$ – Praneet Srivastava May 29 '18 at 9:33

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