I'm struggling with the quotient manifold theorem as exposed in John Lee's book "Introduction to smooth manifolds" (theorem 9.16).

This means we can write the transition map between these coordinates as $(\tilde{x},\tilde{x}) = (A(x,y),B(y))$, where $A$ and $B$ are smooth maps defined on some neighborhood of the origin. The transition map $\tilde{\eta} \circ \eta^{-1}$ is just $\tilde{y}= B(y)$, which is clearly smooth.

I don't get why we can write the transition map as such.

PS: The argument is not that long but requires a lot of prelimenary work so I'm not going to post it here. It can be found here: https://www.mathi.uni-heidelberg.de/~lee/StephanSS16.pdf at the end of page 5.

  • $\begingroup$ "I don't understand " is not a question. Your question must be stated in such a way that somebody can answer.. $\endgroup$ – Thomas Jun 2 '17 at 6:01
  • $\begingroup$ Ok I've added precision on what I don't understand $\endgroup$ – tomak Jun 2 '17 at 8:40

In the text $(x,y)$ and $(\tilde x, \tilde y)$ are the coordinates for two charts of $M$. Since $M$ has a smooth structure there is a smooth transition map $(\tilde x, \tilde y) = (A(x, y), B(x,y))$ for some smooth functions $A$ and $B$. Then from the argument in the text it follows that $B(x,y) = B(x', y)$ for all $x, x'$, so $B$ can be considered to be a function only of $y$ (if you want to be explicit let $B(y) = B(x_0, y)$, for some arbitrary $x_0$) hence $(\tilde x, \tilde y) = (A(x,y), B(y))$ for some smooth functions $A$ and $B$.


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