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Could somebody tell me if this is correct? I'm trying to understand mappings and inverse mappings in introductory differential geometry. The transition functions baffle me.

Suppose our manifold of interest is $M = \mathbb R.$ Suppose further that we have the charts $(U_1,\phi)$ and $(U_2,\psi)$ where $U_1 = \mathbb R$ and $U_2 = \mathbb R$. Let $\phi$ and $\psi$ be given by:

$$\phi(x) = x^3$$ $$\psi(x) = x^2$$

where x is the coordinate in $M$.

  1. What is $\phi^{-1}$?
  2. What is $\psi \circ \phi^{-1}$?

I think the answers are

  1. $\phi^{-1} = y^{1/3}$
  2. $\psi \circ \phi^{-1} = y^{2/3}$ or do I need a separate $y_1$ and $y_2$ where $y_1 = \phi(x)$ and $y_2=\psi(x)$?
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  • $\begingroup$ Note that $\psi$ is not a coordinate chart on $\mathbb R$, because it's neither injective nor everywhere defined. $\endgroup$ – Jack Lee Jul 14 '15 at 17:17
  • $\begingroup$ Yeah, there are lots of pathologies. Also, I believe that $\psi \circ \phi^{-1}$ is not differentiable at the origin. So lets not call them coordinate charts but merely functions on $M$. The point that I am trying to get at is whether I did 1. and 2. correctly for the inverse and the transition map. It's the inverses that have me confused. $\endgroup$ – jeo15 Jul 14 '15 at 19:35

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