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Suppose $U,V$ are open subsets of $R^2$ and $S\subseteq R^3$ . Suppose $f:U\to S$ and $g:V\to S$ are bijective differentiable maps whose Jacobians have rank 2. Is the composite map $g^{-1}f$ from $U$ to $V$ a diffeomorphism?

Drooga
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    Since $g$ is a diffeomorphism, $g^{-1}$ is a diffeomorphism. And since $f$ is a diffeomorphism, a composition such as $g^{-1} \circ f$ is a diffeomorphism. We should definitely hope so because this is exactly how transition functions on smooth atlases are defined. – Osama Ghani May 25 '20 at 00:32
  • $g(V)$ can’t be open in $R^3$ so how is $g$ a diffeomorphism? – Drooga May 25 '20 at 00:42
  • Sorry, I meant diffeomorphism onto its image (i.e. a smooth embedding). Basically $f(U) $ carves out a $2$-dimensional submanifold, and so $f$ is a diffeomorphism from $U$ onto $S$ using the smooth structure of a submanifold. – Osama Ghani May 25 '20 at 01:05

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