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Answer by Robert Bryant for Algebraic atlas on smooth manifolds

The answers to your question in the case $n=1$ are well-known. In higher dimensions, the answers are less complete, but something is known.For example, in the real case when $n=1$, there is only one...

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Algebraic atlas on smooth manifolds

A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition...

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