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Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem".

I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby (especially because the wikipedia article also footnotes to Boothby...), and then in Boothby, he says in a footnote that it is also known as the "Straightening Out Theorem."

Wikipedia also has an article on a "Domain Straightening Theorem".

Which seems vaguely related but does not explicitly discuss anything about rank.

Could someone please help me sort out which theorem is which? My main goal is to find a more in-depth discussion of the "Constant Rank Theorem" (or whatever the true general case of the IFT is) (reading suggestions welcome!), but I would also like to know which of these names refers to the same theorem, and which doesn't.

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    They are different theorems although both are corollaries to the local inversion theorem. The "Domain Straightening Theorem" asserts that all vector fields look the same (in some appropriate chart) near a non singular point, and the "constant rank theorem" tells you that all maps of constant rank look the same when read in appropriate charts for the domain and target manifolds.2012-08-27

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