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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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I think they are both standard theorems in differential manifolds, and they actually occur without the name given now. For example, in other books the constant rank theorem was referred to as the normal form of a submersion/immersion map.

As others said, the two theorems are different and both can be proved using inverse function theorem carefully. The first one tells you locally a differentiable map between two manifolds near a regular point can be viewed as a linear map in appropriate local coordinates. The second tells you that locally a vector field at a regular point can be viewed as given by the coordinate vector field $\{ \partial_{x_i}\}$, where $x_{i}$ are the coordinate functions. Since a vector field is a section of the tangent bundle, and a tangent vector is defined (partly) as maps $C^{\infty}(M)_{p}\rightarrow F$, I do not think the two theorems are the same or can be "translated".