Recently I was reading chapter 4 (p. 60) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof’s of the equivalence of the Implicit Function Theorem (finite-dimensional vector spaces) and the Picard Theorem for ordinary differential equations. We know that the Implicit Function Theorem (finite-dimensional vector spaces) is a particular case of the Constant Rank Theorem. We also know that the Frobenius Theorem is a generalization of Picard’s Theorem for ordinary differential equations. Based on these facts follow my question.
Is there any reference which provides a solution to this question? If the Frobenius theorem does not imply the Constant Rank Theorem there is some explanation for the negative? Conversely, if the Constant Rank Theorem does not imply the Frobenius theorem there is some explanation for the negative?