I think the introduction to this topic should be changed to the following.
In mathematics, a differentiable map f from an m-manifold M to an n-manifold N is said to be a submersion, if its differential is a surjective map at every point p of M, or equivalently, if rank Df(p) = dim N = n.
- Why? The intorduction, as it stands, offers far more detail and explanation that your proposed introduction. Dharma6662000 (talk) 04:00, 25 August 2008 (UTC)
I deleted the submersion theorem. It seemed to have been tacked on with a later edit. Its notation didn't match the original article (e.g. using instead if p to denote a point in the source manifold). Secondly there was absolutely no attempt to say what the notation meant, or to link it. For example writing thinks like without saying that it was the space of linear maps from to . I understood what that means because I understand the idea. But there were other random notations like that even I didn't understand. This theorem needs to be rewriten by someone that has some sympathy for the reader. I'll try to do it tomorrow. Dharma6662000 (talk) 04:00, 25 August 2008 (UTC)
The final paranthetical remark is misleading: in Morse theory N = R so a critical point (in the sense of not being a submersion) is one where df has rank less than 1, that is df = 0. There is therefore no situation where f is neither critical nor a submersion (sorry - too many negatives!) I will remove this if no-one objects. Simplifix (talk) 14:38, 2 March 2009 (UTC)
- I agree with you. 22.214.171.124 (talk) 01:03, 10 March 2009 (UTC)
Two Defs. of Critical Points
I've changed the definitions somewhat. It seems that there are two definitions of "critical point" in the literature. One is more common, e.g. it is the one used in Sard's theorem and in the Critical Point article (where a book of DoCarmo is referenced). This is the one I've used (note: this is what was previously called in the article a "singular point" - which is a non-standard term as far as I can tell. It leads to the terminologically absurd result that if the dimension of M is less than the dimension of N, all the points are signular!).
It is true that some authors use critical point to describe a point where the rank of the Jacobian is non-maximal (which makes critical points more rare in the case where the dimension of M is less than the dimension of N). This is explained in a "word of warning".
I hope you find this acceptable, at least as a first approximation.. :-)