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I have created some genuine content at implicit function theorem. I’d like to hear the comments on the global variant, which is there, taken from Miščenko’s book on vector bundles in Russian (the other similar book of his in English, cited at vector bundle, is in fact quite different).
I disagree with the definition given of regular value. The definition with which I am familiar is that $q$ is regular if for each $p \in f^{-1}(q)$, the map $T_p(f): T_p(M) \to T_q(N)$ has maximal rank (i.e., $\min(\dim(M), \dim(N))$). Perhaps this is not important for your application, but it is important otherwise.
Hm, I quoted here Miščenko, but my 1988 notebook gives the same definition (regular value means that the tangent map from any point in the preimage is surjective). The same is in 5.6 in Bröcker-Jänich, in the standard textbook Hirsch, chapter 1, just before theorem 3.2 and in Bott-Tu just before theorem 4.9. Your definition is weaker, I heard of it as well, but the sample of the standard textbooks agrees with what I said. P.S. also the same in S. Sternberg, Lectures on differential geometry, Ch. II, Def. 3.3.
By the way, is word (?) “dimensionalysis” a typo at exponential map or it is intended ?
Unfortunately, I can’t effectively investigate this further until I get power back at my house. (I’m working from an iPad for my Internet connection, and living like a refugee at other people’s houses.)
My recollection agrees with Zoran’s definition. When $\dim(M) \lt \dim(N)$, I can see the importance of asking whether $T_p(f)$ has rank $dim(M)$, but it seems to me to be a different question. But both are generalisations of the case from ordinary calculus, so I could imagine the same terminology being used.
We discussed “dimensionalysis” before.
It’s annoying to me that I may be misremembering, but I’m just about prepared to admit that I am indeed. I’ll come back if I have anything useful to add.
It looks like I just massively rewrote the article, but really I just added some links!
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