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I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not understand the Holonomic splitting proposition. What does it exactly say? Can one put this in mathematical language? I think it means that over some sufficiently small ball $U\subset V$ one can write the bundle $X^{(r)}\rightarrow V$ as the trivial fibration $U\times \mathbb{R}^{qN_{r}} \rightarrow U$. What I do not understand is why one needs a holonomic section for that? Hope there is somebody who can explain those things to me.

cheers, Ben

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The theorem says that there does not only exist a trivialisation, but even a holonomic trivialisation. In the end, the holonomic sections are the ones that carry geometric meaning, so they are the ones you are interested in. The theorem says that once you have one holonomic section, you get a lot of them (via the holonomic trivialisation). The holonomic splitting theorem is needed in the proof of Thom Transversality (Theorem 2.3.2).

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  • $\begingroup$ I do not understand how such a trivialisation looks like. Does it mean that if one picks a ball $U \subset V$ then one looks at the submanifold $F(U)\subset X^{(r)}$ and this submanifold has a tubular neighbourhood $\nu \subset X^{(r)}$ which at the same time is a fibration $\nu \rightarrow U$ such that $\nu$ is diffeomorphic to $J^{r}(\mathbb{R}^{n},\mathbb{R}^{q})$? I am a bit confused since by the term "section" the authors mean both the section as a map and its image. $\endgroup$
    – Ben
    Commented Feb 10, 2016 at 10:15
  • $\begingroup$ Is it correct what I have said above? If yes how can one prove such a thing? $\endgroup$
    – Ben
    Commented Feb 10, 2016 at 10:16
  • $\begingroup$ For the first two lines of your comment: Yes. The think the fibration property should be part of the definition of "tubular neighbourhood"? $\endgroup$
    – Helene Sigloch
    Commented Feb 10, 2016 at 10:21
  • $\begingroup$ I am not sure about whether $\nu$ is diffeomorphic to $J^r( \mathbb{R}^n, \mathbb{R}^q)$. At least, that is not the point. The point is that the horizontal sections from $\nu$ are holonomic. I understand though that you are confused by the use of the term "section". Still, I think it makes no difference which point of view you take. $\endgroup$
    – Helene Sigloch
    Commented Feb 10, 2016 at 10:26
  • $\begingroup$ What do you mean by horizontal sections from $\nu$? $\endgroup$
    – Ben
    Commented Feb 10, 2016 at 10:36

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