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



Holonomic splitting
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).


