Week 15: The Canonical Flip

So, I've been trying to read through Transversal Mappings and Flows by Abraham and Robbin (Joel Robbin was my linear algebra professor at UW back when I was an undergrad, and he literally wrote the book on the subject: we used a manuscript that I still have to this day that eventually became a book Matrix Algebra Using MINImal MATlab. The manuscript had about a third of it devoted to pure linear algebra, over abstract finite dimensional real and complex vector spaces, not using matrices; the theorems were mainly restatements of theorems from the matrix algebra portion of the book, and referenced their matrix algebra counterparts, but this portion of the book was written like manifold theory using manifold that were all diffeomorphic to R^n, and it really set the tone for a lot of math I was to learn later) in my efforts to learn global analysis (I've had a really hard time finding a good introduction to the subject, and this one seems as good as any, maybe better than most). The book talks about the double tangent bundle and its relations to second order ordinary (autonomous) differential equations, and front and center is the "canonical flip" on the double tangent bundle.

The Wikipedia article on the double tangent bundle also mentions the canonical flip, but I couldn't make heads or tails of it. I have been trying to read through Robbin's presentation of it since, oh, roughly since I made my last blog post, in December 2015, but I just couldn't wrap my head around it.

Then, finally, it came to me yesterday as I was riding the elevator back up to my parents' condo after having pored over the section on the double tangent bundle in Abraham and Robbin at a coffeshop for the umpteen-millionth time. In the book The Geometry of Jet Bundles by D. L. Saunders (a book on partial differential equations on manifolds, yet another book on global analysis I'm nominally trying to work through in my efforts to transition from manifold topology [MSC 57] to global analysis [MSC 58] and beyond that to geometric control theory [no MSC code yet, AFAIK]), he goes over in detail that, for an arbitrary vector bundle \pi: E -> M, in considering the tangent bundle to E, D\pi: TE -> TM, we have that for the horizontal vector bundle - that is, if (p, v1, v2, v3) is an arbitrary point of TE, the horizontal vector bundle is {(p, v2) | (p, v1, v2, v3) \in TE} - is canonically vector bundle isomorphic to TM - and, I understood the canonical flip! It just interchanges TM = {(p, v1) | (p, v1, v2, v3) \in TE} \subseteq TE with the horizontal vector bundle {(p, v2) | (p, v1, v2, v3) \in TE}.

So, persistence and hard work does sometimes pay off.

Now, Abraham and Robbin go on to define a second-order (autonomous) ordinary differential equations as dual sections of \tau(TM) and D\tau(M), and they, just before that, show that D: \Gamma(\tau(M) -> \Gamma(D\tau(M) and \omega \circ D: \Gamma(\tau(M) -> \tau(TM) are linear (over real vector bundles), so if \xi: M -> TM is an first-order (autonomous) ordinary differential equation on M, then (say) (1/2)D\xi + (1/2)(\omega \circ D\xi) is a second-order (autonomous) ordinary differential equation on M - nifty stuff.