Timeline for Understanding exterior differential systems
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 22 at 19:59 | vote | accept | Bilateral | ||
| Jul 19 at 15:37 | answer | added | Robert Bryant | timeline score: 10 | |
| Jul 19 at 13:48 | comment | added | Robert Bryant | @Bilateral: That's a good question. Basically, my reason for formulating it on the coframe bundle $P$ is that this can be done without making any arbitrary choices (such as, for example, local coordinates on $M$) and the solutions to your problem are exactly the same as sections $e:P\to M$ of this bundle that are integral manifolds of the EDS that I described. I don't see any formulation that is more natural than that. Moreover, the 'hidden' compatibility conditions in the equations you have written are uncovered naturally by closing the EDS under exterior derivative. I'll illustrate below. | |
| Jul 19 at 13:11 | comment | added | Bilateral | Thanks a lot for the answer @RobertBryant. I understand but, is there any conceptual reason why that has to be the manifold on which the problem is formulated as an EDS? Or is it simply a choice that works well? Would it be possible that a different choice of "auxiliary manifold" would be better suited for the problem depending on the specific $T_{ij}^k$? | |
| Jul 19 at 9:38 | comment | added | Robert Bryant | The manifold on which this problem is posed as an exterior differential system is the coframe bundle $\pi:P\to M$, where an element $u\in P$ is an isomorphism $u:T_{\pi(u)}M\to\mathbb{R}^n$. Your equations are then interpreted as $2$-forms on $P$ and you seek a section $e:M\to P$ such that $e:M\to P$ is an integral manifold of the ideal generated by these $2$-forms. The $2$-form $\mathrm{d}(F_k e^k)$ is of the form $\tfrac12\,F_{jk}\, e^j\wedge e^k$, where the functions $F_{jk}=-F_{kj}$ are defined on $P$, and your desired section $e$ must take values in the zero locus of these functions, etc. | |
| Jul 18 at 22:55 | comment | added | Deane Yang | This is analogous to how to write a PDE as an exterior differential system. | |
| Jul 18 at 22:53 | comment | added | Deane Yang | The graph of a section of a bundle is a submanifold of the bundle itself. Your equations define an exterior differential system on the bundle You want a $n$-dimensional submanifold on which the system vanishes and is transversal to the fibers. | |
| Jul 18 at 18:44 | history | edited | LSpice | CC BY-SA 4.0 |
Typo; deleted "thanks"
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| Jul 18 at 18:36 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting
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| Jul 18 at 18:34 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jul 18 at 18:26 | history | asked | Bilateral | CC BY-SA 4.0 |