Timeline for Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Current License: CC BY-SA 4.0
3 events
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| Dec 3, 2023 at 10:46 | comment | added | Robert Bryant | Then you need to edit your question. You wrote, "I want to characterize which Riemannian manifolds $(M,h)$ [$\ldots$] admit solutions to the previous system.", which, to me, indicates that you are regarding the metric $h$ as a given that constrains the solutions. Perhaps you meant to write something like, "I want to understand the metrics $h=\sum e^k\otimes e^k$ defined by coframings $e$ that satisfy the above conditions". | |
| Dec 2, 2023 at 22:03 | comment | added | Bilateral | Thanks a lot for the answer! In your formulation of the problem, you are looking for solutions compatible with a given metric $h$. In my formulation, I am looking for coframes that satisfy the given equations without imposing compatibility with any fixed metric. It is only a posteriori, after characterizing the coframes that satisfy the equations, that I was interested in understanding the properties of the Riemannian metrics they naturally define. I guess that your analysis still applies by using the general frame bundle with structure group $Gl(3,\mathbb{R})$ instead of $O(3,\mathbb{R})$ | |
| Dec 2, 2023 at 19:14 | history | answered | Robert Bryant | CC BY-SA 4.0 |