Timeline for In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms
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| when toggle format | what | by | license | comment | |
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| Dec 24, 2013 at 0:36 | answer | added | User0.9999999..... | timeline score: 1 | |
| May 27, 2013 at 0:43 | comment | added | Robert Bryant | @Deane: I took 'the Ricci flow' to mean the PDE system itself which one can think of as a submanifold of an appropriate jet bundle $J$ over $M\times\mathbb{R}$. Then a 'symmetry' would a self-diffeomorphism of $J$ that carries solutions to solutions (thought of via their $k$-jet graphs). @Carlo: You probably want to be more precise about what 'operations' you allow. Let $\mathcal{S}$ be the set of all Ricci-flat metrics on $\mathbb{R}^n$ and let $\sigma:\mathcal{S}\to\mathcal{S}$ be any mapping whatsoever. This is an 'operation that transforms one solution to another solution', no? | |
| May 26, 2013 at 21:12 | comment | added | Carlo Beenakker | Andrews & Hopper (link above, page 66) define a symmetry as a diffeomorphism that is an isometry of the initial metric, and then prove that Ricci flow preserves this symmetry (so the isometry persists for positive time). I imagine one could more generally define a symmetry as any operation that transforms one solution into another solution. For the vacuum Einstein equations it has been proven that diffeomorophisms (with scaling) exhaust all symmetries: arxiv.org/abs/gr-qc/9404030 | |
| May 26, 2013 at 18:22 | comment | added | Deane Yang | Could someone say what the definition of a symmetry of the flow is? My reaction to the question was: isn't that the definition of a symmetry? | |
| May 26, 2013 at 15:53 | comment | added | Robert Bryant | @Carlo: Most textbooks on Ricci flow will indeed show that scalings and diffeomorphisms are symmetries of the PDE, but I am not so sure that there are many books that show the converse, namely, that any symmetry of the PDE system (once this has been properly defined) is necessarily a combination of scalings and diffeomorphisms. | |
| May 26, 2013 at 15:31 | comment | added | Carlo Beenakker | any textbook will have this proof, see for example amazon.com/gp/… | |
| May 26, 2013 at 14:56 | comment | added | Sepideh Bakhoda | crosspost: math.stackexchange.com/questions/401943/… | |
| May 26, 2013 at 14:55 | history | asked | Sepideh Bakhoda | CC BY-SA 3.0 |