# In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms

Can anyone help me and prove that in the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms?

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– Sepideh Bakhoda May 26 '13 at 14:56
any textbook will have this proof, see for example amazon.com/gp/… – Carlo Beenakker May 26 '13 at 15:31
@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. – Robert Bryant May 26 '13 at 15:53
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? – Deane Yang May 26 '13 at 18:22
@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? – Robert Bryant May 27 '13 at 0:43