Definition of steady Ricci soliton

Question

Firstly, my English is poor, so I am not sure that which is solution , and the steady solution is the solution of evolution equation in picture below ?

Secondly, what is the symmetry group of a equation ? As I know , the symmetry group is the all permutations on some symbols. But,seemly, there are nothing between equation and permutation. (the equation in the picture below is Ricci flow i.e. $\partial_tg_{ij}=-2R_{ij}$).

Thirdly , how to get a 1-parameter subgroup ? As I know ,the 1-parameter group is a family $(\varphi_t)_{t\in I}$ ($I$ open interval with $0\in I$) of diffeomorphisms from $M$ to $M$ and satisfy $$ \varphi_t\circ\varphi_s(q)=\varphi_{t+s}(q)~~~~if~~s,t,s+t\in I_q $$ In fact ,I have suspicion about that when $s,t$ is large enough , $s+t\notin I$. So ,it is not a group.

Fourthly , what is the mean of move ? It means that if $g(x,t)$ is solution of equation ,then $g(x,t)\circ\varphi_s$ still be solution of equation?

Answer 1

If $\phi(t) : M\rightarrow M $ is a family of diffeomorphism then assume that $$ g(t)=\phi(t)^\ast g(0),\ \frac{\partial }{ \partial t} g(t)=-2 {\rm Ric} (g(t)) $$

That is $g(t)$ is steady soliton in general sense

If $X(x):=\frac{\partial }{\partial t}\bigg|_{t=0}\phi(t,x)$, then $$ L_X g(0)=\frac{\partial }{ \partial t} g(t) =-2 {\rm Ric} (g(0)) $$

So $g(0)$ is steady Ricci solition if $L_X g(0)=-2 {\rm Ric} (g(0))$

From this, steady soliton $\Rightarrow$ steady Ricci solition Now we will prove opposite direction :

If $\phi (t)$ is flow of $X$ define $$g(t,x):=\phi(t)^\ast g(0) (\phi(t,x))$$ Then

\begin{align*} \frac{\partial }{\partial t}\bigg|_{t=t_0} g(t,x)&=\frac{\partial }{\partial s}\bigg|_{s=0} g(t_0+s,x)\\&= \frac{\partial }{\partial s}\bigg|_{s=0} \phi(s+t_0)^\ast g(0)(\phi(s+t_0,x)) \\&=\frac{\partial }{\partial s}\bigg|_{s=0} \phi(t_0)^\ast \phi(s)^\ast g(0)\bigg(\phi (s, \phi(t_0,x))\bigg) \\&=\frac{\partial }{\partial s}\bigg|_{s=0} \phi(t_0)^\ast g(s)( \phi(t_0,x)) \\&= \phi(t_0)^\ast \bigg(-2{\rm Ric}\ ( g(0) (\phi(t_0,x)) ) \bigg) \\&= -2{\rm Ric}\ (\phi(t_0)^\ast g(0)) \end{align*}

Hence we complete the proof