I am reading Peter Topping's lecture notes on Ricci flow and got into a slight amount of confusion with a computation relating to the Gaussian soliton. One considers the stationary flow of the flat metric on Euclidean space to be a gradient Ricci soliton.
So we start with a Ricci soliton $(\mathbb{R}^n,g_0)$ which gives a self-similar solution to the Ricci flow. If we set $\sigma (t):= 1 - 2 \lambda t$, the time-dependent vector field $X(t)=\frac{1}{\sigma} Y$, and $Y=\lambda \textbf{x}$.
$X(t)=\frac{\lambda}{1 - 2 \lambda t} \textbf{x}$.
$X(t)$ is integrated to obtain a family of diffeomorphisms $\psi_t$:
$\psi_t(\textbf{x}) = (1 - \frac{1}{2}\text{log}(1 - 2 \lambda t))$,
$\frac{\partial}{\partial t} \psi_t(\textbf{x}) = \frac{\lambda}{1 - 2 \lambda t} \textbf{x}$.
We define $\hat{g}(t):= \sigma (t) \psi^{*}_t (g_0)$. If $f$ is the map taking $x$ to $cx$ (ie. which multiplies by a constant $c$), then a computation shows that $f^{*}(g_0) = c^2 g_0$. Use this to obtain the fact that in this case
$\hat{g} = (1 - 2 \lambda t)(1 - \frac{1}{2} \text{log}(1- 2 \lambda t))^2 g_0,$
$\partial_t \hat{g} = \dot{\sigma} (1 - \frac{1}{2} \log \sigma)^2 + 2 \sigma (1 - \frac{1}{2} \log \sigma) \cdot(-\frac{1}{2}) \frac{\dot{\sigma}}{\sigma} = \dot{\sigma}((1 - \frac{1}{2}\log \sigma)^2 - (1 - \frac{1}{2} \log \sigma))$.
We have that $\text{Ric} (\hat{g}) =0$ and the equation for Ricci flow then implies that $\partial_t \hat{g} = 0$. However, one sees above that the derivative of $\hat{g}$, unless some mistake has been made in the calculation.