A compute of Ricci Flow

Question

Let $g(0)=g_0$ and $Ric(g_0)=\lambda g_0,\lambda\in\mathbb{R}$, the $Ric(g)$ is the Ricci curvature,$g$ is Riemannian metric. How to show that :

The $g(t)=(1-2\lambda t)g_0$ is a solution of $$ \frac{\partial g}{\partial t}=-2Ric(g) $$

Thanks for any detail answer or hint.

Answer 1

When you multiply the metric by a factor, the Ricci doesn't scale. One way to see this is to look at its expression in the normal coordinates given here.