Consider a spacetime with metric $$ ds^2 = -dt^2 + a^2(t)d\Omega_k^2, \quad k=0,\pm1$$ where $a(t)$ is any regular function and $d\Omega_k^2$ is the 3-dimensional metric of
- the 3-sphere $S^3$, if $k=1$.
- the flat 3-space $\Bbb{R}^3$, if $k=0$.
- the hyperbolic 3-space $H^3$, if $k=-1$.
Questions: what is the Ricci tensor $R_{\mu\nu}$ of this metric and what is $a(t)$ to be in order to have $$ G_{\mu\nu}+\Lambda g_{\mu\nu}=0,$$ for some $\Lambda\in\Bbb{R}$ ?