Let $M(u,v)=N_1(u) \times N_2(v)$ be a 2-dimensional Riemannian product manifold, so $N_1(u)$ and $N_2(v)$ are 1-dimensional curve.
Is correct that, since $N_1$ is naturally embeded in $\mathbb{R}^2$ (and the same for $N_2$), then $\mathbb{R}^2 \times \mathbb{R}^2=\mathbb{R}^4$, so $M$ has metric: $g_M=g_{N_1} + g_{N_2}$, with $g_{N_1}=g_{11}du^2$ and $g_{N_2}=g_{44}dv^2$ respectively?
Is corrected to say that write $g_{N_2}=g_{22}dv^2$ is wrong?
Because $M$ is Ricci flat and if the metric of $M$ is $g_M=g_{N_1} + g_{N_2}$, with $g_{N_1}=g_{11}du^2$ and $g_{N_2}=g_{22}dv^2$, then $M$ could be not Ricci flat.