= Solution
Put $r^2=y_1^2+y_2^2$ and write the <Riemannian metric> as a <conformal rescaling of a Riemannian metric>:
$$
g=e^{2\omega}\delta,\qquad \omega=-\frac12\log(1+r^2).
$$
The base metric is flat, so its <Ricci scalar> is zero and its <Laplacian> is $\Delta=\partial_1^2+\partial_2^2$. The formula for <scalar curvature under conformal rescaling> simplifies in dimension two to
$$
R_g=-2e^{-2\omega}\Delta\omega.
$$
Differentiate explicitly:
$$
\partial_i\omega=-\frac{y_i}{1+r^2},\qquad
\Delta\omega=-\frac2{1+r^2}+\frac{2r^2}{(1+r^2)^2}
=-\frac2{(1+r^2)^2}.
$$
Since $e^{-2\omega}=1+r^2$, the required <Ricci scalar> is
$$
\boxed{R_g(y_1,y_2)=\frac4{1+y_1^2+y_2^2}.}
$$
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