  Any compact manifold with positive scalar curvature has an associated
asymptotically flat metric constructed using the Green's function of the
conformal Laplacian, and the mass of this metric is an important geometric
invariant. An explicit expression for the mass of the product of spheres $S^2
\times S^2$, both with the same Gaussian curvature, is given. Expressions for
the masses of the quotient spaces $G(2,4)$, and $\mathbb{RP}^2 \times
\mathbb{RP}^2$ are also given. The values of these masses arise in a
construction of critical metrics on certain $4$-manifolds; applications to this
problem will also be discussed.
