The mass of the product of spheres

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.