Fubini-Study measure on a product I read that if $\Omega_2$ is the Fubini-Study form on $\mathbb{P}_1\times\mathbb{P}_1$, and $\Omega$ the Fubini-Study form on $\mathbb{P}_1$, then for all $(x,y)\in\mathbb{P}_1\times\mathbb{P}_1$, one has $\Omega_2(x,y)=\Omega(x)+\Omega(y)$.

I wish to justify this equality. What is the exact definition of the Fubini-Study form on the product space?
In the language of Quantum Mechanics, if we can say that the ket $ |\psi&gt; $ is separable then it can clearly be written as $ \psi&gt; = |\psi_{S_{1}}&gt;\otimes |\psi_{S_{2}}&gt;$ for subspaces $S_{1}$ and $S_{2}$ then the full metric is the sum of the metrics on the subspaces $g = g_{S_{1}}+g_{S_{2}}$.
In the language of Quantum Mechanics, if we can say that the ket $ |\psi&gt; $ is separable then it can clearly be written as $ \psi&gt; = |\psi_{S_{1}}&gt;\otimes |\psi_{S_{2}}&gt;$ for subspaces $S_{1}$ and $S_{2}$ then the full metric is the sum of the metrics on the subspaces $g = g_{S_{1}}+g_{S_{2}}$.