Here we have some (blue) curves $\ln(x)\ln(y)=s$ for several values of $s$ and some (green) curves $\ln(1-x)\ln(y)=s$ for several values of $s.$ The (red) is the multiplication of the green and blue.
The blue and green curves are the invariant "hyperbola."
Let $(M_1,g_1)$ be the blue semi-Riemannian manifold and $(M_2,g_2)$ the green semi-Riemannian manifold with the following metrics:
$g_1=\frac{dxdy}{xy}$ and $g_2=\frac{dudv}{v-uv}.$
Upon taking the Cartesian product of blue curves and green curves, what metric can be put on this new space (of red curves)? Will it be some sort of combination of $g_1$ and $g_2$ or something completely different?
For two semi-Riemannian manifolds $(M_{1}, g_{1})$ and $(M_{2}, g_{2})$, we have the product metric tensor $g_{} = g_{1} \bigoplus g_{2}$. Before doing that, however, we need to obtain $g_{1}$ and $g_{2}$ in terms of $x$, $y$, $dx$, and $dy$. We now have that \begin{equation} ds_{1}^{2} = \frac{dx dy}{xy} \implies xy = \frac{dx dy}{ds_{1}^{2}} \end{equation} \begin{equation} ds_{2}^{2} = \frac{dx dy}{x - xy} \implies x - xy = \frac{dx dy}{ds_{2}^{2}} \end{equation} We can now add them and find a suitable metric \begin{equation} xy + x - xy = x = \frac{dx dy}{ds_{1}^{2}} + \frac{dx dy}{ds_{2}^{2}} = \frac{ds_{2}^{2} dx dy}{ds_{1}^{2}ds_{2}^{2}} + \frac{ds_{1}^{2} dx dy}{ds_{1}^{2}ds_{2}^{2}} = \frac{ds_{2}^{2} dxdy + ds_{2}^{1} dxdy}{ds_{1}^{2}ds_{2}^{2}} \end{equation} \begin{equation} \implies ds_{1}^{2}ds_{2}^{2} = \frac{ds_{2}^{2} dxdy + ds_{2}^{1} dx dy}{x} = dxdy \frac{ds_{2}^{2} + ds_{2}^{1}}{x} = dxdy \left( \frac{dxdy}{x^{2}y} + \frac{dx dy}{x^{2} - x^{2}y}\right) \end{equation} by substituting in the original metrics $ds_{1}^{2}$ and $ds_{2}^{2}$.