$F=p_1times p_2$ a hyperbolic space?

Consider the space of parametrized curves $p_1,$ $x=e^{log(n_i)e^t}$ $y=e^{log(n_i)e^{-t}}$ for $-infty<Re(t)<infty$ and $n_iin(0,1)$ for each $i.$ Revolve the space of parametrized curves about $y=x.$ Create a new parametrization $p_2$, $1-x$ and $y.$ Revolve the space of new parametrized curves about $y=1-x.$ Take the Cartesian product $F=p_1times p_2$ of the curves in each parametrization. Can the space $F$ be equipped with some type of modified hyperbolic metric based on the combination of $p_1$ and $p_2?$ Assume that one can treat the space of $p_1$ as a lorentzian space, likewise for $p_2,$ leading to hyperbolic spaces and metrics for each. But I’m not sure if $F$ is also a hyperbolic space. Since the product of two constant curvature spaces is not constant anymore I have doubts that $F$ could be a hyperbolic space. Thanks very much.