The metric of a sphere is given by:
$$ds^2=R^2\left(\mathrm d\theta^2+\sin^2 \theta\ \mathrm d\phi^2\right)$$
The sphere and the hyperboloid are connected like the sine and hyperbolic sine are connected. Does that mean that the metric of a hyperboloid can be obtained by replacing the sine in the formula above with the hyperbolic sine?