As far as I understand it, it is like this: Imagine a strictly fictional 4D Euclidean space (nothing to do with space-time). Now, let us consider hyper-surfaces (i.e. 3D surfaces in 4D space given by some relation f(x_{1}, x_{2}, x_{3}, x_{4}) = 0) that are rotationally invariant, i.e. the functional dependence above is of the form:
<br />
F(r, x_{4}) = 0, \; r = \sqrt{x^{2}_{1} + x^{2}_{2} + x^{2}_{3}}<br />
Here, we may use an analogy with 3D space and 2D (hyper)surfaces. If they are rotationally invariant around x_{3}, then they are of the form F(\rho, x_{3}) = 0, \; \rho = \sqrt{x^{2}_{1} + x^{2}_{2}}.
One hypersurface is spherical if:
<br />
r^{2} + x^{2}_{4} = a^{2}<br />
in analogy to a 3D sphere \rho^{2} + x^{2}_{3} = a^{2}.
Another surface is hyperbolic (and open along the x_{4} direction) if:
<br />
x^{2}_{4} - r^{2} = a^{2}<br />
in analogy to the 3D hyperboloid x^{2}_{3} - \rho^{2} = a^{2}.
Finally, a flat hyper-surface perpendicular to the x_{4} direction may be written as:
<br />
x_{4} = a \Rightarrow x^{2}_{4} = a^{2}<br />
in analogy to a 3D plane perpendicular to the x_{3} direction x_{3} = a.
A clever observation is that all of these cases correspond to an implicit functional relationship of the form:
<br />
x^{2}_{4} + \Omega \, r^{2} = a^{2}<br />
and the different cases correspond to the values:
\Omega = 1 - spherical
\Omega = 0 - flat
\Omega = -1 - hyperbolic
Now, let us return to the question of the metric along this surface. The squared distance between to infinitesimally distant points is still given by the famous Euclidean form:
<br />
ds^{2} = dx^{2}_{1} + dx^{2}_{2} + dx^{2}_{3} + dx^{2}_{4}<br />
but, not all 4 coordinates are independent because of the above functional relationship (we are fixed on the 3D hyper surface and cannot wander off of it). We can use that relationship to eliminate one of the coordinates, namely x_{4}. Let us differentiate the above relation:
<br />
2 \, x_{4} \, dx_{4} + \Omega \, 2 \, r \, dr = 0 \Rightarrow dx_{4} = -\frac{\Omega \, r \, dr}{x_{4}}<br />
Furthermore, we can still use spherical coordinates for the remaining three Cartesian coordinates and write:
<br />
dx^{2}_{1} + dx^{2}_{2} + dx^{2}_{3} = dr^{2} + r^{2} \, \left( d\theta^{2} + \sin^{2}{\theta} \, d\phi^{2} \right)<br />
everywhere. The squared differential of the fourth Cartesian coordinate is:
<br />
dx^{2}_{4} = \left( -\frac{\Omega \, r \, dr}{x_{4}} \right) = \frac{\Omega^{2} \, r^{2} \, dr^{2}}{x^{2}_{4}} = \frac{\Omega^{2} \, r^{2} \, dr^{2}}{a^{2} - \Omega \, r^{2}}<br />
where we have used the functional relation once more to eliminate the remaining x^{2}_{4}. Combining everything together and simplifying the coefficient in front of dr^{2}, we get:
<br />
ds^{2} = \frac{a^{2} + \Omega \, (\Omega - 1) \, r^{2}}{a^{2} - \Omega \, r^{2}} \, dr^{2} + r^{2} \, \left( d\theta^{2} + \sin^{2}{\theta} \, d\phi^{2} \right)<br />
This metric corresponds to specific classes of non-Euclidean 3D spaces.
