content_image/369081.jpg content_image/369080.jpg content_image/369078.jpg content_image/369077.jpg content_image/369079.jpg content_image/369076.jpg
flat universes. However, there is also an alternative interpretation when we restrict our attention to the thick-line rectangle of Fig.1. In fact, we realize that the upper tunnel (the strip where in the Reissner-Nordstr¨ om case we would expect to have the central singularity) is a copy of the lower tunnel. This suggests that we might identify the two tunnels. If we do that, the manifold becomes finite and cyclic in the timelike coordinate. 

## VI. THE EXTREME CASE 

In the extreme case M = M$_{0}$ the metric (2) becomes 

<img src='content_image/369081.jpg'>

where φ is a differentiable and not vanishing function in the interval [0 , ∞ ). In order to derive the maximal extension we shall follow the procedure adopted in [20]. To this purpose we consider the surface { ϑ = const , ϕ = const } and we write (28) as follows: 

By introducing null coordinates p and q defined as 

our metric becomes 

<img src='content_image/369080.jpg'>

<img src='content_image/369078.jpg'>

<img src='content_image/369077.jpg'>

<img src='content_image/369079.jpg'>

where now r is a function of p and q . It is worth mentioning that the surface { r = r$_{0}$, ϑ = const , ϕ = const } is made of radial null geodesics corresponding to lines parallel to p = const and q = const in the ( p, q )-plane. 

The metric (31) is regular for all real values of p and q . In order to understand where the coordinate singularity r$_{0}$ lies in the ( p, q )-plane we need to study the signs of φ ( r$_{0}$ ) and φ $^{′}$( r$_{0}$ ). First of all, notice that 

<img src='content_image/369076.jpg'>

