Editing Hilbert's fourth problem (section)

===Pogorelov's theorem===
The following theorem was proved by [[Aleksei Pogorelov|Pogorelov]] in 1973<ref name="Pogorelov1973" /><ref name="Pogorelov1974" />

'''Theorem'''. ''Any two-dimensional continuous complete flat metric is  a <math>\sigma</math>-metric.''

Thus Hilbert's fourth problem for the two-dimensional case was completely solved.

A consequence of this is that you can glue boundary to boundary two copies of the same planar convex shape, with an angle twist between them, you will get a 3D object without crease lines, the two faces being [[Developable surface|developable]].
[[File:IllustrationPogorelovTheorem.png|thumb|Illustration of the Pogorelov Theorem]]