Editing Hilbert's fourth problem (section)

==Flat metrics==
[[Image:Desargues theorem alt2.svg|thumb|right|400px|Desargues's theorem]]
[[Desargues's theorem]]:

''If two triangles lie on a plane such that the lines connecting corresponding vertices of the triangles meet at one point, then the three points, at which the prolongations of three pairs of corresponding sides of the triangles intersect, lie on one line.''

The necessary condition for solving Hilbert's fourth problem is the  requirement that a metric space that satisfies the axioms of this problem should be Desarguesian, i.e.,:
* if the space is of dimension 2, then the Desargues's theorem and its inverse should hold;
* if the space is of dimension greater than 2, then any three points should lie on one plane.

For Desarguesian spaces [[Georg Hamel]] proved that  every solution of Hilbert's fourth  problem can be represented in a real [[projective space]] <math>RP^{n}</math> or in a convex domain of <math>RP^{n}</math> if one determines  the congruence of segments by equality of their lengths in a special metric for which the lines of the projective space are geodesics.

Metrics of this type are called '''flat''' or '''projective'''.

Thus, the solution of  Hilbert's fourth problem was reduced to the solution of the problem of constructive determination of all complete flat metrics.

Hamel solved this problem under the assumption of high regularity of the metric.<ref name="Hamel1903" /> However, as simple examples show, the class of regular flat metrics is smaller than the class of all flat metrics. The axioms of geometries under consideration imply only a continuity of the metrics. Therefore, to solve Hilbert's fourth problem completely it is necessary  to determine  constructively all the continuous flat metrics.