Editing Hilbert's fourth problem (section)

{{Short description|Construct all metric spaces where lines resemble those on a sphere}}
In [[mathematics]], '''Hilbert's fourth problem''' in the 1900 list of [[Hilbert's problems]] is a foundational question in [[geometry]]. In one statement derived from the original, it was to find — up to an isomorphism  — all [[geometry|geometries]] that have an [[axiom]]atic system of the classical geometry ([[Euclidean geometry|Euclidean]], [[Hyperbolic geometry|hyperbolic]] and [[Elliptic geometry|elliptic]]), with those axioms of [[congruence (geometry)|congruence]] that involve the concept of the angle dropped, and `[[triangle inequality]]', regarded as an axiom, added.

If one assumes the continuity axiom in addition, then, in the case of the Euclidean plane, we come to the problem posed by [[Jean Gaston Darboux]]: "To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines."<ref>{{cite book
| last1=Darboux | first1=Gaston | authorlink1=Jean Gaston Darboux
| title=Leçons sur la theorie generale des surfaces
| volume=III
| place=Paris
| date=1894}}</ref>

There are several interpretations of the original statement of [[David Hilbert]]. Nevertheless, a solution was sought, with the German mathematician [[Georg Hamel]] being the first to contribute to the solution of Hilbert's fourth problem.<ref name="Hamel1903">{{cite journal
| last1=Hamel | first1=Georg
| title=Uber die Geometrien in denen die Geraden die Kürzesten sind
| journal=Mathematische Annalen
| volume=57
| date=1903
| issue=2
| pages=221–264
| doi=10.1007/BF01444348 | doi-access=free}}</ref>

A recognized solution was given by Soviet mathematician [[Aleksei Pogorelov]] in 1973.<ref name="Pogorelov1973">А.&nbsp;В.&nbsp;Погорелов, ''Полное решение IV проблемы Гильберта'', ДАН СССР №&nbsp;208, т.1 (1973), 46–49. English translation: {{cite journal
| last1=Pogorelov | first1=A. V.
| title=A complete solution of ''Hilbert's fourth problem''
| journal=Doklady Akademii Nauk SSSR
| volume=208
| issue=1
| date=1973
| pages=48–52}}</ref><ref name="Pogorelov1974">А.&nbsp;В.&nbsp;Погорелов, ''Четвертая Проблема Гильберта''. Наука, 1974.
English translation: A.V. Pogorelov, ''Hilbert's Fourth Problem'', Scripta Series in Mathematics, Winston and Sons, 1979.</ref> In 1976, Armenian mathematician [[Rouben V. Ambartzumian]] proposed  another proof of Hilbert's fourth problem.<ref name="Ambartzumian1976">R. V. Ambartzumian, A note on pseudo-metrics on the plane, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
1976, Volume 37, Issue 2, pp 145–155</ref>