David Hilbert's Problem #4

Hilbert’s Fourth Problem

The Straight Line as The Shortest Distance Between Points: Construct all metrics where lines are geodesics.

Hilbert’s fourth problem is about what happens when you relax the rules of Euclidean geometry. Specifically, what geometries can exist in which a straight line is the shortest distance between two points but in which some axioms of Euclidean geometry are abandoned? Some mathematicians consider the problem too vague to have a real resolution, but there are solutions for some interpretations of the question.

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