On the functional equation of periodic lines (Q1428269)

In a metric space \((S,d) \) a metric line is a function \(x\) from \(\mathbb R\) into \(S\) such that \(d(x(u), x(v)) = | u-v|\), for all \(u, v\) in \(\mathbb R\). Given \(k > 0\), a \(k\)-periodic line of \((S,d)\) is a mapping \(x\) from \([0,k]\) into \(S\) such that \(d(x(u), x(v)) = | u-v|\) when \(| u-v| \leq k/2\) and \(d(x(u), x(v)) = k - | u - v |\) otherwise. The author shows that if \( S\) is a real inner product space of (finite or infinite) dimension \(\geq 2\), \(\cosh d (x,y) = \sqrt {1+x^2} \cdot \sqrt {1+y^2} - xy\) and \(k > 0\) then the equation of the \(k\)-periodic lines has no solution. This paper completes a study of the author on metric lines and \(k\)-periodic lines in euclidean, hyperbolic, spherical or elliptic geometry over real inner product spaces (\(\dim S \geq 2\)).