Metric and Periodic Lines in Real Inner Product Space Geometries

Metric and Periodic Lines in de Sitter’s World

“…A subset k ⊂ S(X) is called a ρ-periodic line of S(X), [4], if there exists a bijection ϕ : k → [0, ρ[ such that, for all x, y ∈ k,…”

“…Benz [4,5] determined the metric lines of hyperbolic geometry, the metric and periodic lines of Euclidean geometry, the 2π-periodic lines of spherical geometry, and the π-periodic lines of elliptic geometry. The metric and periodic lines of Lorentz-Minkowski geometry have been determined in [10,11].…”

“…If x 0 , x 4 = 1, then x 4 = p cos ϕ4 k2 + q sin ϕ4 k2 + r. If x 0 , x 4 > 1, then (11) also holds for i = 4. If x 0 , x 4 2 < 1, then…”

“…For example, distance functions can be characterized as two-point invariants which are additive on lines [2]. On the other hand, the notion of metric and periodic lines are examples of lines characterized by distance functions [4][5][6].…”

Metric lines in Lorentz–Minkowski geometry

“…Benz determined in [3] the metric lines of hyperbolic geometry, the metric and periodic lines of Euclidean geometry, the 2π-periodic lines of spherical geometry, and the π-periodic lines of elliptic geometry. He then posed the problem to determine all metric lines of (X, d).…”