Four vertex theorems on Lorentzian spheres \(S^ 2_ 1\) and \(H^ 2_ 1\) (Q1842029)

The author proves a four vertex theorem for the geodesic curvature of simple closed curves in de Sitter space \(S^2_1 = \{x = (x_1, x_2, x_3) \mid \langle x, x\rangle_1 = x^2_1 + x^2_2 - x^2_3 = 1\}\) and anti-de Sitter space \(H^2_1 = \{x = (x_1, x_2, x_3) \mid \langle x, x\rangle_2 = x^2_1 - x^2_2 - x^2_3 = -1\}\). The curves are assumed to be spacelike (for \(S^2_1)\) resp. timelike (for \(H^2_1)\) and convex in the sense that the sign of the geodesic curvature does not change. The proof consists of a straightforward application of the well-known four vertex theorem for simple closed convex curves in the hyperbolic plane [\textit{G. W. M. Kallenberg}, Nieuw Arch. Wiskunde, III. Ser. 9, 1-15 (1961; Zbl 0192.279)].