Prince Rupert's Problem in Non-Euclidean Geometries

We formulate a centered version of Prince Rupert's problem for exponential cubes in three-dimensional spaces of constant sectional curvature. Transvections provide exact rigid motions, while Fermi coordinates reduce the existence of a passage to inclusions between two-dimensional transverse footprints. We establish quantitative stability bounds and a Euclidean limit with an error of order $O(|\kappa|)$, where $\kappa=Kr^2$. For $|\kappa|\le 1/100$, we determine the capacities of the diagonal and Nieuwland configurations and construct a passage allowing a uniform $5.8\%$ radial enlargement, with a solid torus as the remaining material. Moreover, for sufficiently small positive curvature, we prove that the full centered index of the cube remains equal to $3\sqrt2/4$, without imposing any fixed inclination on the moving cube. Throughout, we carefully distinguish the existence of a passage, optimality within the centered protocol, and the questions that remain open in hyperbolic geometry and for more general rigid motions.