An Exactly Solvable Isochronous Geodesic Flow on a Non-Compact Paraboloid

This work presents an exactly solvable example of a geodesic flow on a non-compact surface of revolution exhibiting global isochrony. For the Riemannian metric ds2=(1+r2) dr2+r2 dθ2,ds^2 = (1+r^2)\,dr^2 + r^2\,d\theta^2,ds2=(1+r2)dr2+r2dθ2, all bounded geodesics are closed and possess identical angular advance Δθ=2π\Delta\theta = 2\piΔθ=2π, independent of angular momentum. The analysis is fully elementary and avoids elliptic integrals, relying instead on a precise cancellation in the orbit equation. The result provides a rare instance of a superintegrable geodesic system on a curved, non-compact manifold, extending classical notions of isochrony—traditionally associated with flat or compact settings—into intrinsic differential geometry. This example serves both as a theoretical contribution to the study of geodesic dynamics and as a benchmark for analytical, symbolic, and numerical investigations of exactly solvable geometric systems