2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74810Involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface $X$ of even genus with an arbitrary Riemannian metric $d$ admitting an involution $τ$, it is known that $\min_{p\in X}d(p,τ(p))$ is bounded by a constant which depends on the genus of $X$. The equivalent result is proved to be false in odd genus, and the optimal constant for hyperbolic Riemann surfaces is calculated in genus 2.12 pages, 8 figuresDifferential Geometry30F45; 30F20, 53C23text