In this paper, involutions 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 tau, it is known that min (p is an element of X) d(p, tau(p)) is bounded by a constant which depends on the area of X. The corresponding claim is proved to be false in odd genus, and the optimal constant for hyperbolic Riemann surfaces is calculated in genus 2.