  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
$\min_{p\in X}d(p,\tau(p))$ 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.
