Assume that $M$ is a simply connected closed Riemannian manifold with no boundary and nonnegative sectional curvaure Assume that ${\bf Z}\_n=(g),\ n\geq 3$ acts on $M$ isometrically. Then if $gx=x$, i.e., it is a fixed point, clearly $g$ acts on ${\rm cut}\ x$ Here I have a question : $g\cdot x \in {\rm cut}\ x$ can happen for some $x$ ? 


(Background : In the paper CRITICAL POINTS OF THE DISPLACEMENT FUNCTION OF AN ISOMETRY - VILNIS OZOLS, he consider an isometry whose displacement is small enough so that it takes each point into the complement of its cut locus. )



How isometric action on Riemannian manifold acts on cut locus
It can happen that $g.x\in cut(x)$ for some $x$. This is what happens for $S\_{n+1}$ acting by permutation of homogeneous coordinates on $\mathbb{CP}^n$.


