  Given any two K\"ahler manifolds $X_1$ and $X_2$, L. Lempert recently proved
that if their spaces of K\"ahler potentials are isometric with respect to the
Mabuchi metric, then $X_1$ and $X_2$ must be diffeomorphic. We prove that this
is no longer the case for Sasaki manifolds. Then, considering regular Sasaki
manifolds $M_1$ and $M_2$, we prove that if the spaces of potentials are
isometric, then $M_1$ and $M_2$ must have, among others, the same universal
covering space. Finally, getting rid of the regularity assumption on $M_1$ and
$M_2$, we investigate the consequences of the existence of affine Mabuchi
isometries: this leads to a family of Sasaki isospectral structures.

Differential Geometry;Complex Variables