  It is known that the moduli space of Einstein structures is an isolated point
so that an Einstein structure has no infinitesimal Einstein deformations. We
examine the rigidity of the Einstein structure by considering deformations of
the round four-sphere $\mathbb{S}^4$. We show that any deformation of the round
four-sphere $\mathbb{S}^4$ causes it to deviate from the Einstein structure,
except trivial deformations at most only changing the size of the sphere.
