It is known that the moduli space of Einstein structures in four dimensions is generally considered to be rigid so that Einstein metrics tend to be isolated modulo diffeomorphisms under infinitesimal Einstein deformations. We examine the rigidity of the Einstein structure by considering deformations of the round four-sphere. We show that any deviation from the standard metric of the round four-sphere (except for scaling) breaks the Einstein condition. This further supports the idea of rigidity. We analyze the Einstein structure of four-manifolds based on the irreducible decomposition of the self-dual structure of Einstein manifolds.

v3: 27 pages, 1 figure, Sections 4 & 5 added; to appear in Journal of Geometry and Physics