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The twistor space of self-dual positive Einstein manifolds naturally admitstwo 1-parameter families of Riemannian metrics, one is the family of canonicaldeformation metrics and the other is the family introduced by B. Chow and D.Yang in 1989. The purpose of this paper is to compare these two families. Inparticular we compare the Ricci tensor and the behavior under the Ricci flow ofthese families. As an application, we propose a new proof to the fact that alocally irreducible self-dual positive Einstein 4-manifold is isometric toeither $S^4$ with a standard metric or $\PP^2(\C)$ with a Fubini-Study metric.

Moving frames on the twistor space of self-dual positive Einstein 4-manifolds