Moving Frames on the Twistor Space of Self-dual Positive Einstein 4-manifolds

The twistor space Z of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in [C-Y]. The purpose of this paper is to compare these two families. In particular we compare the Ricci tensor and the behavior under the Ricci flow of these families. As an application, we propose a new proof to the fact that a locally irreducible self-dual positive Einstein 4-manifold is isometric to either S with a standard metric or P(C) with a Fubini-Study metric. 0. Introduction Let R be the oriented Euclidean 4-space which is identified with H and we consider the right multiplication of the group Sp(1) of unit quaternions. This defines a subgroup Sp(1)− in SO(4) which is isomorphic to Sp(1). The centralizer of this subgroup in SO(4) is again isomorphic to Sp(1) identified with the left multiplication of unit quaternions, which we denote by Sp(1)+. We thus have the group homomorphism Sp(1)× Sp(1) → SO(4) defined by the left/right multiplication of unit quaternions on H. Its kernel is ±(id, id) ∼= Z2 and therefore we have the decomposition of the Lie algebra so(4) = sp(1)+ ⊕ sp(1)−. Here, sp(1)± are copies of sp(1) and sp(1)+ (resp. sp(1)−) corresponds to the left (resp. right) multiplication of unit quaternions. In terms of the identification so(4) = ΛR, the component sp(1)+ (resp. sp(1)−) corresponds to the self-dual (resp. anti-self-dual) 2-forms. A Riemannian 4-manifold (M, gM) is said to be self-dual positive Einstein if its self-dual part W− of the Weyl tensor vanishes and Einstein condition is satisfied with positive scalar curvature. The twistor space of a self-dual Riemannian 4manifold (M, g) is defined as follows. Let P be the holonomy reduction of the bundle of oriented orthonormal frames of (M, g). For instance, if M = (S, gstd), P is a principal SO(4)-bundle over S and if M = (P, gFS), P is a principal S(U(1)× U(2))-bundle. The holonomy group of a self-dual positive Einstein manifold (M, g) is a subgroup of SO(4) and we write it as H. The fiber of P over m ∈ M consists 1 The twistor space of a self-dual positive Einstein 4-manifold is a positive Kähler-Einstein manifold and hence simply connected. This implies that any self-dual positive Einstein 4-manifold is simply connected. Typeset by AMS-TEX 1