Conformal Flatness and Self-duality of Thurston-geometries

We show which Thurston-geometries in dimensions 3 and 4 admit invariant conformally flat or half-conformally flat metrics. 1. Threeand four-dimensional geometries A geometry in the sense of Thurston is a pair (X, G) where X is a simply connected smooth manifold on which the connected Lie group G acts smoothly in a transitive fashion such that the following hold: (1) the stabilizer of a point is compact; (2) G has a discrete subgroup Γ such that Γ \ X has finite volume for some G-invariant smooth measure; (3) G is maximal with these properties. Maximality means that if there is a pair (X ′, G′) satisfying (1) and (2) together with a diffeomorphism f : X → X ′ and an injective homomorphism of Lie groups φ : G → G′ such that f is φ-equivariant, then φ must in fact be an isomorphism. We shall refer to G as the structure group of the geometry, and X will be called model space. Of course, such a manifold X can be furnished with a Riemannian metric such that G acts via isometries. But there is a whole range of possible choices provided the isotropy groups are small. In dimension three, such geometries have been classified by Thurston (for an account see [Sco]), and in dimension four this has been done by Filipkiewicz [Fil], see also [Wall1]. The reader is refered to [Pat] for a general approach to the classification using Cartan triples. For later reference we shall collect here the possible different spaces X which appear in these lists together with the identity component of the stabilizers of the action (see Table 1). For simplicity we shall refer to the geometries S̃L2(R), Nil, Sol, Sol 0, F , Nil, Sol m,n, Sol 4 1 as Lie group geometries. Observe that for any Lie group geometry (X,G) the model-space X is itself a Lie group, and the group G contains X as a subgroup. F 4 is the only geometry which does not admit compact models. A manifold M is said to have a geometry of type (X,G) provided M is covered by a collection of open sets each diffeomorphic to some open set in X such that the transition functions are given by elements of G. If the manifold X in a pair (X,G) admits a geometric structure such as a metric or an almost complex structure invariant under the action of G, this structure Received by the editors July 23, 1996. 1991 Mathematics Subject Classification. Primary 53A35. c ©1998 American Mathematical Society 1165 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use