arXiv:math/0105041v1 [math.DG] 5 May 2001 WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS. A STATE OF THE ART. LIVIU ORNEA This is a survey on quaternion Hermitian Weyl (locally conformally quaternion K¨ahler) and hyper- hermitian Weyl (locally conformally hyperk¨ ahler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a study is two-fold: it comes from the constantly growing interest in Weyl (and Einstein-Weyl) geometry and, on the other hand, from the necessity of understanding the existing classes of quaternion Hermitian manifolds. Various geometries are involved in the following discussion. The first sections give the minimal back- ground on Weyl geometry, quaternion Hermitian geometry and 3-Sasakian geometry. The reader is sup- posed familiar with Hermitian (K¨ ahler and, if possible, locally conformally K¨ahler) and metric contact (mainly Sasakian) geometry. All manifolds and geometric objects on them are supposed differentiable of class C ∞ . 1. Weyl structures We present here the necessary background concerning Weyl structures on conformal manifolds. We refer to [16], [18], [21] or to the most recent survey [14] for more details and physical interpretation (motivation) for Weyl and Einstein-Weyl geometry. Let M be a n-dimensional, paracompact, smooth manifold, n ≥ 2. A CO(n) ≃ O(n) × R + structure on M is equivalent with the giving of a conformal class c of Riemannian metrics. The pair (M,c) is a conformal manifold. For each metric g ∈ c one can consider the Levi-Civita connection ∇ g , but this will not be compatible with the conformal class. Instead, we shall work with CO(n)-connections. Precisely: Definition 1.1. A Weyl connection D on a conformal manifold (M,c) is a torsion-free connection which preserves the conformal class c. We say that D defines a Weyl structure on (M,c) and (M,c,D) is a Weyl manifold. Preserving the conformal class means that for any g ∈ c, there exists a 1-form θ g (called the Higgs field) such that Dg = θ g ⊗ g. This formula is conformally invariant in the following sense: if h = e f g, f ∈C ∞ (M ), then θ h = θ g − df. (1.1) Conversely, if one starts with a fixed Riemannian metric g on M and a fixed 1-form θ (with T = θ ♯ ), the connection D = ∇ g − 1 2 {θ ⊗ Id + Id ⊗ θ − g ⊗ T } 1991 Mathematics Subject Classification. 53C15, 53C25, 53C55, 53C10. Key words and phrases. Weyl structure, quaternion Hermitian manifold, locally conformally K¨ ahler geometry, 3-Sasakian geometry, Einstein manifold, homogeneous manifold, foliation, complex structure, Riemannian submersion, QKT structure. The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme. 1