The Ricci form of Kähler manifolds

Abstract

Kahler metrics as geometric U m -structures We start with a short review on G -structures which will help us to characterize Kahler and Ricci-flat Kahler metrics. Let M be an n -dimensional manifold and let G be any closed subgroup of Gl n (ℝ). D efinition 17.1. A topological G-structure on M is a reduction of the principal frame bundle Gl( M ) to G. A geometrical G-structure is given by a topological G-structure G ( M ) together with a torsion-free connection on G ( M ). Let us give some examples. An orientation on M is a -structure. An almost complex structure is a Gl m (ℂ)-structure, for n = 2 m . A Riemannian metric is an O n -structure. Recall (Proposition 4.7) that if the group G is the stabilizer of an element ξ of some representation ρ : Gl n (ℝ) ξ End( V ), then a G -structure is simply a section σ in the associated bundle Gl( M )× ρ , where O denotes the Gl n (ℝ)-orbit of ξ in V . By Theorem 5.16, the G -structure is geometrical if and only if there exists a torsion-free linear connection on M with respect to which σ is parallel. P roposition 17.2. The U m -structure defined by an almost complex structure J together with a Hermitian metric h on a manifold M is geometrical if and only if the metric is Kahler . P roof . The point here is that if G is a closed subgroup of O n then there exists at most one torsion-free connection on any G -structure (by the uniqueness of the Levi-Civita connection).