By Helga Baum
Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy teams etc.) are of relevant importance in differential geometry and physics. recognized examples of such operators are the Yamabe-, the Paneitz-, the Dirac- and the twistor operator. the purpose of the seminar was once to provide the fundamental rules and a few of the hot advancements round Q-curvature and conformal holonomy. The half on Q-curvature discusses its foundation, its relevance in geometry, spectral concept and physics. the following the impression of rules that have their beginning within the AdS/CFT-correspondence turns into obvious. The half on conformal holonomy describes contemporary class effects, its relation to Einstein metrics and to conformal Killing spinors, and comparable precise geometries.
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Extra resources for Conformal Differential Geometry: Q-Curvature and Conformal Holonomy
Of Q-curvatures. In the following, we shall be interested mostly in manifolds of even dimension. The critical GJMS-operators and the critical Q-curvatures play a distinguished role. In fact, similarly as for n = 2 and n = 4, the critical GJMS-operators govern the conformal transformation laws of the critical Q-curvatures. 1. 2 (The fundamental identity). On Riemannian manifolds (M, g) of even dimension n, n enϕ Qn(e2ϕ g) = Qn (g) + (−1) 2 Pn (g)(ϕ). 7) Proof. 2) to the function n u = e( 2 −N )ϕ .
One of the main features of these metrics on X is that, near the boundary M , they are determined by their conformal inﬁnities (at least to some extent). For more details concerning the following discussion we refer to [FG07]. 2. A Poincar´e-Einstein metric, or just Poincar´e metric, for (M, c) is a conformally compact metric g+ on X so that 1. g+ has conformal inﬁnity c. 2. For odd n, Ric(g+ ) + ng+ vanishes to inﬁnite order along M . 3. 30) trh (i∗ (ρ−(n−2) (Ric(g+ ) + ng+ ))) = 0, h ∈ c. 31) Note that the hyperbolic metric on Bn+1 is a Poincar´e-Einstein metric with conformal inﬁnity given by the conformal class of the round metric.
1. , ρ > 0 on X, ρ = 0 on M deﬁning function ρ ∈ C ∞ (X) and dρ = 0 on M , so that ρ2 g extends to a smooth metric up to the boundary. 1, the metrics g¯ = ρ2 g are called conformal compactiﬁcations of g. Any conformal compactiﬁcation induces a metric on M by i∗ (ρ2 g), ¯ A change of the deﬁning function yields a metric in the same where i : M → X. conformal class on the boundary. The conformal class c = [i∗ (ρ2 g)] is called the conformal inﬁnity of g. The function |dρ|2g¯ on M does not depend on the choice of ρ.
Conformal Differential Geometry: Q-Curvature and Conformal Holonomy by Helga Baum