Conformal Differential Geometry: Q-Curvature and Conformal Holonomy (Oberwolfach Seminars, Vol. 40) (Oberwolfach Seminars, 40)

Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics. Well-known examples of such operators are the Yamabe-, the Paneitz-, the Dirac- and the twistor operator. The aim of the seminar was to present the basic ideas and some of the recent developments around Q-curvature and conformal holonomy. The part on Q-curvature discusses its origin, its relevance in geometry, spectral theory and physics. Here the influence of ideas which have their origin in the AdS/CFT-correspondence becomes visible. The part on conformal holonomy describes recent classification results, its relation to Einstein metrics and to conformal Killing spinors, and related special geometries. "Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics. Well-known examples of conformally covariant operators are the Yamabe, the Paneitz, the Dirac and the twistor operator. These operators are intimately connected with the notion of Branson's Q-curvature. The aim of these lectures is to present the basic ideas and some of the recent developments around Q-curvature and conformal holonomy." "The part on Q-curvature starts with a discussion of its origins and its relevance in geometry and spectral theory. The subsequent lectures describe the fundamental relation between Q-curvature and scattering theory on asymptotically hyperbolic manifolds. Building on this, they introduce the recent concept of Q-curvature polynomials and use these to reveal the recursive structure of Q-curvatures. The part on conformal holonomy starts with an introduction to Cartan connections and their holonomy groups. Then we define holonomy groups of conformal manifolds, discuss their relation to Einstein metrics and recent classification results in Riemannian and Lorentzian signature. In particular, we explain the connection between conformal holonomy and conformal Killing forms and spinors, and describe Fefferman metrics in CR geometry as Lorentzian manifolds with conformal holonomy SU(1,m)."--Jacket Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics. Well-known examples of conformally covariant operators are the Yamabe, the Paneitz, the Dirac and the twistor operator. These operators are intimely connected with the notion of Branson's Q-curvature. The aim of these lectures is to present the basic ideas and some of the recent developments around Q -curvature and conformal holonomy. The part on Q -curvature starts with a discussion of its origins and its relevance in geometry and spectral theory. The following lectures describe the fundamental relation between Q -curvature and scattering theory on asymptotically hyperbolic manifolds. Building on this, they introduce the recent concept of Q -curvature polynomials and use these to reveal the recursive structure of Q -curvatures. The part on conformal holonomy starts with an introduction to Cartan connections and its holonomy groups. Then we define holonomy groups of conformal manifolds, discuss its relation to Einstein metrics and recent classification results in Riemannian and Lorentzian signature. In particular, we explain the connection between conformal holonomy and conformal Killing forms and spinors, and describe Fefferman metrics in CR geometry as Lorentzian manifold with conformal holonomy SU(1,m) Title Page 4 Copyright Page 5 Table of Contents 6 Preface 7 Chapter 1 Q-curvature 11 1.1 The flat model of conformal geometry 11 1.2 Q-curvature of order 4 15 1.3 GJMS-operators and Branson’s Q-curvatures 31 1.4 Scattering theory 41 1.5 Residue families and the holographic formula for Qn 56 1.6 Recursive structures 69 Chapter 2 Conformal holonomy 89 2.1 Cartan connections and holonomy groups 89 2.2 Holonomy groups of conformal structures 99 2.2.1 The first prolongation of the conformal frame bundle 100 2.2.2 The normal conformal Cartan connection – invariant form 104 2.2.3 The normal conformal Cartan connection – metric form 107 2.2.4 The tractor connection and its curvature 109 2.3 Conformal holonomy and Einstein metrics 113 2.4 Classification results for Riemannian and Lorentzian conformal holonomy groups 119 2.5 Conformal holonomy and conformal Killing forms 121 2.6 Conformal holonomy and conformal Killing spinors 125 2.7 Lorentzian conformal structures with holonomy group SU(1,m) 138 2.7.1 CR geometry and Fefferman spaces 139 2.7.2 Conformal holonomy of Fefferman spaces 145 2.8 Further results 147 Bibliography 149 Index 158