Conformal Differential Geometry: Q-Curvature and Conformal Holonomy

✍ Scribed by Helga Baum, Andreas Juhl

• Publisher: Birkhäuser Basel
• Year: 2010
• Tongue: English
• Leaves: 164
• Series: Oberwolfach Seminars
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Oberwolfach Seminars
Volume 40......Page 3
Conformal
Differential Geometry......Page 4
ISBN 9783764399085......Page 5
Table of Contents......Page 6
Preface......Page 8
1.1 The flat model of conformal geometry......Page 13
1.2 Q-curvature of order 4......Page 17
1.3 GJMS-operators and Branson’s Q-curvatures......Page 33
1.4 Scattering theory......Page 43
1.5 Residue families and the holographic formula for Qn......Page 58
1.6 Recursive structures......Page 71
2.1 Cartan connections and holonomy groups......Page 91
2.2 Holonomy groups of conformal structures......Page 101
2.2.1 The first prolongation of the conformal frame bundle......Page 102
2.2.2 The normal conformal Cartan connection – invariant form......Page 106
2.2.3 The normal conformal Cartan connection – metric form......Page 109
2.2.4 The tractor connection and its curvature......Page 111
2.3 Conformal holonomy and Einstein metrics......Page 115
2.4 Classification results for Riemannian and Lorentzian conformal holonomy groups......Page 121
2.5 Conformal holonomy and conformal Killing forms......Page 123
2.6 Conformal holonomy and conformal Killing spinors......Page 127
2.7 Lorentzian conformal structures with holonomy group SU(1,m)......Page 140
2.7.1 CR geometry and Fefferman spaces......Page 141
2.7.2 Conformal holonomy of Fefferman spaces......Page 147
2.8 Further results......Page 149
Bibliography......Page 151
Index......Page 161