Conformal Differential Geometry Q-Curvature and Conformal Holonomy
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
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Helga Baum Andreas Juhl
Conformal Differential Geometry Q-Curvature and Conformal Holonomy
Birkhäuser Basel · Boston · Berlin
Authors: Helga Baum Humboldt-Universität Institut für Mathematik 10099 Berlin Germany e-mail: [email protected]
Andreas Juhl Humboldt-Universität Institut für Mathematik 10099 Berlin Germany e-mail: [email protected] and Universitet Uppsala Matematiska Institutionen Box-480 75106 Uppsala Sweden e-mail: [email protected]
2000 Mathematics Subject Classification 53A30, 53B15, 53B20, 53B25, 53B30, 53B50, 53C05, 53C10, 53C15, 53C21, 53C25, 53C27, 53C28, 53C29, 53C50, 53C80, 58 J50, 32V05
Library of Congress Control Number: 2009942367
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-7643-9908-5 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
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e-ISBN 978-3-7643-9909-2
987654321
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Contents Preface 1 Q-curvature 1.1 The flat model of conformal geometry . . . . . . . . 1.2 Q-curvature of order 4 . . . . . . . . . . . . . . . . . 1.3 GJMS-operators and Branson’s Q-curvatures . . . . 1.4 Scattering theory . . . . . . . . . . . . . . . . . . . . 1.5 Residue families and the holographic formula for Qn 1.6 Recursive structures . . . . . . . . . . . . . . . . . .
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2 Conformal holonomy 2.1 Cartan connections and holonomy groups . . . . . . . . . . . . . 2.2 Holonomy groups of conformal structures . . . . . . . . . . . . . 2.2.1 The first prolongation of the conformal frame bundle . . . 2.2.2 The normal conformal Cartan connection – invariant form 2.2.3 The normal conformal Cartan connection – metric form . 2.2.4 The tractor connection and its curvature . . . . . . . . . . 2.3 Conformal holonomy and Einstein metrics . . . . . . . . . . . . . 2.4 Classification results for Riemannian and Lorentzian conformal holonomy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conformal holonomy and conformal Killing forms . . . . . . . . . 2.6 Conformal holonomy and conformal Killing spinors . . . . . . . . 2.7 Lorentzian conformal structures with holonomy group SU(1, m) . 2.7.1 CR geometry and Fefferman spaces . . . . . . . . . . . . . 2.7.2 Conformal holonomy of Fefferman spaces . . . . . . . . . 2.8 Further result
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