Conformal Differential Geometry

Q-Curvature and Conformal Holonomy

  • Helga Baum
  • Andreas Juhl

Part of the Oberwolfach Seminars book series (OWS, volume 40)

Table of contents

  1. Front Matter
    Pages i-x
  2. Pages 1-78
  3. Pages 79-138
  4. Back Matter
    Pages 139-152

About this book

Introduction

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).

Keywords

Spinor Tensor conformal invariants curvature differential geometry holonomy

Authors and affiliations

  • Helga Baum
    • 1
  • Andreas Juhl
    • 1
    • 2
  1. 1.Institut für MathematikHumboldt-UniversitätBerlinGermany
  2. 2.Matematiska InstitutionenUniversitet UppsalaUppsalaSweden

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-7643-9909-2
  • Copyright Information Birkhäuser Basel 2010
  • Publisher Name Birkhäuser Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-7643-9908-5
  • Online ISBN 978-3-7643-9909-2
  • About this book