Conformal differential geometry
Subtitle: Q-curvature and conformal holonomy
By: Helga Baum, Andreas Juhl.
Contributors: Juhl, Andreas, 1955-
Series: Oberwolfach seminars -- 41
Publisher: Birkhäuser
Published: 2010
Place of publication: Basel; Boston
Language: eng
Number of pages: 152
ISBN-13: 9783764399085
Subjects: Conformal geometry; Geometry, Differential
Description: 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).
Notes: Includes bibliographical references (p. [139]-147) and index.