Communications in Mathematical Physics

, Volume 235, Issue 2, pp 339–378

Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus

  • A. Rod Gover
  • Lawrence J. Peterson


 We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6 and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on [12], where the relationship of the normal standard tractor bundle to the ambient construction is described.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • A. Rod Gover
    • 1
  • Lawrence J. Peterson
    • 2
  1. 1.Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand. E-mail:
  2. 2.Department of Mathematics, The University of North Dakota, Grand Forks, ND 58202-8376, USA. E-mail: lawrence.peterson@und.nodak.eduUS

Personalised recommendations