Monatshefte für Mathematik

, Volume 152, Issue 4, pp 265–282 | Cite as

Ambient connections realising conformal Tractor holonomy

  • Stuart Armstrong
  • Thomas Leistner


For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham.

2000 Mathematics Subject Classification: 53C29, 53A30 
Key words: Holonomy groups, conformal holonomy, ambient construction, C-spaces 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Stuart Armstrong
    • 1
  • Thomas Leistner
    • 2
  1. 1.University of OxfordOxfordUK
  2. 2.University of AdelaideSAAustralia

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