In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle).

The term tractor is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan conformal connection, and later rediscovered within the formalism of local twistors and generalized to projective connections by Michael Eastwood et al. in Tractor bundles can be defined for arbitrary parabolic geometries.

Conformal manifolds

equipped with the following data:

• ,

• ,

• a linear connection,

is a linear isomorphism at each point from the tangent bundle.

To go the other way, and construct a tractor bundle from a conformal structure, requires more work. The tractor bundle is then an associated bundle.

is the Schouten tensor

is invariant under the change in gauge, and the connection can be shown to be invariant using the conformal change in the Levi-Civita connection and Schouten tensor.

Projective manifolds

is a linear isomorphism of the tangent space.

and using the aforementioned isomorphism.

.

(by Bianchi).

References