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,[1] and later rediscovered within the formalism of local twistors and generalized to projective connections by Michael Eastwood et al. in [2]

Conformal manifolds

The tractor bundle for a -dimensional conformal manifold of signature is a rank vector bundle equipped with the following data:

  • a metric , of signature ,
  • a line subbundle ,
  • a linear connection , preserving the metric , and satisfying the nondegeneracy property that, for any local non-vanishing section of the bundle ,

is a linear isomorphism at each point from the tangent bundle of () to the quotient bundle , where denotes the orthogonal complement of in relative to the metric .

Given a tractor bundle, the metrics in the conformal class are given by fixing a local section of , and defining for ,

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 of the Cartan geometry determined by the conformal structure. The conformal group for a manifold of signature is , and one obtains the tractor bundle (with connection) as the connection induced by the Cartan conformal connection on the bundle associated to the standard representation of the conformal group. Because the fibre of the Cartan conformal bundle is the stabilizer of a null ray, this singles out the line bundle .

More explicitly, suppose that is a metric on , with Levi-Civita connection . The tractor bundle is the space of 2-jets of solutions to the eigenvalue equation

where is the Schouten tensor. A little work then shows that the sections of the tractor bundle (in a fixed Weyl gauge) can be represented by -vectors

The connection is

The metric, on and is:

The preferred line bundle is the span of

Given a change in Weyl gauge , the components of the tractor bundle change according to the rule

where , and the inverse metric has been used in one place to raise the index. Clearly the bundle 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

Let be a projective manifold of dimension . Then the tractor bundle is a rank vector bundle , with connection , on equipped with the additional data of a line subbundle such that, for any non-vanishing local section of , the linear operator

is a linear isomorphism of the tangent space to .

One recovers an affine connection in the projective class from a section of by defining

and using the aforementioned isomorphism.

Explicitly, the tractor bundle can be represented in a given affine chart by pairs , where the connection is

where is the projective Schouten tensor. The preferred subbundle is that spanned by .

Here the projective Schouten tensor of an affine connection is defined as follows. Define the Riemann tensor in the usual way (indices are abstract)

Then

where the Weyl tensor is trace-free, and (by Bianchi).

References

  1. Thomas, T. Y., "On conformal differential geometry", Proc. N.A.S. 12 (1926), 352–359; "Conformal tensors", Proc. N.A.S. 18 (1931), 103–189.
  2. Bailey, T. N.; Eastwood, M. G.; Gover, A. R. (1994), "Thomas's structure bundle for conformal, projective and related structures", Rocky Mountain J, 24: 1191–1217
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.