Cartan geometry: jet space perspective on the tractor bundle

Let G a Lie group and HG a Lie subgroup. For simplicity we assume that the adjoint action of H on g/h is faithful.

Let M a differentiable manifold of the same dimension as G/H.
A (H,G)-Cartan geometry on M is defined as a reduction of the structure group of the frame bundle of M to H (more accurately to the adjoint action of H on g/h), along with a “Cartan connection one-form” which is a g-valued equivariant one-form which is non degenerate. Call the H-principal bundle P.

Here are two things one can do with Cartan geometries:

  • Construct the associated (right) principal G-bundle P×HG on which the Cartan connection can be extended to a G-principal connection. For a G-module V, one can construct the associated bundle P×HG×GVP×HV (called tractor bundle) which is in particular endowed with a induced G-principal connection called the tractor connection.

  • Considering GG/H as a reference space with (H,G)-Cartan geometry, one can construct the space of 1-jets of maps from M to G/H preserving the H-structure (between the chosen points), in other words linear isomorphisms between tangent spaces which preserve the class of “H-frames”. It is readily constructed as (P×G)/H for the diagonal action on the right. It is a bundle over M×G/H with fibre diffeomorphic to H; the fibre over M is G. It inherits an action of G (naturally) on the left through the germs of the associated diffeomorphisms acting on the 1-jets and is hence a left principal G-bundle over M.

Now, the two constructed G-principal bundles are isomorphic. They can be easily related by the inversion of the component G of P×G before taking the quotient by H. If one is rather willing to “invert” the principal bundle P, using the inverse right action, then the first construction can be related to 1-jets of diffeomorphisms between M and HG.

My question is: how can the tractor bundle construction be naturally expressed with the 1-jet interpretation of the “extended” G-principal bundle?
To me the natural idea would be to consider applications G/HV which are equivariant under the (left) action of G, but it is not a construction on the coset space that is familiar to me.

Answer

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Source : Link , Question Author : jpdm , Answer Author : Community

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