# Cartan geometry: jet space perspective on the tractor bundle

Let $$GG$$ a Lie group and $$H⊂GH\subset G$$ a Lie subgroup. For simplicity we assume that the adjoint action of $$HH$$ on $$g/h\mathfrak g/\mathfrak h$$ is faithful.

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

Here are two things one can do with Cartan geometries:

• Construct the associated (right) principal $$GG$$-bundle $$P×HGP\times_H G$$ on which the Cartan connection can be extended to a $$GG$$-principal connection. For a $$GG$$-module $$VV$$, one can construct the associated bundle $$P×HG×GV≃P×HVP\times_H G\times_G V\simeq P\times_H V$$ (called tractor bundle) which is in particular endowed with a induced $$GG$$-principal connection called the tractor connection.

• Considering $$G→G/HG\to G/H$$ as a reference space with $$(H,G)(H,G)$$-Cartan geometry, one can construct the space of $$11$$-jets of maps from $$MM$$ to $$G/HG/H$$ preserving the $$HH$$-structure (between the chosen points), in other words linear isomorphisms between tangent spaces which preserve the class of “$$HH$$-frames”. It is readily constructed as $$(P×G)/H(P\times G)/H$$ for the diagonal action on the right. It is a bundle over $$M×G/HM\times G/H$$ with fibre diffeomorphic to $$HH$$; the fibre over $$MM$$ is $$GG$$. It inherits an action of $$GG$$ (naturally) on the left through the germs of the associated diffeomorphisms acting on the $$11$$-jets and is hence a left principal $$GG$$-bundle over $$MM$$.

Now, the two constructed $$GG$$-principal bundles are isomorphic. They can be easily related by the inversion of the component $$GG$$ of $$P×GP\times G$$ before taking the quotient by $$HH$$. If one is rather willing to “invert” the principal bundle $$PP$$, using the inverse right action, then the first construction can be related to $$11$$-jets of diffeomorphisms between $$MM$$ and $$H∖GH\backslash G$$.

My question is: how can the tractor bundle construction be naturally expressed with the $$11$$-jet interpretation of the “extended” $$GG$$-principal bundle?
To me the natural idea would be to consider applications $$G/H→VG/H \to V$$ which are equivariant under the (left) action of $$GG$$, but it is not a construction on the coset space that is familiar to me.