# Cartan geometry: jet space perspective on the tractor bundle

Let $$G$$ a Lie group and $$H\subset G$$ a Lie subgroup. For simplicity we assume that the adjoint action of $$H$$ on $$\mathfrak g/\mathfrak h$$ 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 $$\mathfrak g /\mathfrak h$$), along with a "Cartan connection one-form" which is a $$\mathfrak g$$-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 $$P\times_H G$$ on which the Cartan connection can be extended to a $$G$$-principal connection. For a $$G$$-module $$V$$, one can construct the associated bundle $$P\times_H G\times_G V\simeq P\times_H V$$ (called tractor bundle) which is in particular endowed with a induced $$G$$-principal connection called the tractor connection.

• Considering $$G\to G/H$$ 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 $$(P\times G)/H$$ for the diagonal action on the right. It is a bundle over $$M\times G/H$$ 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 $$P\times G$$ 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 $$H\backslash G$$.

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 $$G/H \to V$$ which are equivariant under the (left) action of $$G$$, but it is not a construction on the coset space that is familiar to me.

• Consider projective connections, where $G$ acts with stabilizer $H$, but $H$ acts on first order frames with stabilizer not trivial, a first prolongation subgroup. Any vector bundle with structure group $H$, associated to the Cartan geomety $H$-bundle via a representation of $H$, cannot be associated to the first order structure, the subbundle of the first order frame bundle, unless the representation is trivial on that prolongation subgroup. (I forget whether tractor bundles are defined to be trivial on the first prolongation; but I think you will remember.) Aug 14, 2021 at 13:17
• If I got it right, you are saying that one needs the whole $H$-structure and not its projection on $\operatorname{End}(\mathfrak g/\mathfrak H)$ to handle the general representations of $H$? Indeed my construction is imprecise in the case $(H,G)$ is not first order ($P\times_H G$ only projects on the $1$-jet space). I will add a first order assumption to the question so as to avoid introducing one more bundle. Thank you for the comment.
– jpdm
Aug 14, 2021 at 15:26