I've been trying to develop an intuition for higher order frame bundles to help me understand them and this is what I've come up with. Criticisms welcome, as I'm not sure it's valid?
NOTE: Always I am considering the neighborhood $U_x$ of a point $x\in M$.
Consider $M$ an n-dimensional smooth manifold, $Diff(M)$ The group of local diffeomorphisms $M\rightarrow \mathbb{R}^n$, and $Diff_{0}(M)$ the closed subgroup of $Diff(M)$ fixing a point $x\in M$.
We may view $Diff(M)$ as a (sort of principal) $Diff_{0}(M)$ bundle over $M$, with target projection:
$$\pi:Diff(M)\rightarrow M$$
and structure group $Diff_{0}(M)$. Equivalently we might consider $M$ as the homogenous space
$$M\backsimeq\frac{Diff(M)}{Diff_{0}(M)}$$ Because both groups in question are infinite dimensional, this does not lend itself easily to rigorous analysis. For useful constructions, we wish to work with finite dimensional spaces.
We seek an approximation to the structure group $Diff_{0}(M)$ of $Diff(M)$.
For all $\phi\in Diff_{0}(M)$, If we consider only 1-jets $j^{1}\phi$ of $\phi$, then in local coordinates $j^{1}\phi$ is simply the first order Taylor expansion of $\phi$ around $x=0$:
$$\phi=\phi_{0}+\frac{\partial\phi}{\partial x^{i}}_{0}x+\cdots$$
We are already familiar with such a bundle. This is nothing more than the frame bundle $FM$ of $M$.
FM can be considered as the set of 1-jets at $0 \in \mathbb{R}^n$ of local diffeomorphisms of open neighbourhoods of $0\in \mathbb{R}^n$ into $M$.
In fact, the differentiable manifold structure defined over $J^1 _n M$ induces on the open submanifold $FM$ its usual structure with respect to which $$\pi:FM \rightarrow M$$ is a principal fibre bundle with structure group $Gl(n,\mathbb{R})$.
Differential Geometry of Frame bundles, Cordero,Dodson and de Leon pg 8.
We can in this sense consider $FM$ and $Gl(n,\mathbb{R})$ as first order approximations to $Diff(M)$ and $Diff_{0}(M)$ respectively.
Equivalently we can construct a homogenous space as the quotient of the the frame bundle by the General Linear group in the neighbourhood $U_{x}$ of a point $x\in M$. $$\frac{G}{H}=\frac{FU_{x}}{Gl(n,\mathbb{R})}$$
and view it as a first order approximation to $Diff(M)/Diff_{0}(M)$ at $U_{x}$.
Many use cases of differential geometry stop here, utilizing only this first order approximation to $Diff(M)$.
If we wish for higher order approximations, we need only consider higher order jets $j^{k}\phi$ of $\phi$. Such a construction is well-known and the set of all k-jets $j^{k}\phi$ of $\phi$ on $M$ is known as the k-th order frame bundle $F^{k}M$ of $M$ with structure group the jet group $G_{n}^{k}$.
In this sense it is reasonable to view $F^{k}M$ and $G_{n}^{k}$ as kth order approximations to $Diff(M)$ and $Diff_{0}(M)$ respectively.
The $F^{k}M$ play a special role as any natural bundle on $M$ is an associated bundle to some $F^{k}M$ (see for example natural operations in differential geometry Kolar, Michor,Slovak)
Specifically a natural bundle of $M$ is a bundle $E$ over $M$ on which the lift $\phi^{E}$ of a local diffeomorphism $\phi\in Diff_{0}(M)$ is an automorphism $Aut(E)$ of $E$. Considering the route we took in defining our $F^{k}M$ this property hardly surprising.
Afterward: I haven't considered the analycity of $\phi$ or convergence in the limit $k \rightarrow \infty$ to these pseudogroups.
I'm really just trying to understand these bundles and would love for anyone to tell me why this is or isn't a valid interpretation. I Never hear of jet groups as approximations to $Diff_0 (M)$, is this a valid interpretation? Also, this should probably all be considered within a neighborhood of the origin.