How is $F(FM)$ related to 2-Jet bundle $F^{2} M$?

Question

I get that the bundle of invertible 2-jets $F^{2}M$ over a manifold $M$ holds second order derivative information about smooth functions in $M$. The Wikipedia article speaks of a close relation between the double tangent bundle $TTM$ and $F^{2}M$. I'm looking at a principal frame bundle, over another principal frame bundle over a Riemannian manifold (denoted $F(FM)$ ).

Given a manifold $M$, We examine it's (tangent) frame bundle $FM$. This is known to be in bijection with the bundle of invertible 1-jets $F^{1}(M)$ on $M$. Considering the total space $FM$, let us now examine it's frame bundle $F(FM)$.

Similarly, we have a bijection between $F^{1}\left(FM\right)$ and $F(FM)$. But(as I understand it) this is just the 1-jet prolongation of $FM$, namely $J^{1}FM$. In principal bundle structure on jet prolongation of frame bundles page 1288, it's stated that $J^{1}FM$ is diffeomorphic to $\tilde{F}^{2}\left(M\right)$,the bundle of semi-holonomic 2-jets over $M$. Further there is a bundle reduction to the holonomic 2-jet bundle such that:

$$F^{2}(M)\rightarrow J^{1}FM$$

So I would expect that the frame bundle of the frame bundle $F(FM)$ of $M$ is diffeomorphic to the semi-holonomic bundle of 2-jets over $M$, $\tilde{F}^{2}(M)$. is this right or am I misunderstanding something? I would be satisfied if anyone can explain the relationship between the two bundles in the title.

Example:

Without getting into individual frames and coordinate patches, suppose I have some n-dimensional Riemannian manifold $M$. $M$ will accordingly have a principle $GL(n)$ frame bundle $FM$ over $M$ that locally looks like $FM\approx\mathbb{\mathbb{R}}^{n}\times gl(n)$. $FM$ is then a Riemannian manifold of dimension $n+n^{2}$.

Now if we take it's frame bundle we have a principle $GL(n+n^{2})$ frame bundle $FFM$ over $FM$.

Without getting into the Homotopy principle too much, we can say this $GL(n+n^{2})$ bundle is the bundle of aholonomic frames over $FM$. We could also say it's a $GL(n+n^{2})\rtimes GL(n)$ bundle over $M$. Does this coincide with the (aholonomic) second order frame bundle $F^{2}$ of $M$?

Some of our frame components here are first and second order derivatives of our original frame components on $M$ (this is because connections on $M$ correspond to frames on $gl(n)$ ) When we start enforcing those relations, we move to semi holonomic (by enforcing the first order derivative conditions) and finally holonomic (by enforcing second order derivative conditions) frames.

As We do this we will have a reduction of the $GL(n+n^{2})$ to some subgroup. This is right along the lines of what is done in this paper but there they're using Jet bundles and calling them higher order frame bundles $F^{2}M$. How are they related to what I'm looking at? I apologize for any sloppiness, I'm new and trying to learn this, it is admittedly out if my main area of study.

Answer 1

I would ask someone "in the know" to please look this over and make their own answer, which I would happily accept if somewhat complete.

After several textbooks and much reading this is what I've come up with. Please please, feel free to point out errors.

In the book The Differential Geometry of Frame bundles by Cordero, Dodson, and De Leon. I found sections 2.1 and 2.2 to be relevant:

Let $\pi_{M}:FM\rightarrow M$ and $\pi_{FM}:FFM\rightarrow FM$ be the frame bundles of $M$ and $FM$ respectively. Then, $j_{M}$ induces a bundle homomorphism of $J_{n}^{1}FM|_{FM}$ into $FFM$ with respect to $j_{n}$, ie.

$$j_{M}\left(X\cdot Y\right)=j_{M}\left(X\right)\cdot j_{n}\left(Y\right)$$

for $X\in J_{n}^{1}FM|_{FM}$ and $Y\in J_{n}^{1}GL(n,\mathbb{R})$, and the following diagram

$$\begin{array}{ccccc} J_{n}^{1}FM & & \overset{j_{M}}{\rightarrow} & & FFM\\ \\ \downarrow^{\pi_{M}^{1}} & & & & \downarrow^{\pi_{FM}}\\ \\ FM & & \overset{1_{FM}}{\rightarrow} & & FM \end{array}$$

is commutative.

Let us suppose the manifold $M$ above is already the frame bundle of some manifold. Using the substitution $M\rightarrow FM$ we can write the above diagram as:

$$\begin{array}{ccccc} J_{n}^{1}FFM & & \overset{j_{FM}}{\rightarrow} & & FFFM\\ \\ \downarrow^{\pi_{FM}^{1}} & & & & \downarrow^{\pi_{FFM}}\\ \\ FFM & & \overset{1_{FFM}}{\rightarrow} & & FFM \end{array}$$

Putting these two diagrams together we get:

$$\begin{array}{ccccccccc} J_{n}^{1}J_{n}^{1}FM & & \overset{j_{M}}{\rightarrow} & & J_{n}^{1}FFM & & \overset{j_{FM}}{\rightarrow} & & FFFM\\ \\ \downarrow^{\pi_{FM}^{2}} & & & & \downarrow^{\pi_{FM}^{1}} & & & & \downarrow^{\pi_{FFM}}\\ \\ J_{n}^{1}FM & & \overset{j_{M}}{\rightarrow} & & FFM & & \overset{1_{FFM}}{\rightarrow} & & FFM\\ \\ \downarrow^{\pi_{M}^{1}} & & & & \downarrow^{\pi_{FM}} & & & & \downarrow^{\pi_{FM}}\\ \\ FM & & \overset{1_{FM}}{\rightarrow} & & FM & & \overset{1_{FM}}{\rightarrow} & & FM \end{array}$$

Let us not forget that the nonholonomic prolongation is given via iterative application of the 1-jet holonomic prolongation: $\tilde{J}_{n}^{k}=\left(J_{n}^{1}\right)^{k}$. Then by extension, it appears one can always relate the kth order non-holonomic prolongation of a vector bundle to taking it's kth order iterative frame bundle (via a bundle homomorphism).

Note: I realize we can take this one step further by noting that (same book page 8):

The total space $FM$ of the frame bundle of $M$ is an open submanifold of $J_{n}^{1}M$, because $FM$ can be considered as the set of 1-jets at $0\in\mathbb{R}^{n}$ of local diffeomorphisms of open neighborhoods of $0\in\mathbb{R}^{n}$ into $M$.

In fact, the differentiable manifold structure defined over $J_{n}^{1}M$ induces on the open submanifold $FM$ its usual structure with respect to which

$\pi_{M}:FM\rightarrow M$

Then we could add a leftmost column to lattermost diagram whose terms would be written with all F's replaced by prolongations of $M$.