# Injective homomorphism of bundle of invertible 1-jets into the Frame bundle? Seeking representations of nonholonomic Jet Groups

I'm pretty new to this so here goes: Herein we will consider always the sub-bundle of invertible jets. Given an n-dimensional smooth manifold $$M^n$$ I know that that the bundle of 1-jets of the $$C^{\infty}$$ map $$\phi:M^{n}\rightarrow\mathbb{R}^{n}$$ is simply the frame bundle, that is:

$$J_{n}^{1}M^{n}\backsimeq FM^{n}$$

But what if we consider 1-jets of the map $$\psi:M^{n}\rightarrow\mathbb{R}^{d}$$ where $$d\leq n$$? Intuitively I would think that $$\psi$$ would have a smooth embedding (i.e. injective immersion) into $$\phi$$ since $$\mathbb{R}^d$$ has a smooth embedding into $$\mathbb{R}^n$$ (In applications here $$\mathbb{R}^n =J^1 _d \mathbb{R}^d$$, see below) such that there is a(n) (injective) homomorphism of bundles:

$$\Psi:J_{d}^{1}M^{n}\rightarrow FM^{n}$$

Being an isomorphism in the case $$d=n$$. Does anyone know if that's the case?

For the interested: seeking representations of the jet groups of such bundles ( which would be given faithfully by such a homomorphism)

Further Note In my use case $$M^n$$ itself will be a principal bundle as I'm actually looking at nonholonomic invertible jets (i.e. $$\tilde{J}^2 _n M=J^1 _n J^1 _n M^n$$) a.k.a higher order nonholonomic frame bundles. The above homomorphism should be enough to find faithful representations. For second order note if $$J^1 _d M^d=N$$ is n-dimensional, then we have $$J^1 _d N$$ is really $$J^1 _d J^1 _d M^d$$ (which is why we will always have that $$d \leq n$$)

Then a representation of the structure group for $$\tilde{J}^k _n M^n = J^1 _n J^1 _n \cdots M^n$$ over $$\tilde{J}^{k-1}M^n$$ could be given by specifying a homomorphism into $$F\tilde{J}^{k-1} M^n$$, the frame bundle of $$\tilde{J}^{k-1} M^n$$. Then you could go order by order to obtain representations of the whole structure group over $$M^n$$; However I'm not sure if the homomorphism in my question is true?

Also since Jet bundles are natural bundles, the bundle of 1-jets over an arbitrary manifold should always be associated to the frame bundle of that manifold. Implying there's a homomorphism between them, which if the action is effective is an injective homomorphism(not quite sure how to say this properly)

From a different direction We could again look at the map on an n-dimensional base space $$M$$ $$\phi:M\rightarrow\mathbb{R}^{n}$$ and consider its 1-jet prolongation w.r.t. $$\phi$$, that is $$j^{1}\phi:J_{n}^{1}M\rightarrow J_{n}^{1}\mathbb{R}^{n}\backsimeq\mathbb{R}^{n+n^{2}}$$

(again we are always considering the subset of invertible jets) Then this gives a chart on $$J_{n}^{1}M\backsimeq FM$$, the frame bundle of $$M$$. Taking another prolongation w.r.t. the map $$j^{1}\phi$$ now $$j_{n+n^{2}}^{1}j_{n}^{1}\phi:J_{n+n^{2}}^{1}J_{n}^{1}M\rightarrow J_{n+n^{2}}^{1}\mathbb{R}^{n+n^{2}}\backsimeq\mathbb{R}^{n+n^{2}+(n+n^{2})^{2}}$$

Would seem to now give us a chart on the frame bundle $$FFM\backsimeq J_{n+n^{2}}^{1}FM$$ of the frame bundle $$FM$$ of $$M$$. If we consider only the subset $$\phi\subset j^{1}\phi$$ in the second prolongation then we have a reduction

$$J_{n+n^{2}}^{1}J_{n}^{1}M\rightarrow J_{n}^{1}J_{n}^{1}M$$ Which might be viewed as a homomorphism of $$J_{n}^{1}J_{n}^{1}M \backsimeq J^1 _n FM$$ into $$J_{n+n^{2}}^{1}J_{n}^{1}M \backsimeq FFM$$ (i.e. $$J^1 _n FM$$ into $$FFM$$ which is what my question is all about)?

This is an attempt. I welcome other answers or comments. We are seeking a homomorphism of bundles $$\phi:J_{d}^{1}N\rightarrow FN$$ from the 1-jet bundle of an n-dimensional manifold $$N$$ to the frame bundle of $$N$$. We have also the target jet projection $$\pi:J_{d}^{1}N\rightarrow N$$. We can define a local trivialization of $$FN$$ as a choice of section $$\sigma:N\rightarrow FN$$. Then we have the composition

$$\pi\circ\sigma:J_{d}^{1}N\rightarrow FN$$

Suppose $$N=J^1 _d M$$ where $$M$$ is d-dimensional. Then we have $$\pi_{1}^{2}:\tilde{J_{d}^{2}}M=J_{d}^{1}J_{d}^{1}M\rightarrow J_{d}^{1}M$$ and a choice of frames on $$J_{d}^{1}M$$ is a section $$\sigma:J_{d}^{1}M\rightarrow FJ^{1}_{d}M$$. The composition is then a bundle morphism:

$$\sigma\circ\pi_{1}^{2}:\left(\tilde{J_{d}^{2}=}J_{d}^{1}J_{d}^{1}M\right)\rightarrow FJ_{d}^{1}M$$

For a Given choice of local trivializations of $$J_{d}^{1}J_{d}^{1}M$$, $$FJ_{d}^{1}M$$, and $$J_{d}^{1}M$$; we have that $$\pi_{1}^{2}$$ and $$\sigma$$ are injective (that is for a given trivialization they are one to one maps). I think this is sufficient to conclude that we have an injective homomorphism of principal bundles?

$$\begin{array}{ccc} J_{d}^{1}J_{d}^{1}M & \overset{\pi_{1}^{2}\circ\sigma}{\rightarrow} & FJ_{d}^{1}M\\ \downarrow\pi & & \downarrow Id_{FJ_{d}^{1}M}\\ J_{d}^{1}M & \overset{\sigma}{\rightarrow} & FJ_{d}^{1}M \end{array}$$

If true this leads immediately to a more general injective homomorphism

$$\sigma\circ\pi_{k-1}^{k}:\left(\tilde{J^{k}_n =}J^{1} _n \tilde{J^{k-1}} _n M\right)\rightarrow F\tilde{J^{k} _n }M$$

Allowing us to construct representations of the nonholonomic jet groups order by order. Can anyone weigh in on this?