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)?


1 Answer 1


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?


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