I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold.
Explanation
Suppose we have a 3D smooth manifold $\mathcal{B}$, which can be covered by a single chart, and a smooth 2D manifold $\mathcal{S}$ embedded inside it. Consider the following bundle structure $P(\mathcal{S})$ over $\mathcal{S}$: at each $p\in\mathcal{S}$, we attach the set of all frames at (with an abuse of notation) $p\in\mathcal{B}$, such that the first $2$ vectors of each frame spans $T_p\mathcal{S}$ (i.e. the third vector is always transverse to $\mathcal{S}$). $P(\mathcal{S})$ is clearly a principal subbundle of the restriction of the frame bundle $F\mathcal{B}$ of $\mathcal{B}$ to $\mathcal{S}$. The structure group $G_0$ (and the standard fibre) is clearly a subset of $Gl(3,\mathbb{R})$, with elements of the form: the upper-left $2\times 2$ block invertible, the $(3,3)$ element non-zero, the $(1,3)$ and $(2,3)$ elements zero.
Now, consider a $G$-structure over $\mathcal{B}$, where $G$ is a proper Lie subgroup of $Gl(3,\mathbb{R})$. Also, consider a non-holonomic $\mathcal{G}$-structure over $\mathcal{S}$ (a non-holonomic $\mathcal{G}$-structure over $\mathcal{S}$ is a $\mathcal{G}$-reduction of the non-holonomic frame bundle of $P(\mathcal{S})$). $\mathcal{G}$ is a proper subgroup of the (outer) semi-direct product $G_0\times Gl(2,\mathbb{R})\times Lin(\mathbb{R}^2,\mathfrak{g}_0)$, where $\mathfrak{g}_0$ is the Lie algebra of $G_0$.
My question is, could the latter $\mathcal{G}$-structure be seen as immersed (or possibly embedded) inside the former $G$-structure? If not,then is there any way (associated bundles or anything) to relate these two structures?