Let $M$ be a smooth (finite dimensional, Hausdorff and second countable) manifold. Let $T^kM$ be the manifold of equivalence class of curves that their derivates (in charts) agree up to order $k$. Let $j^k_0\gamma \in T^kM $ denote the equivalence class of $\gamma$.
Then the following maps are surjective submersions: $$ \pi^k_r:T^kM \longrightarrow T^rM, \;\; k\geq r$$ $$ j^k_0\gamma \mapsto j^r_0\gamma $$ Obviously $\pi^k_s=\pi^r_s\circ \pi^k_r $ for $k\geq r\geq s$.
Question: Is there a canonical way to describe the vertical bundle $V\pi^k_r:=\ker T\pi^k_r$ in terms of the spaces $T^sM$ and $T(T^sM)$?
We can do it easily in the case of $r=k-1$ as the map: $$ \pi^k_{k-1}:T^kM \longrightarrow T^{k-1}M$$ It's an affine bundle with associated vector bundle $T^{k-1}M \times_M TM \longrightarrow T^{k-1}M$. Then we have the canonical isomorphism: $$ \mathfrak J: T^kM\times_M TM \longrightarrow V\pi^k_{k-1}$$ $$ (j^k_0\gamma, v) \mapsto \frac{d}{dt}|_{t=0} (j^k_0\gamma+tv), \;\; v\in T_{\gamma(0)}M $$
I found myself an extension that Crampin gaves (in Higher-order differential equations and higher-order lagrangian mechanics) for the case $r=0$ (or $r=k$ in the following notation): $$ \mathfrak J: T^kM\times_{T^{r-1}M} T(T^{r-1}M) \longrightarrow V\pi^k_{k-r}$$ $$ (j^k_0 \chi_0,\frac{d}{ds}|_{s=0}j^{r-1}_0\chi_s)\mapsto \frac{d}{ds}|_{s=0}j^k_0\hat \chi_s $$ For any smooth map $\chi:\mathbb R^2 \longrightarrow M$, $\chi_s(t):=\chi(s,t)$, $\hat\chi_s(t):=\chi(st^{k-r+1},t) $. These are well defined isomorphism.
I'm looking for a more geometric equivalent description of these map more similar to the affine case $r=k-1$.
