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According to Liebermann's Introduction to the theory of semi-holonomic jets p.177: a local section $s:U\subset M\to J^1E$ is said to be adapted at $x\in U$ if $s(x)=j^1_x(\beta \circ s)$􏰆 where $\ … Intuitively this condition means the following: think of a non-holonomic jet $s\in (J^1 J^1 E)_x$ as specifying a family of jets $s(x_1)\in (J^1E)_{x_1}$, one for every $x_1 \sim_1 x$. …

The definition of $k$-th order jet as an equivalence class $[f]_x^k$ of a function $f\in C^\infty M$ at point $x\in M$, gives you a natural map \begin{align} \mathcal{j}^k\colon C^\infty M &\to \maths …

(In fact, in the case of jets of submanifolds we probably need to allow vector bundles $V$ over $J^1 E$ instead of over $E$, since the vertical bundle $VE$ doesn't exist and is replaced by a "normal" bundle …

But your objects are more general since they don't have to be invariant under the action of the automorphism group of $D_k(n)$, while jets will be. …

Is there a canonical way to identify an Element... ? Yes: an element $j\in J^1E$ is the same as subspace $R\subset T_{\phi}E$ of dimension $\dim(M)$ transversal to $VE$. Since your metric connect …