Background
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I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to specify the Euler-Lagrange equations and the Noether theorem for this case.




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Consider a vector bundle $(E,\pi,M,\mathbb R^n)$ with a bundle metric $g$ and a metric connection $\nabla$. Let $J^1E$ be the 1-jet bundle associated to $E$. Is there a canonical way to identify an element $j \in J^1E$ with an element $(\phi,\nabla \phi) \in E\times(E\otimes TM^\*)$? I would also be grateful for some bibliography on that subject.


What I know is that there exists a 1:1 correspondence between sections of $J^1E \to E$ and connections on $E$. Furthermore the connection leads uniquely to a splitting $TE = VE \oplus HE$ of the tangent bundle of $E$.



1-jet bundle on vector bundle with metric connection

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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 connection gives a splitting $T\_\phi E=V\_\phi E\oplus H\_\phi E$ and since $V\_{\phi}E\cong E\_{\pi(\phi)}$ and $H\_\phi E\cong T\_{\pi(\phi)}M$ canonically, you may interpret $R$ as the graph of a linear map $T\_{\pi(\phi)}M\to E\_{\pi(\phi)}$, hence as an element in $E\otimes T^\*M$.


A reference which might be useful: [Symmetries and Conservation Laws for Differential Equations of Mathematical Physics](http://books.google.ch/books?id=tJSobeeSYHQC&lpg=PR12&ots=-JUxzNE0_5&dq=symmetries%2520and%2520conservation%2520laws%2520in%2520equations%2520of%2520mathematical%2520physics&hl=de&pg=PP1#v=onepage&q=symmetries%2520and%2520conservation%2520laws%2520in%2520equations%2520of%2520mathematical%2520physics&f=false) 


**Edit:** (in response to the comment) I assume your definition of jet is as follows: two sections $\phi,\tilde\phi$ of the bundle have the same 1st jet at $p\in M$ iff their values and their first derivatives coincide at $p$ (one then checks that this is independent of the coordinates). Geometrically this means that the two sections are tangent (picture them as submanifolds in the total space), so the plane $R$ is their tangent space at $p$. In local coordinates $(x\_1,\ldots,x\_m,\phi\_1,\ldots,\phi\_n)$ the plane $R$ is spanned by the vectors $\partial\_{x\_1}+\sum\partial\_{x\_1}(\phi\_j)e\_j,\ldots,\partial\_{x\_m}+\sum\partial\_{x\_m}(\phi\_j)e\_j$ and your tensor is $\sum \partial\_{x\_k}(\phi\_j)e\_j\otimes dx\_j$.


