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Background

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.

Given a vector bundle $(E,\pi,M,\mathbb R^n)$ with a bundle metrig $g$ 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 correpondence between sections of $J^1E \to E$ and connections on $E$. Furthermore the connection leads uniquley to a splitting $TE=VE \oplus HE$ of the Tangent Bundle of $E$.

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1 Answer

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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 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

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 $\sum \partial_{x_k}(\phi_j)e_j\otimes dx_j$.

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Thank you! What i still don't understand is that an element $j\in J^1E$ builds an subspace $R \subset T_\phi E$. With a given local trivialization $\psi$, $j$ could be represented by a point in the corresponding horizontal space $HE^\psi_\phi \subset T_\phi E$ of $\psi$. Why does the set of those points in $T_\phi E$ according to the set of all trivializations build a subspace? I tried to prove that, but what I need is that the trivializations build a vector space themselves, which is obviously wrong (take [$M=\mathbb R^n, E=TM, \alpha = id =-\beta$ local trivialisations] as a counterexample). – Tobias Ohrmann Sep 12 at 22:59
Thank you very much, this helped me a lot, unfortunately i'm not autherized to click you answer up :) – Tobias Ohrmann Sep 14 at 14:29
You're welcome. – Michael Sep 14 at 19:38
I still have one question: Does such an element $(\phi,\nabla \phi)$ exist for every $j_x=(x_i,y_j,y^j_i) \in (J^1E)_x$? As the condition, which have to be fulfilled for such an $\phi$, i get: $y_i^j-\sum_{k=1}^n \Gamma_{ik}^j y_k = \partial_{x_i}\phi^j - \sum_{k=1}^n\Gamma_{ik}^j \phi^k$ So, if $j$ is holonomic, one can choose the prolonged section of $j$ and the condition is fulfilled. Is holonomy also a necessary condition for the existence of such a section $\phi$? – Tobias Ohrmann Sep 15 at 14:07
I'm not sure I understand the question. The identification we were talking about is bijective: to every section of $J^1 E$ corresponds a section of $E\oplus(E\otimes T^*M)$. By holonomic section in $J^1 E$ you mean a section which is the prolongation of a section in $E$? – Michael Sep 15 at 17:37
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