# 1-jet bundle on vector bundle with metric connection

## 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.

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 $$E$$.

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

• 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 '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 '12 at 14:29
• 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 '12 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 Bächtold Sep 15 '12 at 17:37
• What I've done: 1. Expressing $j_x:T_x→T_{ϕ(x)}E$ in local coordinates: $j_x(∂_{x_i})=∂_{x_i}+∑^n_{j=1}y^j_i ∂_{y_j}$ 2. Splitting $j_x$ in horizontal and vertical part, according to ∇: $j_x=$ (horizontal part) $+ ∑^n_{j=1}(y^j_i−\sum^n_{k=1}Γ^j_{ik}y_k)∂_{y_j}$ 3. Project $j_x$ on the vertical part 4. Expressing $∇ϕ:T_xX→V_yY$ in local coordinates: $∇_∂_{x_i}\phi=∑^n_{j=1}(∂_{x_i}\phi_j−∑^n_{k=1}Γ^j_{ik}\phi_k)∂_{y_j}$5. Compare both expressions – Tobias Ohrmann Sep 16 '12 at 7:29