1-jet bundle on vector bundle with metric connection

2012-09-12T10:40:18Z

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 $E$.

Answer by Michael for 1-jet bundle on vector bundle with metric connection

2012-09-12T17:11:11Z

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

1-jet bundle on vector bundle with metric connection - MathOverflow

1-jet bundle on vector bundle with metric connection

Background

Answer by Michael for 1-jet bundle on vector bundle with metric connection