1-jet bundle on vector bundle with metric connection - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:36:30Z http://mathoverflow.net/feeds/question/106999 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106999/1-jet-bundle-on-vector-bundle-with-metric-connection 1-jet bundle on vector bundle with metric connection Tobias Ohrmann 2012-09-12T10:40:18Z 2012-09-13T06:37:10Z <h2>Background</h2> <p>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.</p> <hr> <p>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.</p> <p>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 $E$.</p> http://mathoverflow.net/questions/106999/1-jet-bundle-on-vector-bundle-with-metric-connection/107033#107033 Answer by Michael for 1-jet bundle on vector bundle with metric connection Michael 2012-09-12T17:11:11Z 2012-09-13T06:37:10Z <blockquote> <p>Is there a canonical way to identify an Element... ?</p> </blockquote> <p>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 $E\otimes T^*M$.</p> <p>A reference which might be useful: <a href="http://books.google.ch/books?id=tJSobeeSYHQC&amp;lpg=PR12&amp;ots=-JUxzNE0_5&amp;dq=symmetries%2520and%2520conservation%2520laws%2520in%2520equations%2520of%2520mathematical%2520physics&amp;hl=de&amp;pg=PP1#v=onepage&amp;q=symmetries%2520and%2520conservation%2520laws%2520in%2520equations%2520of%2520mathematical%2520physics&amp;f=false" rel="nofollow">Symmetries and Conservation Laws for Differential Equations of Mathematical Physics</a> </p> <p><strong>Edit:</strong> (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 $\sum \partial_{x_k}(\phi_j)e_j\otimes dx_j$.</p>