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Community Bot
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##Background##

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 $E$.

##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 $E$.

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 $E$.

replaced tag 'noethertheorem'
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Ricardo Andrade
Ricardo Andrade
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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.

GivenConsider a vector bundle $(E,\pi,M,\mathbb R^n)$ with a bundle metrigmetric $g$ and a metric connection $\nabla$. Let $J^1E$ be the 1-Jetjet 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 iI know is that there exists a 1:1 correpondencecorrespondence between sections of $J^1E \to E$ and connections on $E$. Furthermore the connection leads uniquleyuniquely to a splitting $TE=VE \oplus HE$$TE = VE \oplus HE$ of the Tangent Bundletangent bundle of $E$.

##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 $E$.

##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 $E$.

deleted 10 characters in body; edited title
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Tobias Ohrmann
Tobias Ohrmann
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1-Jet Bundlejet bundle on Vector Bundlevector bundle with metric Connectionconnection

##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 Bundlevector bundle $(E,\pi,M,\mathbb R^n)$ with a bundle metrig $g$ and a metric connection $\nabla$. Let $\mathfrak J^1E$$J^1E$ be the 1-Jet bundle associated to $E$. Is there a canonical way to identify an Elementelement $j \in J^1E$ 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 $E$.

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.

Given a Vector Bundle $(E,\pi,M,\mathbb R^n)$ with a bundle metrig $g$ and a metric connection $\nabla$. Let $\mathfrak J^1E$ 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 $E$.

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.

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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Tobias Ohrmann
Tobias Ohrmann
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