Tensor

Revision as of 10:50, 15 May 2013

1 Definition

Let ${{Authors|Jost Eschenburg}}{{Stub}} == Definition == <wikitex>; Let $M$ be a smooth manifold and $E,E'$ vector bundles over $M$. A ''tensor'' (''field'') is a section in the bundle $\textup{Hom}(E,E')$ of bundle homomorphisms between $E$ and $E'$. Alternatively, a tensor $T \in \Gamma\textup{Hom}(E,E')$ can be viewed as a $C^\infty(M)$-linear map $T : \Gamma E \to \Gamma E'$ which means \begin{equation} T(fs) = fT(s) \end{equaiton} for any smooth function $f \in C^\infty(M)$ and any section $s\in \Gamma E$. The bundle $E$ may be itself a tensor product $E = E_1\otimes \dots\otimes E_k$ of vector bundles $E_1,\dots,E_k$. Then a tensor $T \in \textup{Hom}(E,E')$ may be viewed as a $C^\infty(M)$-multilinear map $T : \Gamma E_1 \times ...\times \Gamma E_k \to \Gamma E$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]] [[Category:Connections and curvature]]M$ be a smooth manifold and $E,E'$ vector bundles over $M$. A tensor (field) is a section in the bundle $\textup{Hom}(E,E')$ of bundle homomorphisms between $E$ and $E'$. Alternatively, a tensor $T \in \Gamma\textup{Hom}(E,E')$ can be viewed as a $C^\infty(M)$-linear map $T : \Gamma E \to \Gamma E'$ which means

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$T(fs) = fT(s) \end{equaiton} for any smooth function$f \in C^\infty(M)$and any section$s\in \Gamma E$.

The bundle $E$ may be itself a tensor product $E = E_1\otimes \dots\otimes E_k$ of vector bundles $E_1,\dots,E_k$. Then a tensor $T \in \textup{Hom}(E,E')$ may be

viewed as a $C^\infty(M)$-multilinear map $T : \Gamma E_1 \times ...\times \Gamma E_k \to \Gamma E$.

2 References