Tensor
From Manifold Atlas
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$T \in \Gamma\textup{Hom}(E,E')$ can be viewed as a $C^\infty(M)$-linear map | $T \in \Gamma\textup{Hom}(E,E')$ can be viewed as a $C^\infty(M)$-linear map | ||
$T : \Gamma E \to \Gamma E'$ which means | $T : \Gamma E \to \Gamma E'$ which means | ||
− | \begin{equation} T(fs) = fT(s) \end{ | + | \begin{equation} T(fs) = fT(s) \end{equation} |
for any smooth function $f \in C^\infty(M)$ and any section $s\in \Gamma E$. | for any smooth function $f \in C^\infty(M)$ and any section $s\in \Gamma E$. | ||
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viewed as a $C^\infty(M)$-multilinear map $T : \Gamma E_1 \times ...\times \Gamma E_k \to \Gamma E$. | viewed as a $C^\infty(M)$-multilinear map $T : \Gamma E_1 \times ...\times \Gamma E_k \to \Gamma E$. | ||
</wikitex> | </wikitex> | ||
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} |
Latest revision as of 10:50, 15 May 2013
The user responsible for this page is Jost Eschenburg. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
Let be a smooth manifold and vector bundles over . A tensor (field) is a section in the bundle of bundle homomorphisms between and . Alternatively, a tensor can be viewed as a -linear map which means
(1)
for any smooth function and any section .
The bundle may be itself a tensor product of vector bundles . Then a tensor may be viewed as a -multilinear map .