---- > [!definition] Definition. ([[tensor product of vector bundles]]) > Let $E,E'$ be [[vector bundle|vector bundles]] of rank $n,n'$ over common base $B$ and with typical fibers $V,V'$. WLOG we may assume $E,E'$ share a common trivializing cover $\{ U_{\alpha} \}$. > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQBREAX1PU1z5CKMsWp0mrdgCEefEBmx4CRAEykxNBizaJOAcjn8lQouQrjtUvQB0bwAOYB9YHaZoAFvQAEdgEYwOPTcdtxGCgLKwsjqqpaSuiB2jvouboyePv6BwaE84jBQDvBEoABmAE4QALZI5iA4EEhkEjrsdmhYIDSM9AGMAAqRpnoVWA4eOOGVNUjqDU2ILVaJHViG3JTcQA > \begin{tikzcd} > \{g_{\alpha \beta}\} & E \arrow[d, "\pi"'] & \\ > & B & E' \arrow[l, "\pi'"] \\ > & & \{g'_{\alpha \beta}\} > \end{tikzcd} > \end{document} > ``` > > Suppose $E$ is determined by [[transition functions for a vector bundle over a smooth manifold|transition functions]] $\{ g_{\alpha \beta} : U_{\alpha} \cap U_{\beta} \to \text{GL}_{n}(V)\}$. Likewise suppose $E'$ is determined by $\{ g_{\alpha \beta}' \}$. Then the [[tensor functor|functions]] $\{ g_{\alpha \beta} \otimes g_{\alpha \beta}' : U_{\alpha} \cap U_{\beta} \to \text{GL}_{nn'} (V) \}$ > determine, via the [[the Steenrod construction of a vector bundle over a smooth manifold|Steenrod construction]], a rank-$nn'$ [[vector bundle]] $E \otimes E'$ over $B$ with typical fiber $V \otimes V'$, called the **tensor product** of the vector bundles $E$, $E'$. > > If $b \in B$, the fiber $(E \otimes E')_{b }$ is isomorphic to $E_{b} \otimes E_{b}'$. - [ ] including tensor powers, exterior powers, symmetric powers (obvious) ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` # Old and deprecated > [!definition] Definition. ([[tensor field]]) > Let $A \subset \rrn$ be [[open set|open in]] $\rrn$. A **$k$-tensor field** is a function $F: A \to \text{Mult}^{k}(\rrn)$. - [ ] (a section of the tensor bundle) > [!intuition] > By the [[linear isomorphism|isomorphism]] between $2$-[[multilinear map]]s and [[matrix|matrices]], one can think of a $2$**-tensor field** as assigning a [[matrix]] to every point in $A$. As the order $k$ increases, so does the order of the array assigned to every point in $A$. > [!definition] Definition. ([[continuous tensor field]]) > A $k$-[[tensor field]] $F$ on an [[open set]] $A \subset \rrn$ is said to be [[continuous]]/$C^{r}$/[[smooth]] if the function $A \times \rrn \times \dots \times \rrn \to \rr$ given by $F(\v x)(\v v_{1},\dots, \v v_{k})$ is [[continuous]]/${C^{r}}$/[[smooth]]. > We denote the [[vector space]] of [[smooth]] $k$-[[k-form on Euclidean space|forms]] on $A$ by $\Omega^{k}(A)$, and assume that all $k$-[[k-form on Euclidean space|forms]] we see are [[smooth]] unless spoken otherwise. > [!basicexample] > ![[CleanShot 2023-02-03 at 17.44.03.jpg]]