Difference between revisions of "Dual bundle"
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For a [[vector bundle]] $\pi:E\to B$ with a vector space $F\simeq \R^n$ as a generic fiber, the dual bundle is a vector bundle $\pi^*:E^*\to B$ over the same base $B$ with the fiber $F^*$ [[bundle#dual|dual]] to the fiber $F$. | For a [[vector bundle]] $\pi:E\to B$ with a vector space $F\simeq \R^n$ as a generic fiber, the dual bundle is a vector bundle $\pi^*:E^*\to B$ over the same base $B$ with the fiber $F^*$ [[bundle#dual|dual]] to the fiber $F$. | ||
| − | The natural bilinear pairing $F\times F^*\to\R$, $(v,v^*)\mapsto\left<v,v^*\right>$ induces the natural pairing between the modules of sections $\Gamma(E)$ and $\Gamma(E^*)$ of the initial bundle and its dual, $ | + | The natural bilinear pairing $F\times F^*\to\R$, $(v,v^*)\mapsto\left<v,v^*\right>$ induces the natural pairing between the modules of sections $\Gamma(E)$ and $\Gamma(E^*)$ of the initial bundle and its dual, |
| − | \left<\cdot,\cdot\right>:\Gamma(E)\times\Gamma(E^*)\to C^\infty(B),\qquad (s,s^*)\mapsto \left<s,s^*\right>(b)=\left<s(b),s^*(b)\right>. | + | $$ |
| − | $ | + | \left<\cdot,\cdot\right>:\Gamma(E)\times\Gamma(E^*)\to C^\infty(B),\qquad (s,s^*)\mapsto \left< s,s^*\right>(b)=\left<s(b),s^*(b)\right>. |
| + | $$ | ||
Revision as of 13:19, 20 May 2012
For a vector bundle $\pi:E\to B$ with a vector space $F\simeq \R^n$ as a generic fiber, the dual bundle is a vector bundle $\pi^*:E^*\to B$ over the same base $B$ with the fiber $F^*$ dual to the fiber $F$.
The natural bilinear pairing $F\times F^*\to\R$, $(v,v^*)\mapsto\left<v,v^*\right>$ induces the natural pairing between the modules of sections $\Gamma(E)$ and $\Gamma(E^*)$ of the initial bundle and its dual, $$ \left<\cdot,\cdot\right>:\Gamma(E)\times\Gamma(E^*)\to C^\infty(B),\qquad (s,s^*)\mapsto \left< s,s^*\right>(b)=\left<s(b),s^*(b)\right>. $$
How to Cite This Entry:
Dual bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dual_bundle&oldid=26752
Dual bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dual_bundle&oldid=26752