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Tensor product bundle

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In differential geometry, the tensor product of vector bundles E, F (over the same space X) is a vector bundle, denoted by EF, whose fiber over each point xX is the tensor product of vector spaces ExFx.[1]

Example: If O is a trivial line bundle, then EO = E for any E.

Example: EE is canonically isomorphic to the endomorphism bundle End(E), where E is the dual bundle of E.

Example: A line bundle L has a tensor inverse: in fact, LL is (isomorphic to) a trivial bundle by the previous example, as End(L) is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space X forms an abelian group called the Picard group of X.

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Variants

One can also define a symmetric power and an exterior power of a vector bundle in a similar way. For example, a section of is a differential p-form and a section of is a differential p-form with values in a vector bundle E.

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See also

Notes

References

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