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In differential geometry, the tensor product of vector bundles E, F (over same space ) is a vector bundle, denoted by EF, whose fiber over a point 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 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.

VariantsEdit

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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NotesEdit

  1. ^ To construct a tensor-product bundle over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose E' such that EE' is trivial. Choose F' in the same way. Then let EF be the subbundle of (EE') ⊗ (FF') with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.

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