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Tensor product bundle
In differential geometry, the tensor product of vector bundles E, F is a vector bundle, denoted by E ⊗ F, whose fiber over a point x is the tensor product of vector spaces Ex ⊗ Fx.[1]
Example: If O is a trivial line bundle, then E ⊗ O = E for any E.
Example: E ⊗ E* 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, L ⊗ L* 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 the some topological space X forms an abelian group called the Picard group of X.
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 \( \Lambda^p T^* M \) is a differential p-form and a section of \( \Lambda^p T^* M \otimes E \) is a differential p-form with values in a vector bundle E.
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See also
tensor product of modules
Notes
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 E ⊕ E ' is trivial. Choose F ' in the same way. Then let E ⊗ F be the subbundle of (E ⊕ E ') ⊗ (F ⊕ F ') with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.
References
Hatcher, Vector Bundles and K-Theory
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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