# 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 *E*_{x} ⊗ *F*_{x}.^{[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 some topological space *X* forms an abelian group called the Picard group of *X*.

## Contents

## Variants[edit]

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*.

## See also[edit]

## Notes[edit]

**^**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[edit]

- Hatcher, Vector Bundles and
*K*-Theory

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