# Tensor product bundle

In differential geometry, the tensor product of vector bundles E, F (over same space ${\displaystyle X}$) is a vector bundle, denoted by EF, whose fiber over a point ${\displaystyle x\in X}$ 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.

## 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 ${\displaystyle \Lambda ^{p}T^{*}M}$  is a differential p-form and a section of ${\displaystyle \Lambda ^{p}T^{*}M\otimes E}$  is a differential p-form with values in a vector bundle E.