Tensor product bundle

In differential geometry, the tensor product of vector bundles, is a vector bundle, denoted by, whose fiber over each point is the tensor product of vector spaces.
Example: If is a trivial line bundle, then for any.
Example: is canonically isomorphic to the endomorphism bundle, where is the dual bundle of.
Example: A line bundle has a tensor inverse: in fact, is a trivial bundle by the previous example, as is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space forms an abelian group called the Picard group of.

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 -form and a section of is a differential -form with values in a vector bundle.