In differential geometry, the tensor product of vector bundles , (over the same space ) is a vector bundle, denoted by , whose fiber over each point is the tensor product of vector spaces .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 such that is trivial. Choose in the same way. Then let be the subbundle of with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.
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 (isomorphic to) 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 .
See also Tensor product of modules
Notes
References Hatcher, Vector Bundles and -Theory