In differential geometry, the tensor product of vector bundles E, F (over the same space X) is a vector bundle, denoted by E ⊗ F, whose fiber over each point x ∈ X is the tensor product of vector spaces Ex ⊗ Fx. 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 a 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.
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 Λ p T ∗ M {\displaystyle \Lambda ^{p}T^{*}M} is a differential p-form and a section of Λ p T ∗ M ⊗ E {\displaystyle \Lambda ^{p}T^{*}M\otimes E} is a differential p-form with values in a vector bundle E.
See also Tensor product of modules
Notes
References Hatcher, Vector Bundles and K-Theory
