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 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, where E ∗ is the dual bundle of E.
Example: A line bundle L has tensor inverse: in fact, L ⊗ L ∗ is a trivial bundle by the previous example, as End 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 is a differential p-form and a section of is a differential p-form with values in a vector bundle E.