Tensor product bundle

In differential geometry, the tensor product of vector bundles E, F (over same space $$X$$) is a vector bundle, denoted by E ⊗ F, whose fiber over a point $$x \in X$$ 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 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 $$\Lambda^p T^* M$$ is a differential p-form and a section of $$\Lambda^p T^* M \otimes E$$ is a differential p-form with values in a vector bundle E.