Tensor product bundle

Not to be confused with a tensor bundle, a vector bundle whose section is a tensor field.

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 E_x ⊗ F_x.

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.

See also

• Tensor product of modules

References

• Hatcher, Vector Bundles and K-Theory