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 E_x ⊗ F_x.^[1]

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.

Notes

^ 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 E' such that E ⊕ E' is trivial. Choose F' in the same way. Then let E ⊗ F be the subbundle of (E ⊕ E') ⊗ (F ⊕ F') with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.

References

Hatcher, Vector Bundles and K-Theory

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