Fine Art


In differential geometry, the tensor product of vector bundles E, F is a vector bundle, denoted by E ⊗ F, whose fiber over a point x is the tensor product of vector spaces Ex ⊗ Fx.[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 the some topological space X forms an abelian group called the Picard group of X.


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


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.


Hatcher, Vector Bundles and K-Theory

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World