Tensor product bundle
In differential geometry, the tensor product of vector bundles Template:Mvar, Template:Mvar (over the same space Template:Mvar) is a vector bundle, denoted by Template:Math, whose fiber over each point Template:Math is the tensor product of vector spaces Template:Math.[1]
Example: If Template:Mvar is a trivial line bundle, then Template:Math for any Template:Mvar.
Example: Template:Math is canonically isomorphic to the endomorphism bundle Template:Math, where Template:Math is the dual bundle of Template:Mvar.
Example: A line bundle Template:Mvar has a tensor inverse: in fact, Template:Math is (isomorphic to) a trivial bundle by the previous example, as Template:Math is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space Template:Mvar forms an abelian group called the Picard group of Template:Mvar.
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 form|differential Template:Mvar-form]] and a section of is a [[vector-valued differential form|differential Template:Mvar-form with values in a vector bundle Template:Mvar]].
See also
Notes
References
- Hatcher, Vector Bundles and Template:Mvar-Theory
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- ↑ 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 Template:Mvar such that Template:Math is trivial. Choose Template:Mvar in the same way. Then let Template:Math be the subbundle of Template:Math with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.