differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the tensor product of

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

s ''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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...

''E''_''x'' ⊗ ''F''_''x''.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. 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 In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

End(''E''), where ''E''^∗ is the

dual bundle In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. Definition The dual bundle of a vector bundle \pi: E \to X is the vector bundle \pi^*: E^* \to X whose fibers are the dual sp ...

of ''E''. Example: A

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

''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 In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...

of ''X''.

Variants

One can also define a

symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in the ...

and an

exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...

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

References

* Hatcher
Vector Bundles and ''K''-Theory
Differential geometry {{differential-geometry-stub

Variants

See also

Notes

References