differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, 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 eve ...

s , (over the same space ) is a vector bundle, denoted by , whose fiber over each point is the tensor product of vector spaces .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 such that is trivial. Choose in the same way. Then let be the subbundle of with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach. Example: If is a trivial line bundle, then for any . Example: is canonically isomorphic to the endomorphism bundle , where 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 spa ...

of . 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 organis ...

has a tensor inverse: in fact, is (isomorphic to) a trivial bundle by the previous example, as is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space 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 ver ...

of .

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 th ...

and an

exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...

of a vector bundle in a similar way. For example, a section of

\Lambda^p T^* M

is a differential -form and a section of

\Lambda^p T^* M \otimes E

is a differential -form with values in a vector bundle .

Notes

References

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

Variants

See also

Notes

References