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Vector Bundle

Basic definition

Given a topological space X, a vector bundle is a way of associating a vector space to each point of X in a consistent way.

Detailed definition

A vector bundle is special class of fiber bundle in which the fiber is a vector space V. Technically, a little more is required; namely, if f:E->B is a bundle with fiber R^n, to be a vector bundle, all of the fibers f^(-1)(x) for x element B need to have a coherent vector space structure. One way to say this is that the "trivializations" h:f^(-1)(U)->U×R^n, are fiber-for-fiber vector space isomorphisms. A vector bundle is a total space E along with a surjective map π:E->B to a base manifold B. Any fiber π^(-1)(b) is a vector space isomorphic to V. The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip.

Related terms

bundle rank | fiber | fiber bundle | Hermitian metric | K-theory | Lie algebroid | linear algebra | principal bundle | Riemannian metric | stable equivalence | tangent bundle | tangent map | trivial bundle | vector bundle connection | vector space | Whitney sum

Educational grade level

graduate school level

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