Vector bundle - Wikipedia

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 point 𝑥 of the space 𝑋 we associate (or "attach") a vector space 𝑉 ( 𝑥 ) in such a way that these vector spaces fit together to form another space of the same kind as 𝑋 (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over 𝑋 . The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space 𝑉 such that 𝑉 ( 𝑥 ) = 𝑉 for all 𝑥 in 𝑋 : in this case there is a copy of 𝑉 for each 𝑥 in 𝑋 and these copies fit together to form the vector bundle 𝑋 × 𝑉 over 𝑋 . Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) m

Vector bundle - Wikipedia Jump to content From Wikipedia, the free encyclopedia Mathematical parametrization of vector spaces by another space The (infinitely extended) Möbius strip is a line bundle over the 1-sphere S 1 {\displaystyle {\mathcal {S}}^{1}} . Locally around every point in S 1 {\displaystyle {\mathcal {S}}^{1}} , it looks like U × R {\displaystyle U\times \mathbb {R} } (where U {\displaystyle U} is an open arc including the point), but the total bundle is different from S 1 × R {\displaystyle {\mathcal {S}}^{1}\times \mathbb {R} } (which is a cylinder instead). In mathematics , a

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