A vector bundle is a special class of fiber bundle in which the fiber is a vector space . Technically, a little more is required; namely, if is a bundle with fiber , to be a vector bundle, all of the fibers for need to have a coherent vector space structure. One way to say this is that the "trivializations" , are fiber-for-fiber vector space isomorphisms.

A vector bundle is a total space along with a surjective map to a base space . Any fiber is a vector space isomorphic to .

The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip.

One use for vector bundles is a generalization of vector functions. For instance, the vector space of tangent vectors at a point of an -dimensional manifold is isomorphic to in a coordinate chart. But the isomorphism with depends on the choice of coordinate chart. Nearby , the vector fields look like functions. To define vector fields on the whole manifold requires the tangent bundle, which is a special case of a vector bundle.

A bundle section of a vector bundle is a map whose projection, , is the identity map on . For instance, on a trivial bundle , a section corresponds to a function by .

Near every point in a vector bundle, there is a trivialization. The structure of the vector bundle, as in all bundles, is that it is locally trivial. In the case of a vector bundle, the transition functions between the trivializations take values in invertible linear transformations of the fiber.

Since the zero vector in is fixed by any linear transformation, the zero section always exists. By "nontrivial section," it is meant that it is not the zero section.

There are several adjectives that can specify properties of a vector bundle. A complex vector bundle has a fiber that is a complex vector space. A real vector bundle has a fiber that is a real vector space, which is the default kind of vector bundle. A line bundle has a fiber that is one dimensional.

A continuous vector bundle has continuous local trivializations that are linear on each fiber. A smooth vector bundle over a smooth manifold has local trivializations that are smooth maps and whose restrictions to the fibers are linear transformations. A holomorphic vector bundle over a complex manifold has complex vector spaces as fibers and local trivializations that are holomorphic maps and linear over on each fiber. The corresponding projections have the same regularity. A smooth complex vector bundle need not be a holomorphic vector bundle; holomorphic vector bundles have complex fibers.

Vector bundles can have inner products on their fibers, with Hermitian inner products in the complex case, and vector bundle connections.

See also

Bundle Rank, Fiber, Fiber Bundle, Griffiths' Conjecture, Hermitian Metric, K-Theory, Lie Algebroid, Line Bundle, Linear Algebra, Nef Vector Bundle, Principal Bundle, Projective Bundle, Riemannian Metric, Stable Equivalence, Tangent Bundle, Tangent Map, Trivial Bundle, Vector Bundle Connection, Vector Space, Whitney Sum Explore this topic in the MathWorld classroom

Portions of this entry contributed by Todd Rowland

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Weisstein, Eric W., with contributions by Todd Rowland. "Vector Bundle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VectorBundle.html

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