Vector Bundle

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 in 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 pi:E->B to a base manifold B. Any fiber pi^(-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.

One use for vector bundles is a generalization of vector functions. For instance, the tangent vectors of an n-dimensional manifold are isomorphic to R^n at a point p in a coordinate chart. But the isomorphism with R^n depends on the choice of coordinate chart. Nearby p, 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 E is a map s:B->E whose projection, pi degreess is the identity map on B. For instance, on a trivial bundle E=B×V, a section s corresponds to a function f:B->V by s(b)=(b,f(b)).

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 linear invertible transformations of the fiber.

Since the element zero in V 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 V 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 is a manifold E with a continuous projection map pi. A smooth vector bundle is a smooth manifold E with a smooth projection pi. Finally, a holomorphic vector bundle is a complex manifold E with a holomorphic projection pi. In this last case, the fiber must be a complex vector space. So there could be a smooth complex vector bundle, but not a holomorphic real vector bundle.

Vector bundles can have metrics on their fibers, either Riemannian or Hermitian, and vector bundle connections.

See also

Bundle Rank, Fiber, Fiber Bundle, Hermitian Metric, K-Theory, Lie Algebroid, Linear Algebra, Principal Bundle, Real Vector 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

This entry contributed by Todd Rowland

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Rowland, Todd. "Vector Bundle." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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