vector bundle

Basic definition

A vector bundleMathworldPlanetmath is a fiber bundleMathworldPlanetmath having a vector spaceMathworldPlanetmath as a fiber and the general linear groupMathworldPlanetmath of that vector space (or some subgroupMathworldPlanetmathPlanetmath) as structure group. Common examples of a vector bundle include the tangent bundle of a differentiable manifold and the Möbius strip (of infiniteMathworldPlanetmath width).

Vector bundles in various categories

As with fiber bundles, the idea of a vector bundle exists in many categoriesMathworldPlanetmath. We talk about topological vector bundles (in the category of topological spaces), we talk about differentiableMathworldPlanetmathPlanetmath vector bundles, we talk about complex analytic (or holomorphic) vector bundles, and we talk about algebraic vector bundles. In each case, the fiber must have a structureMathworldPlanetmath from the appropriate category, and the general linear group must also be equipped with a structure from the appropriate category (generally this means it must be a group object and it must act through morphismsMathworldPlanetmath in the category).

Specifically, if we want a topological vector bundle, we must supply a topological spaceMathworldPlanetmath for the base space, a topological space for the whole space, and the projection map must be continuousMathworldPlanetmathPlanetmath; this specifies a topology on each fiber. The general linear group must also act continuously.

If we are in the category of schemes, each local trivialization must be an affine spaceMathworldPlanetmath over the affine ring of the neighborhood on the scheme, and the general linear group scheme must act on it through morphisms of schemes.

Sections of a vector bundle

As with any fiber bundle, a vector bundle may have sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. If a vector bundle on X is defined on an open cover {Uα} with transition functionsMathworldPlanetmath ϕαβ taken from GLn, a section is a collectionMathworldPlanetmath of functions fα:UαUα×V which give the identityPlanetmathPlanetmathPlanetmath when projected down to Uα and such that


Sections may be added and scaled by field elements by simply applying these operationsMathworldPlanetmath to each fiber, so they form a vector space. A very common application of the Riemann-Roch theorem is to count the number of linearly independentMathworldPlanetmath sections on a curve, surface, or higher-dimensional varietyMathworldPlanetmathPlanetmathPlanetmath.

One is often interested in families of sections that are linearly independent in each fiber. If the vector bundle has dimensionMathworldPlanetmathPlanetmath n and there are n sections that are linearly independent on every fiber, then the vector bundle is isomorphicPlanetmathPlanetmathPlanetmath to the Cartesian product of X×V, which is called the trivial vector bundle. Such a family of sections is therefore called a trivialization.

One is sometimes interested in sections of a related vector bundle obtained by restricting the base space to some open subset. In this way, one can obtain a sheaf from a vector bundle, called the sheaf of sections.

Operations on vector bundles

Since the fiber of a vector bundle is a vector space, one can do many operations on vector bundles over a fixed space X; in fact, almost all the usual operations on vector spaces can be applied. However, they are often not quite as simple as in the case of finite-dimensional vector spaces.

One can take direct sumsMathworldPlanetmathPlanetmathPlanetmath and tensor productsPlanetmathPlanetmathPlanetmath of vector bundles; the dimensions (if finite) behave as expected. Morphisms between vector bundles over X are just linear maps on the fibers, with appropriate continuity conditions: since the space of linear maps between two vector spaces is again a vector space, a morphism between vector bundles must be a vector bundle itself.

If one has a short exact sequenceMathworldPlanetmath of vector bundles over X,


then the dimension of U is the sum of the dimensions of T and V, as one might expect; but one often cannot write U as the direct sum of T and V. In this way, vector bundles resemble modules over a ring or abelian groupsMathworldPlanetmath; in fact it is the behaviour of finite-dimensional vector spaces that is “too good to be true”.

Relation to other objects

In the algebraic categoryPlanetmathPlanetmathPlanetmath, that is, vector bundles over schemes, there is a very nice correspondence between vector bundles and locally free sheaves; when the dimension is one and the scheme is nice enough, there is a further correspondence with Cartier divisors.

Title vector bundle
Canonical name VectorBundle
Date of creation 2013-03-22 13:07:15
Last modified on 2013-03-22 13:07:15
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Definition
Classification msc 55R25
Related topic ReductionOfStructureGroup
Related topic SheafOfSections2
Related topic FrameGroupoid
Defines section
Defines trivial vector bundle
Defines sheaf of sections