# vector bundle

## Basic definition

A *vector bundle ^{}* is a fiber bundle

^{}having a vector space

^{}as a fiber and the general linear group

^{}of that vector space (or some subgroup

^{}) as structure group. Common examples of a vector bundle include the tangent bundle of a differentiable manifold and the Möbius strip (of infinite

^{}width).

## Vector bundles in various categories

As with fiber bundles, the idea of a vector bundle exists in many categories^{}. We talk about topological vector bundles (in the category of topological spaces), we talk about differentiable^{} 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 structure^{} 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 morphisms^{} in the category).

Specifically, if we want a topological vector bundle, we must supply a topological space^{} for the base space, a topological space for the whole space, and the projection map must be continuous^{}; 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 space^{} 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 sections^{}. If a vector bundle on $X$ is defined on an open cover $\{{U}_{\alpha}\}$ with transition functions^{} ${\varphi}_{\alpha \beta}$ taken from ${GL}_{n}$, a section is a collection^{} of functions ${f}_{\alpha}:{U}_{\alpha}\to {U}_{\alpha}\times V$ which give the identity^{} when projected down to ${U}_{\alpha}$ and such that

$${{\varphi}_{\alpha \beta}\circ {f}_{\beta}|}_{{U}_{\alpha}\cap {U}_{\beta}}={{f}_{\alpha}|}_{{U}_{\alpha}\cap {U}_{\beta}}.$$ |

Sections may be added and scaled by field elements by simply applying these operations^{} 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 independent^{} sections on a curve, surface, or higher-dimensional variety^{}.

One is often interested in families of sections that are linearly independent in each fiber. If the vector bundle has dimension^{} $n$ and there are $n$ sections that are linearly independent on every fiber, then the vector bundle is isomorphic^{} to the Cartesian product of $X\times V$, 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 sums^{} and tensor products^{} 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 sequence^{} of vector bundles over $X$,

$$0\to T\to U\to V\to 0,$$ |

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 groups^{}; 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 category^{}, 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 |