# vector bundle

## 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  $\phi_{\alpha\beta}$ taken from $\mathop{GL}_{n}$, 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

 $\phi_{\alpha\beta}\circ f_{\beta}|_{U_{\alpha}\cap U_{\beta}}=f_{\alpha}|_{U_{% \alpha}\cap U_{\beta}}.$

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.

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