Vector bundle

In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together , form another topological space (or manifold or variety). A typical example is the tangent bundle of a manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Or consider a smooth curve in R2, and attach to every point of the curve the line normal to the curve at that point; this yields the normal bundle of the curve.

This article deals mostly with real vector bundles, with finite-dimensional fibers. Complex vector bundles are important in many cases, too; they are a special case, meaning that they can be seen as extra structure on an underlying real bundle.

= Definition and first consequences =

A real vector bundle is given by the following data:

topological spaces X (the base space ) and E (the total space )

a continuous map π : E → X (the projection )

for every x in X , the structure of a real number vector space on the Fiber bundle π−1({ x })

satisfying the following compatibility condition: for every point in X there is an open neighborhood U , a natural number n , and a homeomorphism φ : U × R n → π−1( U ) such that for every point x in U :

πφ( x , v ) = x for all vectors v in R n

the map v mapsto φ( x , v ) yields an isomorphism between the vector spaces R n and π−1({ x }).

The open neighborhood U together with the homeomorphism φ is called a local trivialization of the bundle. The local trivialization shows that locally the map π looks like the projection of U × R n on U .

A vector bundle is called trivial if there is a global trivialization , i.e. if it looks like the projection X × R n → X .

Every vector bundle π : E → X is surjective, since vector spaces cannot be empty set.

Every fiber π−1({ x }) is a finite-dimensional real vector space and hence has a Hamel dimension d x . The function x mapsto d x is locally constant, i.e. it is constant on all connectedness of X . If it is constant globally on X , we call this dimension the rank of the vector bundle. Vector bundles of rank 1 are called line bundles.

= Vector bundle morphisms =

A morphism from the vector bundle π1 : E 1 → X 1 to the vector bundle π2 : E 2 → X 2 is given by a pair of continuous maps f : E 1 → E 2 and g : X 1 → X 2 such that

g π1 = π2 f

for every x in X 1, the map π1−1({ x }) → π2−1({ g ( x )}) induced by f is a linear transformation between vector spaces.

The class of all vector bundles together with bundle morphisms forms a category (mathematics). Restricting to smooth manifolds and smooth bundle morphisms we obtain the category of smooth vector bundles.

We can also consider the category of all vector bundles over a fixed base space X . As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the identity map on X . That is, bundle morphisms for which the following diagram commutative diagram:

(Note that this category is not abelian category; the kernel (mathematics) of a morphism of vector bundles is in general not a vector bundle in any natural way.)

= Sections and locally free sheaves =

Given a vector bundle π : E → X and an open subset U of X , we can consider sections of π on U , i.e. continuous functions s : U → E with π s = id U . Essentially, a section assigns to every point of U a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold.

Let F ( U ) be the set of all sections on U . F ( U ) always contains at least one element: the function s that maps every element x of U to the zero element of the vector space π−1({ x }). With the pointwise addition and scalar multiplication of sections, F ( U ) becomes itself a real vector space. The collection of these vector spaces is a sheaf (mathematics) of vector spaces on X .

If s is an element of F ( U ) and α : U → R is a continuous map, then α s is in F ( U ). We see that F ( U ) is a module (mathematics) over the ring of continuous real-valued functions on U . Furthermore, if O X denotes the structure sheaf of continuous real-valued functions on X , then F becomes a sheaf of O X -modules.

Not every sheaf of O X -modules arises in this fashion from a vector bundle: only the locally free sheaf ones do. (The reason: locally we are looking for sections of a projection U × R n → U ; these are precisely the continuous functions U → R n , and such a function is an n -tuple of continuous functions U → R.)

Even more: the category of real vector bundles on X is category theory to the category of locally free and finitely generated sheaves of O X -modules. So we can think of the vector bundles as sitting inside the category of sheaves of O X -modules; this latter category is abelian, so this is where we can compute kernels of morphisms of vector bundles.

= Operations on vector bundles =

Two vector bundles on X , over the same field, have a Whitney sum, with fibre at any point the direct sum of fibres. In a similar way, fibrewise tensor product and dual space bundles may be introduced.

= Variants and generalizations =

Vector bundles are special fiber bundles, loosely speaking those where the fibers are vector spaces.

Smooth vector bundles are defined by requiring that E and X be s.

Replacing real vector spaces with complex number ones, we obtain complex vector bundles. This is a special case of reduction of the structure group of a bundle. Vector spaces over other topological fields may also be used, but that is comparatively rare.

If we allow arbitrary Banach spaces in the local trivialization (rather than only R n ), we obtain Banach bundles.

=References=

Jurgen Jost, Riemannian Geometry and Geometric Analysis , (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See section 1.5 .

Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics , (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.5 .