Vector Bundles

Vector Bundle

A (real) vector bundle of rank over a base topological space (usually a smooth manifold) consists of the following information:

• topological spaces and ;
• a continuous surjection , called the bundle projection;
• for every , the fiber yields a finite-dimensional real vector space,

such that is locally trivialisable. That is, for every point , there is an open neighbourhood of , a natural number , and a homeomorphism that for all , is a linear isomorphism onto .
In other words, the following diagram commutes:

Remark

Intuitively, the local trivialisation means that locally, the vector bundle doesn’t twist or bend in a complicated way. So, local trivialization is the property that allows us to study the vector bundle piece by piece, using the familiar structure of product spaces in each piece, even if the entire bundle has a more complex, “twisted” global structure.

e.g.

• The trivial bundle is a projection ;
• If we let be the quotient space of under the identifications , then the projection induces a map which is a one dimensional vector bundle, or line bundle. Since is homeomorphic to a Möbius band with its boundary circle deleted, we call this bundle the Möbius bundle.
  

Section

A section of a vector bundle over through is essentially an assignment to choose one vector from each fiber of the bundle in a smooth manner, such that .

e.g. A vector bundle over a manifold with section and fibers :

Proposition

Let be a vector bundle. The space of sections, denoted , is a vector space, with a module structure over the ring .

Frame of a Vector Bundle

Let be a vector bundle on a smooth manifold . A local frame over an open subset , is a set of smooth sections of such that for every , the vectors form a basis of the fiber .

Remark

A frame provides a way to trivialise the bundle locally, giving a smooth choice of basis for each vector space fiber over an open set of the base manifold.

On an -manifold, given two coordinate charts the coordinate change induces a corresponding change of basis: And we can identify the change of basis matrix as a smooth map .

Constructions on Vector Bundles

Subbundle

is called a subbundle of if for every , the fiber is a linear subspace of the fiber .

Direct Sum of Vector Bundles

Suppose , are vector bundles,

Quotient Bundle

Given a subbundle of a vector bundle , the quotient bundle is the vector bundle over whose fibers are the quotient spaces , for all .

Dual Bundle

The dual bundle of a vector bundle over is the vector bundle over whose fibers are the dual spaces to the fibers of .

Pullback Bundle & Pullback Section

Suppose is a vector bundle on a manifold , and is a local diffeomorphism map. Then the pullback bundle is a vector bundle over whose fibers are the pullbacks of the fibers of . That is, for each , the fiber is defined as . A section of induces a section of by