Vector Bundle - Definition and First Consequences

Definition and First Consequences

A real vector bundle consists of:

1. topological spaces X (base space) and E (total space)
2. a continuous surjection π : E → X (bundle projection)
3. for every x in X, the structure of a finite-dimensional real vector space on the fiber π−1({x})

where the following compatibility condition is satisfied: for every point in X, there is an open neighborhood U, a natural number k, and a homeomorphism

such that for all x ∈ U,

• (π∘φ)(x,v) = x for all vectors v in Rk, and

• the map v ↦ φ(x,v) is an isomorphism between the vector spaces Rk and π−1({x}).

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

Every fiber π−1({x}) is a finite-dimensional real vector space and hence has a dimension k_x. The local trivializations show that the function x ↦ k_x is locally constant, and is therefore constant on each connected component of X. If k_x is equal to a constant k on all of X, then k is called the rank of the vector bundle, and E is said to be a vector bundle of rank k. Often the definition of a vector bundle includes that the rank is well defined, so that k_x is constant. Vector bundles of rank 1 are called line bundles, while those of rank 2 are less commonly called plane bundles.

The Cartesian product X × Rk, equipped with the projection X × Rk → X, is called the trivial bundle of rank k over X.