Vector bundle: Encyclopedia II - Vector bundle - Definition and first consequences

Vector 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 vector space on the fiber 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 × Rⁿ → π⁻¹(U) such that for every point x in U:

• πφ(x,v) = x for all vectors v in Rⁿ
• the map v φ(x,v) yields an isomorphism between the vector spaces Rⁿ 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 × Rⁿ on U.

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

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

Every fiber π⁻¹({x}) is a finite-dimensional real vector space and hence has a dimension d_x. The function x d_x is locally constant, i.e. it is constant on all connected components 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.

Adapted from the Wikipedia article "Definition and first consequences", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki