Topology MOC

Vector bundle

A (real¹) vector bundle is a 𝐶𝛼-fibre bundle topology

ℝ𝑘→𝐸𝜋↠𝐵

where for every 𝑝 ∈𝐵 the fibre 𝐸𝑝 is a 𝑘-dimensional vector space over ℝ and we have a local trivalization 𝒯 of 𝐸 such that each 𝜑 ∈𝒯 restricts to a linear isomorphism. Usually we denote vector bundles by the entire space 𝐸 where the projection 𝜋 :𝐸 ↠𝐵 onto the base space and the structure of a vector space on each fibre 𝐸𝑝 is understood.

Further terminology

• A Vector bundle morphism is a Bundle map which respects the linear structure. We can thus form the category Category of vector bundles.
• Many of the usual constructions on vector spaces carry over:
  • Direct sum of vector bundles
  • Tensor product of vector bundles
  • Hom vector bundle
  • Dual vector bundle

Examples

• The Tangent bundle for a manifold, as well as vector bundles obtained thence using the constructions above.

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Footnotes

1. See Complex vector bundle. ↩