Knowlpedia

Rank of a vector bundle

The fiber dimension of a real or complex vector bundle, which is locally constant on the base.

Let π:E→M\pi:E\to M be a smooth real or complex vector bundle. The rank of EE at x∈Mx\in M is

rankx(E):=dim⁡F(Ex),\mathrm{rank}_x(E):=\dim_{\mathbb F}(E_x),

where F=R\mathbb F=\mathbb R for real bundles and F=C\mathbb F=\mathbb C for .

Local triviality makes x↦rank⁡x(E)x\mapsto\operatorname{rank}_x(E) locally constant. If MM is connected, its common value is the rank of EE, denoted rank⁡(E)\operatorname{rank}(E).

Examples
  1. If dim⁡M=n\dim M=n, then rank⁡(TM)=rank⁡(T∗M)=n\operatorname{rank}(TM)=\operatorname{rank}(T^*M)=n.
  1. The trivial bundle M×Fr→MM\times\mathbb F^r\to M has rank rr.
  1. If EE and FF are bundles over the same connected base, then
    rank(E⊕F)=rank(E)+rank(F),rank(E⊗F)=rank(E) rank(F),\mathrm{rank}(E\oplus F)=\mathrm{rank}(E)+\mathrm{rank}(F),\qquad \mathrm{rank}(E\otimes F)=\mathrm{rank}(E)\,\mathrm{rank}(F),
    where ⊕\oplus and ⊗\otimes denote the fiberwise direct sum and tensor product.