nLab
dual vector bundle

Context

Bundles

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

Given a vector bundle, its dual is the vector bundle obtained by passing fiber-wise to the dual vector space.

Examples

Proposition

Let XX be a topological space and let E ip iXE_i \overset{p_i}{\to} X be a two topological vector bundles over XX, of finite rank of a vector bundle. Then a homomorphism of vector bundles

f:E 1E 2 f \;\colon\; E_1 \rightarrow E_2

is equivalently a section of the tensor product of vector bundles of E 2E_2 with the dual vector bundle of E 1E_1.

Hom Vect(X)(E 1,E 2)Γ X(E 1 * XE 2). Hom_{Vect(X)}(E_1, E_2) \;\simeq\; \Gamma_X( E_1^\ast \otimes_X E_2 ) \,.

Moreover, this section is a trivializing section (this example) precisely if the corresponding morphism is an isomorphism.

Revised on July 4, 2017 11:45:37 by Urs Schreiber (195.37.209.183)