Knowlpedia

Definition

Vector bundle morphism

A smooth map between total spaces of vector bundles that covers a base map and is linear on each fiber.

Let πE:E→M\pi_E:E\to M and πF:F→N\pi_F:F\to N be smooth real or . A vector bundle morphism (also called a fiberwise linear bundle map) from EE to FF is a pair (Φ,f)(\Phi,f) consisting of a Φ:E→F\Phi:E\to F and a smooth map f:M→Nf:M\to N such that:

  1. Covers the base map: πF∘Φ=f∘πE\pi_F\circ \Phi = f\circ \pi_E.
  1. Fiberwise linearity: for every x∈Mx\in M, the induced map on fibers
    Φx:Ex→Ff(x),Φx(v):=Φ(v),\Phi_x:E_x\to F_{f(x)},\qquad \Phi_x(v):=\Phi(v),
    is F\mathbb F-linear (with F=R\mathbb F=\mathbb R or C\mathbb C according to the bundles).

If M=NM=N and f=idMf=\mathrm{id}_M, one often says Φ:E→F\Phi:E\to F is a bundle map over MM.

Composition of vector bundle morphisms is defined by composition of the total-space maps and the base maps, and yields a category of smooth vector bundles and bundle morphisms.

Fixed-base and varying-base categories

There are two useful categorical conventions.

  • In the VectF(M)\mathbf{Vect}_{\mathbb F}(M), every object lies over one chosen MM, and every morphism covers id⁡M\operatorname{id}_M.
  • In the varying-base category, objects may lie over different manifolds and (Φ,f):E→F(\Phi,f):E\to F may cover an arbitrary smooth map f:M→Nf:M\to N.

Only the first convention sends sections covariantly by simple postcomposition:

s⟼Φ∘s.s\longmapsto\Phi\circ s.

When f≠id⁡Mf\ne\operatorname{id}_M, Φ∘s\Phi\circ s lies over ff, rather than being a section of F→NF\to N. A common alternative is to express a varying-base map as a bundle map E→f∗FE\to f^*F over MM.

Examples
  1. Differential of a smooth map. For any smooth map f:M→Nf:M\to N, the differential
    df:TM→TNdf:TM\to TN
    is a vector bundle morphism covering ff between the .
  1. Projection from a direct sum. For bundles E,F→ME,F\to M, the projection prE:E⊕F→E\mathrm{pr}_E:E\oplus F\to E is a bundle morphism over idM\mathrm{id}_M, fiberwise the linear projection Ex⊕Fx→ExE_x\oplus F_x\to E_x.
  1. Inclusion of a subbundle. If E⊆FE\subseteq F is a over the same base MM, then the inclusion map E↪FE\hookrightarrow F is a vector bundle morphism over idM\mathrm{id}_M.
References
  1. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth vector bundles and bundle homomorphisms.
  2. Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: bundle maps and pullback bundles.