Vector Bundle - Vector Bundle Morphisms

Vector Bundle Morphisms

A morphism from the vector bundle π₁ : E₁ → X₁ to the vector bundle π₂ : E₂ → X₂ is given by a pair of continuous maps f : E₁ → E₂ and g : X₁ → X₂ such that

• g ∘ π₁ = π₂ ∘ f

• for every x in X₁, the map π₁−1({x}) → π₂−1({g(x)}) induced by f is a linear map between vector spaces.

Note that g is determined by f (because π₁ is surjective), and f is then said to cover g.

The class of all vector bundles together with bundle morphisms forms a category. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are also often called (vector) bundle homomorphisms.

A bundle homomorphism from E₁ to E₂ with an inverse which is also a bundle homomorphism (from E₂ to E₁) is called a (vector) bundle isomorphism, and then E₁ and E₂ are said to be isomorphic vector bundles. An isomorphism of a (rank k) vector bundle E over X with the trivial bundle (of rank k over X) is called a trivialization of E, and E is then said to be trivial (or trivializable). The definition of a vector bundle shows that any vector bundle is locally trivial.

We can also consider the category of all vector bundles over a fixed base space X. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the identity map on X. That is, bundle morphisms for which the following diagram commutes:

(Note that this category is not abelian; the kernel of a morphism of vector bundles is in general not a vector bundle in any natural way.)

A vector bundle morphism between vector bundles π₁ : E₁ → X₁ and π₂ : E₂ → X₂ covering a map g from X₁ to X₂ can also be viewed as a vector bundle morphism over X₁ from E₁ to the pullback bundle g*E₂.