Vector bundle

From Academic Kids

In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space (or manifold or variety). A typical example is the tangent bundle of a differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Or consider a smooth curve in R2, and attach to every point of the curve the line normal to the curve at that point; this yields the "normal bundle" of the curve.

This article deals mostly with real vector bundles, with finite-dimensional fibers. Complex vector bundles are important in many cases, too; they are a special case, meaning that they can be seen as extra structure on an underlying real bundle.

Contents

Definition and first consequences

A real vector bundle is given by the following data:

  • topological spaces X (the "base space") and E (the "total space")
  • a continuous map π : EX (the "projection")
  • for every x in X, the structure of a real vector space on the fiber π−1({x})

satisfying the following compatibility condition: for every point in X there is an open neighborhood U, a natural number n, and a homeomorphism φ : U × Rn → π−1(U) such that for every point x in U:

  • πφ(x,v) = x for all vectors v in Rn
  • the map v |-> φ(x,v) yields an isomorphism between the vector spaces Rn and π−1({x}).

The open neighborhood U together with the homeomorphism φ is called a local trivialization of the bundle. The local trivialization shows that "locally" the map π looks like the projection of U × Rn on U.

A vector bundle is called trivial if there is a "global trivialization", i.e. if it looks like the projection X × RnX.

Every vector bundle π : EX is surjective, since vector spaces cannot be empty.

Every fiber π−1({x}) is a finite-dimensional real vector space and hence has a dimension dx. The function x |-> dx is locally constant, i.e. it is constant on all connected components of X. If it is constant globally on X, we call this dimension the rank of the vector bundle. Vector bundles of rank 1 are called line bundles.

Vector bundle morphisms

A morphism from the vector bundle π1 : E1X1 to the vector bundle π2 : E2X2 is given by a pair of continuous maps f : E1E2 and g : X1X2 such that

  • gπ1 = π2f
Missing image
BundleMorphism-01.png
Image:BundleMorphism-01.png

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

The class of all vector bundles together with bundle morphisms forms a category. Restricting to smooth manifolds and smooth bundle morphisms we obtain the category of smooth vector bundles.

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:

Missing image
BundleMorphism-02.png
Image:BundleMorphism-02.png

(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.)

Sections and locally free sheaves

Given a vector bundle π : EX and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s : UE with πs = idU. Essentially, a section assigns to every point of U a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold.

Let F(U) be the set of all sections on U. F(U) always contains at least one element: the function s that maps every element x of U to the zero element of the vector space π−1({x}). With the pointwise addition and scalar multiplication of sections, F(U) becomes itself a real vector space. The collection of these vector spaces is a sheaf of vector spaces on X.

If s is an element of F(U) and α : UR is a continuous map, then αs is in F(U). We see that F(U) is a module over the ring of continuous real-valued functions on U. Furthermore, if OX denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of OX-modules.

Not every sheaf of OX-modules arises in this fashion from a vector bundle: only the locally free ones do. (The reason: locally we are looking for sections of a projection U × RnU; these are precisely the continuous functions URn, and such a function is an n-tuple of continuous functions UR.)

Even more: the category of real vector bundles on X is equivalent to the category of locally free and finitely generated sheaves of OX-modules. So we can think of the vector bundles as sitting inside the category of sheaves of OX-modules; this latter category is abelian, so this is where we can compute kernels of morphisms of vector bundles.

Operations on vector bundles

Two vector bundles on X, over the same field, have a Whitney sum, with fibre at any point the direct sum of fibres. In a similar way, fibrewise tensor product and dual space bundles may be introduced.

Variants and generalizations

Vector bundles are special fiber bundles, loosely speaking those where the fibers are vector spaces.

Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π : EX be a smooth map, and the local trivialization maps φ be diffeomorphisms.

Replacing real vector spaces with complex ones, we obtain complex vector bundles. This is a special case of reduction of the structure group of a bundle. Vector spaces over other topological fields may also be used, but that is comparatively rare.

If we allow arbitrary Banach spaces in the local trivialization (rather than only Rn), we obtain Banach bundles.

References

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