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 vectorspaces, "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 R^{2}, 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.
Table of contents |
2 Morphisms 3 Sections and locally free sheaves 4 Operations on vector bundles 5 Variants and generalizations |
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 : E → X (the "projection")
- for every x in X, the structure of a real vector space on the fiber π^{−1}({x})
- πφ(x,v) = x for all vectors v in R^{n}
- the map v |-> φ(x,v) yields an isomorphism between the vector spaces R^{n} and π^{−1}({x}).
A vector bundle is called tivial if there is a "global trivialization", i.e. if it looks like the projection X × R^{n} → X.
Every vector bundle π : E → X 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 d_{x}. The function x |-> d_{x} 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.
Morphisms
- gπ_{1} = π_{2}f
- for every x in X_{1}, the map π_{1}^{−1}({x}) → π_{2}^{−1}({g(x)}) induced by f is a linear transformation between vector spaces.
We can also consider the category of all vector bundles on 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. (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
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 elememnt 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 vectorspace is a sheaf of vector spaces on X.
If s is an element of F(U) and α : U → R 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 O_{X} denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of O_{X}-modules.
Not every sheaf of O_{X}-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 × R^{n} → U; these are precisely the continuous functions U → R^{n}, and such a function is an n-tuple of continuous functions U → R.)
Even more: the category of real vector bundles on X is equivalent to the category of locally free and finitely generated sheaves of O_{X}-modules. So we can think of the vector bundles as sitting inside the category of sheaves of O_{X}-modules; this latter category is abelian, so this is where we can compute kernels of morphisms of vector bundles.
Operations on vector bundles
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 : E → X 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 R^{n}), we obtain Banach bundles.