# Vector bundle

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 **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.

*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.*

**Complex**## 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 :
*E*→*X*(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* × **R**^{n} → π^{−1}(*U*) such that for every point *x* in *U*:

- πφ(
*x*,*v*) =*x*for all vectors*v*in**R**^{n} - the map
*v***Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mapsto}**φ(*x*,*v*) yields an isomorphism between the vector spaces**R**^{n}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* × **R**^{n} on *U*.

A vector bundle is called **trivial** 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* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mapsto}**
*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.

## Vector bundle morphisms

A **morphism** from the vector bundle π_{1} : *E*_{1} → *X*_{1} to the vector bundle π_{2} : *E*_{2} → *X*_{2} is given by a pair of continuous maps *f* : *E*_{1} → *E*_{2} and *g* : *X*_{1} → *X*_{2} such that

*g*π_{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.

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:

(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 : *E* → *X* and an open subset *U* of *X*, we can consider **sections** of π on *U*, i.e. continuous functions *s* : *U* → *E* with π*s* = id_{U}. 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 : *U* → **R** is a continuous map, then α*s* 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

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 : *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**.

## References

- Jurgen Jost,
*Riemannian Geometry and Geometric Analysis*, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2*See section 1.5*. - Ralph Abraham and Jarrold E. Marsden,
*Foundations of Mechanics*, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X*See section 1.5*.