Vector bundle

In mathematics, a vector bundle is a geometric construction where to each point of a topological space (or manifold, or algebraic manifold) we join a vector space in a compatible way, so that all those vector spaces, "stuck together", form another topological space (or manifold or differentiable manifold).

A typical example is the tangent bundle of a differentiable manifold: to each point of the manifold we associate the tangent space of the manifold at that point. Or consider a differentiable curve in R, and attach to each point on the curve the normal of the line to the curve at that point; this gives the "normal bundle" of the curve This article deals mainly 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 an additional structure in an underlying real bundle.

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 and overlying function π: E → X (the "projection")

• for each x in X the structure of a real vector space in the fiber π^-1(chuckles)x}) that satisfies the following compatibility status: for each point at X There's an open neighborhood. U, a natural number nand a homeomorphism

${displaystyle varphi colon Utimes {mathbb {R} }^{n}to pi ^{-1}(U)}$

such that for every point x in U

* πφ()x, v) = x for all vectors v in Rⁿ

* Function v ◊ φ(x, v) gives an isomorphism between the vector space Rⁿ and π^-1(chuckles)x}).

The open neighborhood U together with the homeomorphism φ is called a local trivialization of the bundle. Local trivialization shows that locally the function π resembles the projection of U x Rⁿ on U .

A vector bundle is called trivial if there is a "global trivialization", that is, if it resembles the projection X x Rⁿ → X. Each vector bundle π: E → X is surjective, since vector spaces cannot be empty. Each fiber π^-1({x}) is a finite-dimensional real vector space and therefore has dimension d _x. The function x |-> d_x is locally constant, i.e. is constant in every connected component of X. If is global constant in X, we call this dimension the range of the vector bundle. A vector bundle of rank 1 is called a line bundle.

Morphisms

A morphism from the vector bundle π₁: E₁ → X₁ to the vector bundle π₂: E₂ → X ₂ is given by a pair of continuous functions f: E₁ → E ₂ and g: X₁ → X₂ such that

• gπ₁ = π₂f

• for each x in X₁, the function π₁^-1(chuckles)x}) → π₂^-1(chuckles)g(x) f is a linear transformation between vectorial spaces.

The composition of two morphisms is again a morphism, and we obtain the category of 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 vector bundle morphisms whose function in the base space is the identity function of X.

(Note that this category is not abelian. The kernel of a vector bundle map is not, in general, a vector bundle naturally.)

Locally free sections and beams

Given a vector bundle π: E → X and an open subset U of X, we can consider sections of π into U, i.e. continuous functions s: U → E with πs = id _U.

Essentially, a section assigns each point of U a vector from the associated vector space, in a continuous manner. For example, the sections of the tangent bundle of a differentiable manifold are nothing other than the vector fields in that manifold.

Let F(U) be the set of all sections in U. F(U) always contains at least one element: the function s that maps each element x of U to the zero element of the space π^-1({x}). With point-to-point addition and scalar multiplication of sections, F(U) also becomes a real vector space. The collection of these vector spaces is a bundle of vector spaces in X.

If s is an element of F(U) and α: U→ R is a continuous function, so 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 bundle of continuous real-valued functions in X, then F will be becomes a bundle of O_X-modules.

Not every bundle of O_X-modules arises in this way from a vector bundle: only the locally free ones. (the reason: locally we are looking for sections of a projection U x Rⁿ → U; these are exactly the continuous functions of U → Rⁿ, and such a function is a < i>n-tuple of continuous functions U → R.)

Further: the category of real vector bundles in X is equivalent to the category of locally free and finitely generated bundles of O_{X-modules. We can think of vector bundles within the category of bundles of OX-modules; this last category is abelian, so this is where we can compute vector bundle morphism kernels.}

Operations on vector bundles

Two vector bundles in X, on the same body, have a sum of Whitney, with the grain at any point given by the direct sum of fibers. In a similar way, the tensor product and the dual space can be introduced fiber by fiber.

Variants and generalizations

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

The differentiable vector bundles are defined by requiring that E and X be differentiable manifolds, π E → < i>X is a differentiable function, and the local trivialization functions φ are diffeomorphisms.

Substituting real vector spaces for complex ones, we obtain complex vector bundles. This is a special case of reducing the structure group of a bundle. Vector spaces on other topological fields can also be used, but that is comparatively rare. If we allow arbitrary Banach spaces in the local trivialization (instead of just Rⁿ), we get Banach bundles.