Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$ (for example $X$ could be a topological space, a manifold, or an algebraic variety): to every point $x$ of the space $X$ we associate (or "attach") a vector space $V(x)$ in such a way that these vector spaces fit together to form another space of the same kind as $X$ (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over $X$ .

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space $V$ such that $V(x)=V$ for all $x$ in $X$ : in this case there is a copy of $V$ for each $x$ in $X$ and these copies fit together to form the vector bundle $X\times V$ over $X$ . Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.

Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.

$(\pi \circ \varphi )(x,v)=x$ for all $v$ in $\mathbb {R} ^{k}$ , and

vectors

the map $v\mapsto \varphi (x,v)$ is a isomorphism between the vector spaces $\mathbb {R} ^{k}$ and $\pi ^{-1}(\{x\})$ .

linear

The Whitney sum (named for ) or direct sum bundle of E and F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum E_x ⊕ F_x of the vector spaces E_x and F_x.

Hassler Whitney

The E ⊗ F is defined in a similar way, using fiberwise tensor product of vector spaces.

tensor product bundle

The Hom-bundle Hom(E, F) is a vector bundle whose fiber at x is the space of linear maps from E_x to F_x (which is often denoted Hom(E_x, F_x) or L(E_x, F_x)). The Hom-bundle is so-called (and useful) because there is a between vector bundle homomorphisms from E to F over X and sections of Hom(E, F) over X.

bijection

Building on the previous example, given a section s of an bundle Hom(E, E) and a function f: X → R, one can construct an eigenbundle by taking the fiber over a point x ∈ X to be the f(x)-eigenspace of the linear map s(x): E_x → E_x. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of s being the zero section and f having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in E, while everywhere else the fiber is the trivial 0-dimensional vector space.

endomorphism

The E* is the Hom bundle Hom(E, R × X) of bundle homomorphisms of E and the trivial bundle R × X. There is a canonical vector bundle isomorphism Hom(E, F) = E* ⊗ F.

dual vector bundle

Most operations on vector spaces can be extended to vector bundles by performing the vector space operation fiberwise.

For example, if E is a vector bundle over X, then there is a bundle E* over X, called the dual bundle, whose fiber at x ∈ X is the dual vector space (E_x)*. Formally E* can be defined as the set of pairs (x, φ), where x ∈ X and φ ∈ (E_x)*. The dual bundle is locally trivial because the dual space of the inverse of a local trivialization of E is a local trivialization of E*: the key point here is that the operation of taking the dual vector space is functorial.

There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles E, F on X (over the given field). A few examples follow.

Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a functorial manner. This is made precise in the language of smooth functors. An operation of a different nature is the pullback bundle construction. Given a vector bundle E → Y and a continuous map f: X → Y one can "pull back" E to a vector bundle f*E over X. The fiber over a point x ∈ X is essentially just the fiber over f(x) ∈ Y. Hence, Whitney summing E ⊕ F can be defined as the pullback bundle of the diagonal map from X to X × X where the bundle over X × X is E × F.

Remark: Let X be a compact space. Any vector bundle E over X is a direct summand of a trivial bundle; i.e., there exists a bundle E' such that E ⊕ E' is trivial. This fails if X is not compact: for example, the tautological line bundle over the infinite real projective space does not have this property.^[1]

C^r then the vector bundle is a C^r vector bundle,

real then the vector bundle is a real analytic vector bundle (this requires the matrix group to have a real analytic structure),

analytic

holomorphic then the vector bundle is a (this requires the matrix group to be a complex Lie group),

holomorphic vector bundle

algebraic functions then the vector bundle is an (this requires the matrix group to be an algebraic group).

algebraic vector bundle

A vector bundle (E, p, M) is smooth, if E and M are smooth manifolds, p: E → M is a smooth map, and the local trivializations are diffeomorphisms. Depending on the required degree of smoothness, there are different corresponding notions of C^p bundles, infinitely differentiable C^∞-bundles and real analytic C^ω-bundles. In this section we will concentrate on C^∞-bundles. The most important example of a C^∞-vector bundle is the tangent bundle (TM, $π$ _TM, M) of a C^∞-manifold M.

A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are smooth functions on overlaps of trivializing charts U and V. That is, a vector bundle E is smooth if it admits a covering by trivializing open sets such that for any two such sets U and V, the transition function

is a smooth function into the matrix group GL(k,R), which is a Lie group.

Similarly, if the transition functions are:

The C^∞-vector bundles (E, p, M) have a very important property not shared by more general C^∞-fibre bundles. Namely, the tangent space T_v(E_x) at any v ∈ E_x can be naturally identified with the fibre E_x itself. This identification is obtained through the vertical lift vl_v: E_x → T_v(E_x), defined as

The vertical lift can also be seen as a natural C^∞-vector bundle isomorphism p*E → VE, where (p*E, p*p, E) is the pull-back bundle of (E, p, M) over E through p: E → M, and VE := Ker(p_*) ⊂ TE is the vertical tangent bundle, a natural vector subbundle of the tangent bundle (TE, $π$ _TE, E) of the total space E.

The total space E of any smooth vector bundle carries a natural vector field V_v := vl_vv, known as the canonical vector field. More formally, V is a smooth section of (TE, $π$ _TE, E), and it can also be defined as the infinitesimal generator of the Lie-group action $(t,v)\mapsto e^{tv}$ given by the fibrewise scalar multiplication. The canonical vector field V characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when X is a smooth vector field on a smooth manifold M and x ∈ M such that X_x = 0, the linear mapping

does not depend on the choice of the linear covariant derivative ∇ on M. The canonical vector field V on E satisfies the axioms

Conversely, if E is any smooth manifold and V is a smooth vector field on E satisfying 1–4, then there is a unique vector bundle structure on E whose canonical vector field is V.

For any smooth vector bundle (E, p, M) the total space TE of its tangent bundle (TE, $π$ _TE, E) has a natural secondary vector bundle structure (TE, p_*, TM), where p_* is the push-forward of the canonical projection p: E → M. The vector bundle operations in this secondary vector bundle structure are the push-forwards +_*: T(E × E) → TE and λ_*: TE → TE of the original addition +: E × E → E and scalar multiplication λ: E → E.

: classifying spaces for vector bundle, among which projective spaces for line bundles

Vector bundle

vectors

linear

Hassler Whitney

tensor product bundle

bijection

endomorphism

dual vector bundle

analytic

holomorphic vector bundle

algebraic vector bundle

Grassmannian

Characteristic class

Splitting principle

Stable bundle

"Vector bundle"

Why is it useful to study vector bundles ?

Why is it useful to classify the vector bundles of a space ?