Connection (vector bundle)
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a connection on a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

 is a device that defines a notion of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

 on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle
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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

, then the notion of parallel transport is required to be linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

. Such a connection is equivalently specified by a covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

, which is an operator that can differentiate sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...

 of that bundle along tangent directions
Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

 in the base manifold. Connections in this sense generalize, to arbitrary vector bundles, the concept of a linear connection
Linear connection
In the mathematical field of differential geometry, the term linear connection can refer to either of the following overlapping concepts:* a connection on a vector bundle, often viewed as a differential operator ;* a principal connection on the frame bundle of a manifold or the induced connection...

 on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 of a smooth manifold, and are sometimes known as linear connections. Nonlinear connections
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...

 are connections that are not necessarily linear in this sense.

Connections on vector bundles are also sometimes called Koszul connections after Jean-Louis Koszul
Jean-Louis Koszul
Jean-Louis Koszul is a mathematician best known for studying geometry and discovering the Koszul complex.He was educated at the Lycée Fustel-de-Coulanges in Strasbourg before studying at the Faculty of Science in Strasbourg and the Faculty of Science in Paris...

, who gave an algebraic framework for describing them .

Formal definition

Let E → M be a smooth vector bundle
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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

 over a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

 M. Denote the space of smooth sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...

 of E by Γ(E). A connection on E is an ℝ-linear map
such that the Leibniz rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...


holds for all smooth functions f on M and all smooth sections σ of E.

If X is a tangent vector field on M (i.e. a section of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 TM) one can define a covariant derivative along X
by contracting X with the resulting covariant index in the connection ∇ (i.e. ∇Xσ = (∇σ)(X)). The covariant derivative satisfies the following properties:
Conversely, any operator satisfying the above properties defines a connection on E and a connection in this sense is also known as a covariant derivative on E.

Vector-valued forms

Let E → M be a vector bundle. An E-valued differential form
Vector-valued form
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms...

 of degree r is a section of the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

 bundle E ⊗ ΛrT*M. The space of such forms is denoted by
An E-valued 0-form is just a section of the bundle E. That is,

In this notation a connection on E → M is a linear map
A connection may then be viewed as a generalization of the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

 to vector bundle valued forms. In fact, given a connection ∇ on E there is a unique way to extend ∇ to a covariant exterior derivative or exterior covariant derivative
Unlike the ordinary exterior derivative one need not have (d)2 = 0. In fact, (d)2 is directly related to the curvature of the connection ∇ (see below).

Affine properties

Every vector bundle admits a connection. However, connections are not unique. If ∇1 and ∇2 are two connections on E → M then their difference is a C
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

-linear operator. That is,
for all smooth functions f on M and all smooth sections σ of E. It follows that the difference ∇1 − ∇2 is induced by a one-form on M with values in the endomorphism bundle End(E) = EE*:
Conversely, if ∇ is a connection on E and A is a one-form on M with values in End(E), then ∇+A is a connection on E.

In other words, the space of connections on E is an affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

 for Ω1(End E).

Relation to principal and Ehresmann connections

Let E → M be a vector bundle of rank k and let F(E) be the principal
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...

 frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...

 of E. Then a (principal) connection
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...

 on F(E) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant
Equivariant
In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant iffor all g ∈ G and all x in X...

 maps F(E) → Rk. (This can be seen by considering the pullback
Pullback bundle
In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...

 of E over F(E) → M, which is isomorphic to the trivial bundle F(E) × Rk.) Given a section σ of E let the corresponding equivariant map be ψ(σ). The covariant derivative on E is then given by
where XH is the horizontal lift of X (recall that the horizontal lift is determined by the connection on F(E)).

Conversely, a connection on E determines a connection on F(E), and these two constructions are mutually inverse.

A connection on E is also determined equivalently by a linear Ehresmann connection on E. This provides one method to construct the associated principal connection.

Local expression

Let E → M be a vector bundle of rank k, and let U be an open subset of M over which E is trivial. Given a local smooth frame (e1, …,ek) of E over U, any section σ of E can be written as (Einstein notation
Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

 assumed). A connection on E restricted to U then takes the form
where
Here ωαβ defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ωαβ determines a one-form ω with values in End(E) and this expression defines ∇ to be the connection d+ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame. In this context ω is sometimes called the connection form
Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....

 of ∇ with respect to the local frame.

If U is a coordinate neighborhood with coordinates (xi) then we can write
Note the mixture of coordinate and fiber indices in this expression. The coefficient functions ωiαβ are tensorial in the index i (they define a one-form) but not in the indices α and β. The transformation law for the fiber indices is more complicated. Let (f1, …,fk) be another smooth local frame over U and let the change of coordinate matrix be denoted t (i.e. fα = eβtβα). The connection matrix with respect to frame (fα) is then given by the matrix expression


Here dt is the matrix of one-forms obtained by taking the exterior derivative of the components of t.

The covariant derivative in the local coordinates and with respect to the local frame field (eα) is given by the expression

Parallel transport and holonomy

A connection ∇ on a vector bundle E → M defines a notion of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

 on E along a curve in M. Let γ : [0, 1] → M be a smooth path
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...

 in M. A section σ of E along γ is said to be parallel if
for all t ∈ [0, 1]. More formally, one can consider the pullback
Pullback bundle
In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...

 γ*E of E by γ. This is a vector bundle over [0, 1] with fiber Eγ(t) over t ∈ [0, 1]. The connection ∇ on E pulls back to a connection on γ*E. A section σ of γ*E is parallel if and only if γ*∇(σ) = 0.

Suppose γ is a path from x to y in M. The above equation defining parallel sections is a first-order ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

 (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector v in Ex there exists a unique parallel section σ of γ*E with σ(0) = v. Define a parallel transport map
by τγ(v) = σ(1). It can be shown that τγ is a linear isomorphism.

Parallel transport can be used to define the holonomy group of the connection ∇ based at a point x in M. This is the subgroup of GL(Ex) consisting of all parallel transport maps coming from loop
Loop (topology)
In mathematics, a loop in a topological space X is a path f from the unit interval I = [0,1] to X such that f = f...

s based at x:
The holonomy group of a connection is intimately related to the curvature of the connection .

Curvature

The curvature of a connection ∇ on E → M is a 2-form F on M with values in the endomorphism bundle End(E) = EE*. That is,
It is defined by the expression
where X and Y are tangent vector fields on M and s is a section of E. One must check that F is C
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

-linear in both X and Y and that it does in fact define a bundle endomorphism of E.

As mentioned above, the covariant exterior derivative d need not square to zero when acting on E-valued forms. The operator (d)2 is, however, strictly tensorial (i.e. C-linear). This implies that it is induced from a 2-form with values in End(E). This 2-form is precisely the curvature form given above. For an E-valued form σ we have

A flat connection is one whose curvature form vanishes identically.

Examples

  • A classical covariant derivative
    Covariant derivative
    In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

     or affine connection
    Affine connection
    In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

     defines a connection on the tangent bundle
    Tangent bundle
    In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

     of M, or more generally on any tensor bundle
    Tensor bundle
    In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed....

     formed by taking tensor products of the tangent bundle with itself and its dual.
  • The Levi-Civita connection
    Levi-Civita connection
    In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...

     is a connection on the tangent bundle of a Riemannian manifold
    Riemannian manifold
    In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

    .
  • The exterior derivative
    Exterior derivative
    In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

     is a flat connection on E = M × Rn (the trivial vector bundle over M).
  • More generally, there is a canonical flat connection on any flat vector bundle
    Flat vector bundle
    In mathematics, a vector bundle is said to be flat if it is endowed with a affine connection with vanishing curvature, ie. a flat connection.-de Rham cohomology of a flat vector bundle:...

    (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.
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