Connection (vector bundle)

Summary

In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle 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. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950).

This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).

MotivationEdit

Let M be a differentiable manifold, such as Euclidean space. A vector-valued function   can be viewed as a section of the trivial vector bundle   One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on M.

 
A section of a bundle may be viewed as a generalized function from the base into the fibers of the vector bundle. This can be visualized by the graph of the section, as in the figure above.

The model case is to differentiate a function   on Euclidean space  . In this setting the derivative   at a point   in the direction   may be defined by the standard formula

 

For every  , this defines a new vector  

When passing to a section   of a vector bundle   over a manifold  , one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term   makes no sense on  . Instead one takes a path   such that   and computes

 

However this still does not make sense, because   and   are elements of the distinct vector spaces   and   This means that subtraction of these two terms is not naturally defined.

The problem is resolved by introducing the extra structure of a connection to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.

  1. (Parallel transport) A connection can be viewed as assigning to every differentiable path   a linear isomorphism   for all   Using this isomorphism one can transport   to the fibre   and then take the difference; explicitly,
     
    In order for this to depend only on   and not on the path   extending   it is necessary to place restrictions (in the definition) on the dependence of   on   This is not straightforward to formulate, and so this notion of "parallel transport" is usually derived as a by-product of other ways of defining connections. In fact, the following notion of "Ehresmann connection" is nothing but an infinitesimal formulation of parallel transport.
  2. (Ehresmann connection) The section   may be viewed as a smooth map from the smooth manifold   to the smooth manifold   As such, one may consider the pushforward   which is an element of the tangent space   In Ehresmann's formulation of a connection, one chooses a way of assigning, to each   and every   a direct sum decomposition of   into two linear subspaces, one of which is the natural embedding of   With this additional data, one defines   by projecting   to be valued in   In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of   moves as e is varied over a fiber.
  3. (Covariant derivative) The standard derivative   in Euclidean contexts satisfies certain dependencies on   and   the most fundamental being linearity. A covariant derivative is defined to be any operation   which mimics these properties, together with a form of the product rule.

Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding choice of how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certain partial differential equations. In the case of the tangent bundle, any pseudo-Riemannian metric (and in particular any Riemannian metric) determines a canonical connection, called the Levi-Civita connection.

Formal definitionEdit

Let   be a smooth real vector bundle over a smooth manifold  . Denote the space of smooth sections of   by  . A covariant derivative on   is either of the following equivalent structures:

  1. an  -linear map   such that the product rule
     
    holds for all smooth functions   on   and all smooth sections   of  
  2. an assignment, to any smooth section s and every x, of a  -linear map   which depends smoothly on x and such that
     
    for any two smooth sections   and any real numbers   and such that for every smooth function f,   is related to   by
     
    for any   and  

Beyond using the canonical identification between the vector space   and the vector space of linear maps   these two definitions are identical and differ only in the language used.

It is typical to denote   by   with   being implicit in   With this notation, the product rule in the second version of the definition given above is written

 

Remark. In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "ℝ" everywhere they appear to "complex" and " " This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.

Induced connectionsEdit

Given a vector bundle  , there are many associated bundles to   which may be constructed, for example the dual vector bundle  , tensor powers  , symmetric and antisymmetric tensor powers  , and the direct sums  . A connection on   induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory of principal bundle connections, but here we present some of the basic induced connections.

Dual connectionEdit

Given   a connection on  , the induced dual connection   on   is defined implicitly by

 

Here   is a smooth vector field,   is a section of  , and   a section of the dual bundle, and   the natural pairing between a vector space and its dual (occurring on each fibre between   and  ). Notice that this definition is essentially enforcing that   be the connection on   so that a natural product rule is satisfied for pairing  .

Tensor product connectionEdit

Given   connections on two vector bundles  , define the tensor product connection by the formula

 

Here we have  . Notice again this is the natural way of combining   to enforce the product rule for the tensor product connection. By repeated application of the above construction applied to the tensor product  , one also obtains the tensor power connection on   for any   and vector bundle  .

Direct sum connectionEdit

The direct sum connection is defined by

 

where  .

Symmetric and exterior power connectionsEdit

Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power,  , the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside the tensor algebra as direct summands, and the connection   respects this natural splitting, one can simply restrict   to these summands. Explicitly, define the symmetric product connection by

 

and the exterior product connection by

 

for all  . Repeated applications of these products gives induced symmetric power and exterior power connections on   and   respectively.

Endomorphism connectionEdit

Finally, one may define the induced connection   on the vector bundle of endomorphisms  , the endomorphism connection. This is simply the tensor product connection of the dual connection   on   and   on  . If   and  , so that the composition   also, then the following product rule holds for the endomorphism connection:

 

By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying

 

for any  , thus avoiding the need to first define the dual connection and tensor product connection.

Any associated bundleEdit

Given a vector bundle   of rank  , and any representation   into a linear group  , there is an induced connection on the associated vector bundle  . This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of   and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.

Exterior covariant derivative and vector-valued formsEdit

Let   be a vector bundle. An  -valued differential form of degree   is a section of the tensor product bundle:

 

The space of such forms is denoted by

 

where the last tensor product denotes the tensor product of modules over the ring of smooth functions on  .

An  -valued 0-form is just a section of the bundle  . That is,

 

In this notation a connection on   is a linear map

 

A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms. In fact, given a connection   on   there is a unique way to extend   to an exterior covariant derivative

 

This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the form   and extended linearly:

 

where   so that  ,   is a section, and   denotes the  -form with values in   defined by wedging   with the one-form part of  . Notice that for  -valued 0-forms, this recovers the normal Leibniz rule for the connection  .

Unlike the ordinary exterior derivative, one generally has  . In fact,   is directly related to the curvature of the connection   (see below).

Affine properties of the set of connectionsEdit

Every vector bundle over a manifold admits a connection, which can be proved using partitions of unity. However, connections are not unique. If   and   are two connections on   then their difference is a  -linear operator. That is,

 

for all smooth functions   on   and all smooth sections   of  . It follows that the difference   can be uniquely identified with a one-form on   with values in the endomorphism bundle  :

 

Conversely, if   is a connection on   and   is a one-form on   with values in  , then   is a connection on  .

In other words, the space of connections on   is an affine space for  . This affine space is commonly denoted  .

Relation to principal and Ehresmann connectionsEdit

Let   be a vector bundle of rank   and let   be the principal frame bundle of  . Then a (principal) connection on   induces a connection on  . First note that sections of   are in one-to-one correspondence with right-equivariant maps  . (This can be seen by considering the pullback of   over  , which is isomorphic to the trivial bundle  .) Given a section   of   let the corresponding equivariant map be  . The covariant derivative on   is then given by

 

where   is the horizontal lift of   from   to  . (Recall that the horizontal lift is determined by the connection on  .)

Conversely, a connection on   determines a connection on  , and these two constructions are mutually inverse.

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

The induced connections discussed in #Induced connections can be constructed as connections on other associated bundles to the frame bundle of  , using representations other than the standard representation used above. For example if   denotes the standard representation of   on  , then the associated bundle to the representation   of   on   is the direct sum bundle  , and the induced connection is precisely that which was described above.

Local expressionEdit

Let   be a vector bundle of rank  , and let   be an open subset of   over which   trivialises. Therefore over the set  ,   admits a local smooth frame of sections

 

Since the frame   defines a basis of the fibre   for any  , one can expand any local section   in the frame as

 

for a collection of smooth functions  .

Given a connection   on  , it is possible to express   over   in terms of the local frame of sections, by using the characteristic product rule for the connection. For any basis section  , the quantity   may be expanded in the local frame   as

 

where   are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by

 

called the local connection form of   over  . The action of   on any section   can be computed in terms of   using the product rule as

 

If the local section   is also written in matrix notation as a column vector using the local frame   as a basis,

 

then using regular matrix multiplication one can write

 

where   is shorthand for applying the exterior derivative   to each component of   as a column vector. In this notation, one often writes locally that  . In this sense a connection is locally completely specified by its connection one-form in some trivialisation.

As explained in #Affine properties of the set of connections, any connection differs from another by an endomorphism-valued one-form. From this perspective, the connection one-form   is precisely the endomorphism-valued one-form such that the connection   on   differs from the trivial connection   on  , which exists because   is a trivialising set for  .

Relationship to Christoffel symbolsEdit

In pseudo-Riemannian geometry, the Levi-Civita connection is often written in terms of the Christoffel symbols   instead of the connection one-form  . It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition to   being a trivialising open subset for the vector bundle  , that   is also a local chart for the manifold  , admitting local coordinates  .

In such a local chart, there is a distinguished local frame for the differential one-forms given by  , and the local connection one-forms   can be expanded in this basis as

 

for a collection of local smooth functions  , called the Christoffel symbols of   over  . In the case where   and   is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.

The expression for how   acts in local coordinates can be further expanded in terms of the local chart   and the Christoffel symbols, to be given by

 

Contracting this expression with the local coordinate tangent vector   leads to

 

This defines a collection of   locally defined operators

 

with the property that

 

Change of local trivialisationEdit

Suppose   is another choice of local frame over the same trivialising set  , so that there is a matrix   of smooth functions relating   and  , defined by

 

Tracing through the construction of the local connection form   for the frame  , one finds that the connection one-form   for   is given by

 

where   denotes the inverse matrix to  . In matrix notation this may be written

 

where   is the matrix of one-forms given by taking the exterior derivative of the matrix   component-by-component.

In the case where   is the tangent bundle and   is the Jacobian of a coordinate transformation of  , the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.

Parallel transport and holonomyEdit

A connection   on a vector bundle   defines a notion of parallel transport on   along a curve in  . Let   be a smooth path in  . A section   of   along   is said to be parallel if

 

for all  . Equivalently, one can consider the pullback bundle   of   by  . This is a vector bundle over   with fiber   over  . The connection   on   pulls back to a connection on  . A section   of   is parallel if and only if  .

Suppose   is a path from   to   in  . The above equation defining parallel sections is a first-order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector   in   there exists a unique parallel section   of   with  . Define a parallel transport map

 

by  . It can be shown that   is a linear isomorphism, with inverse given by following the same procedure with the reversed path   from   to  .

 
How to recover the covariant derivative of a connection from its parallel transport. The values   of a section   are parallel transported along the path   back to  , and then the covariant derivative is taken in the fixed vector space, the fibre   over  .

Parallel transport can be used to define the holonomy group of the connection   based at a point   in  . This is the subgroup of   consisting of all parallel transport maps coming from loops based at  :

 

The holonomy group of a connection is intimately related to the curvature of the connection (AmbroseSinger 1953).

The connection can be recovered from its parallel transport operators as follows. If   is a vector field and   a section, at a point   pick an integral curve   for   at  . For each   we will write   for the parallel transport map traveling along   from   to  . In particular for every  , we have  . Then   defines a curve in the vector space  , which may be differentiated. The covariant derivative is recovered as

 

This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms   between fibres of   and taking the above expression as the definition of  .

CurvatureEdit

The curvature of a connection   on   is a 2-form   on   with values in the endomorphism bundle  . That is,

 

It is defined by the expression

 

where   and   are tangent vector fields on   and   is a section of  . One must check that   is  -linear in both   and   and that it does in fact define a bundle endomorphism of  .

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

 

A flat connection is one whose curvature form vanishes identically.

Local form and Cartan's structure equationEdit

The curvature form has a local description called Cartan's structure equation. If   has local form   on some trivialising open subset   for  , then

 

on  . To clarify this notation, notice that   is a endomorphism-valued one-form, and so in local coordinates takes the form of a matrix of one-forms. The operation   applies the exterior derivative component-wise to this matrix, and   denotes matrix multiplication, where the components are wedged rather than multiplied.

In local coordinates   on   over  , if the connection form is written   for a collection of local endomorphisms  , then one has

 

Further expanding this in terms of the Christoffel symbols   produces the familiar expression from Riemannian geometry. Namely if   is a section of   over  , then

 

Here   is the full curvature tensor of  , and in Riemannian geometry would be identified with the Riemannian curvature tensor.

It can be checked that if we define   to be wedge product of forms but commutator of endomorphisms as opposed to composition, then  , and with this alternate notation the Cartan structure equation takes the form

 

This alternate notation is commonly used in the theory of principal bundle connections, where instead we use a connection form  , a Lie algebra-valued one-form, for which there is no notion of composition (unlike in the case of endomorphisms), but there is a notion of a Lie bracket.

In some references (see for example (MadsenTornehave1997)) the Cartan structure equation may be written with a minus sign:

 

This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.

Bianchi identityEdit

A version of the second (differential) Bianchi identity from Riemannian geometry holds for a connection on any vector bundle. Recall that a connection   on a vector bundle   induces an endomorphism connection on  . This endomorphism connection has itself an exterior covariant derivative, which we ambiguously call  . Since the curvature is a globally defined  -valued two-form, we may apply the exterior covariant derivative to it. The Bianchi identity says that

 .

This succinctly captures the complicated tensor formulae of the Bianchi identity in the case of Riemannian manifolds, and one may translate from this equation to the standard Bianchi identities by expanding the connection and curvature in local coordinates.

There is no analogue in general of the first (algebraic) Bianchi identity for a general connection, as this exploits the special symmetries of the Levi-Civita connection. Namely, one exploits that the vector bundle indices of   in the curvature tensor   may be swapped with the cotangent bundle indices coming from   after using the metric to lower or raise indices. For example this allows the torsion-freeness condition   to be defined for the Levi-Civita connection, but for a general vector bundle the  -index refers to the local coordinate basis of  , and the  -indices to the local coordinate frame of   and   coming from the splitting  . However in special circumstance, for example when the rank of   equals the dimension of   and a solder form has been chosen, one can use the soldering to interchange the indices and define a notion of torsion for affine connections which are not the Levi-Civita connection.

Gauge transformationsEdit

Given two connections   on a vector bundle  , it is natural to ask when they might be considered equivalent. There is a well-defined notion of an automorphism of a vector bundle  . A section   is an automorphism if   is invertible at every point  . Such an automorphism is called a gauge transformation of  , and the group of all automorphisms is called the gauge group, often denoted   or  . The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle   of the frame bundle of the vector bundle  . This is not to be confused with the lowercase a adjoint bundle  , which is naturally identified with   itself. The bundle   is the associated bundle to the principal frame bundle by the conjugation representation of   on itself,  , and has fibre the same general linear group   where  . Notice that despite having the same fibre as the frame bundle   and being associated to it,   is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised as  

A gauge transformation   of   acts on sections  , and therefore acts on connections by conjugation. Explicitly, if   is a connection on  , then one defines   by

 

for  . To check that   is a connection, one verifies the product rule

 

It may be checked that this defines a left group action of   on the affine space of all connections  .

Since   is an affine space modelled on  , there should exist some endomorphism-valued one-form   such that  . Using the definition of the endomorphism connection   induced by  , it can be seen that

 

which is to say that  .

Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space   is the moduli space of all connections on  . In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on  , which is of significant interest in gauge theory and physics.

ExamplesEdit

  • A classical covariant derivative or affine connection defines a connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor products of the tangent bundle with itself and its dual.
  • A connection on   can be described explicitly as the operator
 
where   is the exterior derivative evaluated on vector-valued smooth functions and   are smooth. A section   may be identified with a map
 
and then
 
  • If the bundle is endowed with a bundle metric, an inner product on its vector space fibers, a metric connection is defined as a connection that is compatible with the bundle metric.
  • A Yang-Mills connection is a special metric connection which satisfies the Yang-Mills equations of motion.
  • A Riemannian connection is a metric connection on the tangent bundle of a Riemannian manifold.
  • A Levi-Civita connection is a special Riemannian connection: the metric-compatible connection on the tangent bundle that is also torsion-free. It is unique, in the sense that given any Riemannian connection, one can always find one and only one equivalent connection that is torsion-free. "Equivalent" means it is compatible with the same metric, although the curvature tensors may be different; see teleparallelism. The difference between a Riemannian connection and the corresponding Levi-Civita connection is given by the contorsion tensor.
  • The exterior derivative is a flat connection on   (the trivial line bundle over M).
  • More generally, there is a canonical flat connection on any flat vector bundle (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.

See alsoEdit

ReferencesEdit

  • Chern, Shiing-Shen (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes
  • Darling, R. W. R. (1994), Differential Forms and Connections, Cambridge, UK: Cambridge University Press, ISBN 0-521-46800-0
  • Kobayashi, Shoshichi; Nomizu, Katsumi (1996) [1963], Foundations of Differential Geometry, Vol. 1, Wiley Classics Library, New York: Wiley Interscience, ISBN 0-471-15733-3
  • Koszul, J. L. (1950), "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique, 78: 65–127
  • Wells, R.O. (1973), Differential analysis on complex manifolds, Springer-Verlag, ISBN 0-387-90419-0
  • Ambrose, W.; Singer, I.M. (1953), "A theorem on holonomy", Transactions of the American Mathematical Society, 75: 428–443, doi:10.2307/1990721
  • Donaldson, S.K. and Kronheimer, P.B., 1997. The geometry of four-manifolds. Oxford University Press.
  • Tu, L.W., 2017. Differential geometry: connections, curvature, and characteristic classes (Vol. 275). Springer.
  • Taubes, C.H., 2011. Differential geometry: Bundles, connections, metrics and curvature (Vol. 23). OUP Oxford.
  • Lee, J.M., 2018. Introduction to Riemannian manifolds. Springer International Publishing.
  • Madsen, I.H.; Tornehave, J. (1997), From calculus to cohomology: de Rham cohomology and characteristic classes, Cambridge University Press