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).
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.
(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.
(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.
(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.
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 , 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.
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.
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 connection
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 connection
The direct sum connection is defined by
Symmetric and exterior power connections
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.
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.
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 forms
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 connections
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 connections
Let be a vector bundle of rank and let be the principalframe 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.
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 symbols
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 trivialisation
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 holonomy
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 .
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 equation
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
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.
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.
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
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.
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.