In
differential geometry, an Ehresmann connection (after the French mathematician
Charles Ehresmann who first formalized this concept) is a version of the notion of a
connection, which makes sense on any smooth
fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless,
linear connections may be viewed as a special case. Another important special case of Ehresmann connections are
principal connections on
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s, which are required to be
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
in the principal
Lie group action.
Introduction
A
covariant derivative in differential geometry is a
linear differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
which takes the
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
of a section of a
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 p ...
in a
covariant manner. It also allows one to formulate a notion of a
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
section of a bundle in the direction of a vector: a section ''s'' is parallel along a vector ''X'' if
. So a covariant derivative provides at least two things: a differential operator, ''and'' a notion of what it means to be parallel in each direction. An Ehresmann connection drops the differential operator completely and defines a connection axiomatically in terms of the sections parallel in each direction . Specifically, an Ehresmann connection singles out a
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of each
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to the total space of the fiber bundle, called the ''horizontal space''. A section ''s'' is then horizontal (i.e., parallel) in the direction ''X'' if
lies in a horizontal space. Here we are regarding ''s'' as a function
from the base ''M'' to the fiber bundle ''E'', so that
is then the
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
of tangent vectors. The horizontal spaces together form a vector subbundle of
.
This has the immediate benefit of being definable on a much broader class of structures than mere vector bundles. In particular, it is well-defined on a general
fiber bundle. Furthermore, many of the features of the covariant derivative still remain: parallel transport,
curvature, and
holonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
.
The missing ingredient of the connection, apart from linearity, is ''covariance''. With the classical covariant derivatives, covariance is an ''a posteriori'' feature of the derivative. In their construction one specifies the transformation law of the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
– which is not covariant – and then general covariance of the ''derivative'' follows as a result. For an Ehresmann connection, it is possible to impose a generalized covariance principle from the beginning by introducing a
Lie group acting on the fibers of the fiber bundle. The appropriate condition is to require that the horizontal spaces be, in a certain sense,
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
with respect to the group action.
The finishing touch for an Ehresmann connection is that it can be represented as a
differential form, in much the same way as the case of a
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.
Historically, connection forms were introduced by Élie Carta ...
. If the group acts on the fibers and the connection is equivariant, then the form will also be equivariant. Furthermore, the connection form allows for a definition of curvature as a
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algeb ...
as well.
Formal definition
![Ehresmann connection](https://upload.wikimedia.org/wikipedia/commons/6/62/Ehresmann_connection.png)
Let
be a smooth
fiber bundle. Let
:
be the
vertical bundle
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
consisting of the vectors "tangent to the fibers" of ''E'', i.e. the fiber of ''V'' at
is
. This subbundle of
is canonically defined even when there is no canonical subspace tangent to the base space ''M''. (Of course, this asymmetry comes from the very definition of a fiber bundle, which "only has one projection"
while a product
would have two.)
Definition via horizontal subspaces
An Ehresmann connection on ''E'' is a smooth subbundle ''H'' of
, called the
horizontal bundle
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the vertical bundle VE and horizontal bundle HE are subbundles of ...
of the connection, which is complementary to ''V'', in the sense that it defines a
direct sum decomposition
. In more detail, the horizontal bundle has the following properties.
* For each point
,
is a
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of the tangent space
to ''E'' at ''e'', called the ''horizontal subspace'' of the connection at ''e''.
*
depends
smoothly on ''e''.
* For each
,
.
* Any tangent vector in ''T''
''e''''E'' (for any ''e''∈''E'') is the sum of a horizontal and vertical component, so that ''T''
''e''''E'' = ''H''
''e'' + ''V''
''e''.
In more sophisticated terms, such an assignment of horizontal spaces satisfying these properties corresponds precisely to a smooth section of the
jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...
''J''
1''E'' → ''E''.
Definition via a connection form
Equivalently, let be the projection onto the vertical bundle ''V'' along ''H'' (so that ''H'' = ker ). This is determined by the above ''direct sum'' decomposition of ''TE'' into horizontal and vertical parts and is sometimes called the
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.
Historically, connection forms were introduced by Élie Carta ...
of the Ehresmann connection. Thus is a
vector bundle homomorphism
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 ...
from ''TE'' to itself with the following properties (of projections in general):
*
2 = ;
* is the identity on ''V'' =Im .
Conversely, if is a vector bundle
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
of ''TE'' satisfying these two properties, then ''H'' = ker is the horizontal subbundle of an Ehresmann connection.
Finally, note that , being a linear mapping of each tangent space into itself, may also be regarded as a ''TE''-valued 1-form on ''E''. This will be a useful perspective in sections to come.
Parallel transport via horizontal lifts
An Ehresmann connection also prescribes a manner for lifting curves from the base manifold ''M'' into the total space of the fiber bundle ''E'' so that the tangents to the curve are horizontal. These horizontal lifts are a direct analogue of
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
for other versions of the connection formalism.
Specifically, suppose that ''γ''(''t'') is a smooth curve in ''M'' through the point ''x'' = ''γ''(0). Let ''e'' ∈ ''E''
''x'' be a point in the fiber over ''x''. A lift of ''γ'' through ''e'' is a curve
in the total space ''E'' such that
:
, and
A lift is horizontal if, in addition, every tangent of the curve lies in the horizontal subbundle of ''TE'':
:
It can be shown using the
rank–nullity theorem
The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theo ...
applied to ''π'' and that each vector ''X''∈''T''
''x''''M'' has a unique horizontal lift to a vector
. In particular, the tangent field to ''γ'' generates a horizontal vector field in the total space of the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
''γ''*''E''. By the
Picard–Lindelöf theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cau ...
, this vector field is
integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. Thus, for any curve ''γ'' and point ''e'' over ''x'' = ''γ''(0), there exists a ''unique horizontal lift'' of ''γ'' through ''e'' for small time ''t''.
Note that, for general Ehresmann connections, the horizontal lift is path-dependent. When two smooth curves in ''M'', coinciding at ''γ''
1(0) = ''γ''
2(0) = ''x''
0 and also intersecting at another point ''x''
1 ∈ ''M'', are lifted horizontally to ''E'' through the same ''e'' ∈ ''π''
−1(''x''
0), they will generally pass through different points of ''π''
−1(''x''
1). This has important consequences for the differential geometry of fiber bundles: the space of sections of ''H'' is not a
Lie subalgebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of the space of vector fields on ''E'', because it is not (in general) closed under the
Lie bracket of vector fields. This failure of closure under Lie bracket is measured by the ''curvature''.
Properties
Curvature
Let be an Ehresmann connection. Then the curvature of is given by
: