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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 \nabla_X s = 0. 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 s(X) lies in a horizontal space. Here we are regarding ''s'' as a function s\colon M\to E from the base ''M'' to the fiber bundle ''E'', so that s\colon TM\to s^*TE 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 TE. 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

Let \pi\colon E\to M be a smooth fiber bundle. Let :V= \ker (\operatorname \pi \colon TE\to TM) 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 e\in E is V_e =T_e(E_). This subbundle of TE 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" \pi\colon E\to M while a product E=M\times F would have two.)


Definition via horizontal subspaces

An Ehresmann connection on ''E'' is a smooth subbundle ''H'' of TE, 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 TE=H\oplus V. In more detail, the horizontal bundle has the following properties. * For each point e\in E, H_e 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 T_e E to ''E'' at ''e'', called the ''horizontal subspace'' of the connection at ''e''. * H_e depends smoothly on ''e''. * For each e\in E, H_e \cap V_e = \. * 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 \tilde(t) in the total space ''E'' such that :\tilde(0) = e, and \pi(\tilde(t)) = \gamma(t). A lift is horizontal if, in addition, every tangent of the curve lies in the horizontal subbundle of ''TE'': :\tilde'(t) \in H_. 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 \tilde \in T_e E. 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 :R = \tfrac ,v/math> where ,-denotes the Frölicher-Nijenhuis bracket of ∈ Ω1(''E'',''TE'') with itself. Thus ''R'' ∈ Ω2(''E'',''TE'') is the two-form on ''E'' with values in ''TE'' defined by :R(X,Y) = v\left( \mathrm - v)X,(\mathrm - v)Yright), or, in other terms, :R\left(X,Y\right) = \left _H,Y_H\rightV, where ''X'' = ''X''H + ''X''V denotes the direct sum decomposition into ''H'' and ''V'' components, respectively. From this last expression for the curvature, it is seen to vanish identically if, and only if, the horizontal subbundle is Frobenius integrable. Thus the curvature is the
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
for the horizontal subbundle to yield transverse sections of the fiber bundle ''E'' → ''M''. The curvature of an Ehresmann connection also satisfies a version of the Bianchi identity: :\left , R\right= 0 where again ,-is the Frölicher-Nijenhuis bracket of v ∈ Ω1(''E'',''TE'') and ''R'' ∈ Ω2(''E'',''TE'').


Completeness

An Ehresmann connection allows curves to have unique horizontal lifts
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...
. For a complete Ehresmann connection, a curve can be horizontally lifted over its entire domain.


Holonomy

Flatness of the connection corresponds locally to the Frobenius integrability of the horizontal spaces. At the other extreme, non-vanishing curvature implies the presence of
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 ...
of the connection.


Special cases


Principal bundles and principal connections

Suppose that ''E'' is a smooth principal ''G''-bundle over ''M''. Then an Ehresmann connection ''H'' on ''E'' is said to be a principal (Ehresmann) connection if it is invariant with respect to the ''G'' action on ''E'' in the sense that :H_=\mathrm d(R_g)_e (H_) for any ''e''∈''E'' and ''g''∈''G''; here \mathrm d(R_g)_e denotes the differential of the right action of ''g'' on ''E'' at ''e''. The one-parameter subgroups of ''G'' act vertically on ''E''. The differential of this action allows one to identify the subspace V_e with the Lie algebra g of group ''G'', say by map \iota\colon V_e\to \mathfrak g. The connection form ''v'' of the Ehresmann connection may then be viewed as a 1-form ''ω'' on ''E'' with values in g defined by ''ω''(''X'')=''ι''(''v''(''X'')). Thus reinterpreted, the connection form ''ω'' satisfies the following two properties: * It transforms
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 ...
ly under the ''G'' action: R_h^*\omega=\hbox(h^)\omega for all ''h''∈''G'', where ''R''''h''* is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
under the right action and ''Ad'' is the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
of ''G'' on its Lie algebra. * It maps vertical vector fields to their associated elements of the Lie algebra: ''ω''(''X'')=''ι''(''X'') for all ''X''∈''V''. Conversely, it can be shown that such a g-valued 1-form on a principal bundle generates a horizontal distribution satisfying the aforementioned properties. Given a local trivialization one can reduce ''ω'' to the horizontal vector fields (in this trivialization). It defines a 1-form ''ω' '' on ''B'' via
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
. The form ''ω determines ''ω'' completely, but it depends on the choice of trivialization. (This form is often also called 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 ...
and denoted simply by ''ω''.)


Vector bundles and covariant derivatives

Suppose that ''E'' is a smooth
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 ...
over ''M''. Then an Ehresmann connection ''H'' on ''E'' is said to be a linear (Ehresmann) connection if ''H''''e'' depends linearly on ''e'' ∈ ''E''''x'' for each ''x'' ∈ ''M''. To make this precise, let ''S''''λ'' denote scalar multiplication by ''λ'' on ''E''. Then ''H'' is linear if and only if H_ = \mathrm d(S_)_e (H_)for any ''e'' ∈ ''E'' and scalar λ. Since ''E'' is a vector bundle, its vertical bundle ''V'' is isomorphic to ''π''*''E''. Therefore if ''s'' is a section of ''E'', then ''v''(d''s''):''TM''→''s''*''V''=''s''*''π''*''E''=''E''. It is a vector bundle morphism, and is therefore given by a section ∇''s'' of the vector bundle Hom(''TM'',''E''). The fact that the Ehresmann connection is linear implies that in addition it verifies for every function f on M the Leibniz rule, i.e. \nabla(f s) = f\nabla (s) + d(f)\otimes s, and therefore is a covariant derivative of ''s''. Conversely a covariant derivative ''∇'' on a vector bundle defines a linear Ehresmann connection by defining ''H''''e'', for ''e'' ∈ ''E'' with ''x''=''π''(''e''), to be the image d''s''''x''(''T''''x''''M'') where ''s'' is a section of ''E'' with ''s''(''x'') = ''e'' and ∇''X''''s'' = 0 for all ''X'' ∈ ''T''''x''''M''. Note that (for historical reasons) the term ''linear'' when applied to connections, is sometimes used (like the word ''affine'' – see
Affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
) to refer to connections defined on the tangent bundle or
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
.


Associated bundles

An Ehresmann connection on a fiber bundle (endowed with a structure group) sometimes gives rise to an Ehresmann connection on an
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
. For instance, a (linear) connection in a vector bundle ''E'', thought of giving a parallelism of ''E'' as above, induces a connection on the associated bundle of frames P''E'' of ''E''. Conversely, a connection in P''E'' gives rise to a (linear) connection in ''E'' provided that the connection in P''E'' is equivariant with respect to the action of the general linear group on the frames (and thus a
principal connection In mathematics, and especially differential geometry and gauge theory, 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. A principal ''G''-conne ...
). It is ''not always'' possible for an Ehresmann connection to induce, in a natural way, a connection on an associated bundle. For example, a non-equivariant Ehresmann connection on a bundle of frames of a vector bundle may not induce a connection on the vector bundle. Suppose that ''E'' is an associated bundle of ''P'', so that ''E'' = ''P'' ×G ''F''. A ''G''-connection on ''E'' is an Ehresmann connection such that the parallel transport map τ : ''F''x → ''F''x′ is given by a ''G''-transformation of the fibers (over sufficiently nearby points ''x'' and ''x''′ in ''M'' joined by a curve). Given a principal connection on ''P'', one obtains a ''G''-connection on the associated fiber bundle ''E'' = ''P'' ×G ''F'' via
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
. Conversely, given a ''G''-connection on ''E'' it is possible to recover the principal connection on the associated principal bundle ''P''. To recover this principal connection, one introduces the notion of a ''frame'' on the typical fiber ''F''. Since ''G'' is a finite-dimensionalFor convenience, we assume that ''G'' is finite-dimensional, although this assumption can safely be dropped with minor modifications. Lie group acting effectively on ''F'', there must exist a finite configuration of points (''y''1,...,''y''m) within ''F'' such that the ''G''-orbit ''R'' = is a principal homogeneous space of ''G''. One can think of ''R'' as giving a generalization of the notion of a frame for the ''G''-action on ''F''. Note that, since ''R'' is a principal homogeneous space for ''G'', the fiber bundle ''E''(''R'') associated to ''E'' with typical fiber ''R'' is (equivalent to) the principal bundle associated to ''E''. But it is also a subbundle of the ''m''-fold product bundle of ''E'' with itself. The distribution of horizontal spaces on ''E'' induces a distribution of spaces on this product bundle. Since the parallel transport maps associated to the connection are ''G''-maps, they preserve the subspace ''E''(''R''), and so the ''G''-connection descends to a principal ''G''-connection on ''E''(''R''). In summary, there is a one-to-one correspondence (up to equivalence) between the descents of principal connections to associated fiber bundles, and ''G''-connections on associated fiber bundles. For this reason, in the category of fiber bundles with a structure group ''G'', the principal connection contains all relevant information for ''G''-connections on the associated bundles. Hence, unless there is an overriding reason to consider connections on associated bundles (as there is, for instance, in the case of
Cartan connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
s) one usually works directly with the principal connection.


Notes


References

* * * * * * * * * * *


Further reading

*
Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...
(1970) "Topological obstruction to integrability", ''Proc. Symp. Pure Math.'', 16 Amer. Math. Soc., Providence, RI. * {{DEFAULTSORT:Ehresmann Connection Connection (mathematics)