View source for Ehresmann connection

The genesis of the general concept of connection on an arbitrary fibred manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200602.png" />, was inspired by a paper by Ch. Ehresmann, [[#References|[a1]]], where he analyzed the classical approaches to connections from the global point of view (cf. also [[Connections on a manifold|Connections on a manifold]]; [[Fibre space|Fibre space]]; [[Manifold|Manifold]]). The main idea is that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200603.png" /> one prescribes an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200604.png" />-dimensional linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200605.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200607.png" /> which is complementary to the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200608.png" /> of the fibre passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e1200609.png" />. These spaces are called the horizontal spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006010.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006011.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006012.png" />-dimensional distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006013.png" />.

There are three main ways to interpret an Ehresmann connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006014.png" />:

1) As the lifting mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006015.png" />, transforming every vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006016.png" /> into the unique vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006017.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006019.png" />. So, every vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006021.png" /> is lifted into a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006023.png" />. The parallel transport on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006024.png" /> along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006026.png" /> is determined by the integral curves of the lifts of the tangent vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006027.png" />.

2) As the connection form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006028.png" />, transforming every vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006029.png" /> into its first component with respect to the direct sum decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006030.png" />. Since the vertical tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006031.png" /> is a subbundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006032.png" />, the connection form is a special tangent-valued one-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006033.png" />.

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006034.png" /> is identified with an element of the first [[Jet|jet]] prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006036.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006037.png" /> is interpreted as a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006038.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006039.png" /> is a [[Vector bundle|vector bundle]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006040.png" /> is a linear morphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006041.png" /> is called a [[Linear connection|linear connection]]. (From this viewpoint, an Ehresmannn connection is also said to be a non-linear connection.) A classical connection on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006042.png" /> corresponds to a linear connection on the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006044.png" /> is a [[Principal fibre bundle|principal fibre bundle]] with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006045.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006046.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006047.png" />-invariant, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006048.png" /> is called a principal connection. These connections have been used most frequently. On the other hand, a big advantage of connections without any additional structure is that prolongation procedures of functorial character can be applied to them with no restriction.

The main geometric object determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006049.png" /> is its curvature. This is a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006050.png" />, whose definition varies according to the above three cases.

1) This is the obstruction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006051.png" /> for lifting the bracket of vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006053.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006054.png" />.

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006055.png" /> is one half of the [[Frölicher–Nijenhuis bracket|Frölicher–Nijenhuis bracket]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006056.png" /> of the tangent-valued one-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006057.png" /> with itself.

3) Consider the jet prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006058.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006059.png" /> characterizes the deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006060.png" /> from the second jet prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006062.png" />, which is a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006063.png" />.

The curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006064.png" /> vanishes if and only if the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006065.png" /> is a [[Foliation|foliation]].

Every Ehresmann connection satisfies the [[Bianchi identity|Bianchi identity]]. In the second approach, this is the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006066.png" /></td> </tr></table>

which is one of the basic properties of the Frölicher–Nijenhuis bracket. For a classical connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006067.png" />, this relation coincides with the second Bianchi identity.

For every section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006068.png" />, one defines its absolute differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006069.png" /> as the projection of the tangent mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006070.png" /> in the direction of the horizontal spaces. Iterated absolute differentiation is based on the fact that every Ehresmann connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006071.png" /> induces canonically an Ehresmann connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006072.png" />, [[#References|[a2]]].

If a tangent-valued one-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006074.png" /> is given, then the Frölicher–Nijenhuis bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006075.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006077.png" />-torsion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006078.png" />. This leads to a far-reaching generalization of the concept of [[Torsion|torsion]] of a classical connection, [[#References|[a3]]]. Even in this case, the basic properties of the Frölicher–Nijenhuis bracket yield a relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120060/e12006079.png" /></td> </tr></table>

which generalizes the first Bianchi identity of a classical connection.

A systematic presentation of the theory of Ehresmann connections (under the name of general connections) can be found in [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Ehresmann, "Les connections infinitésimales dans un espace fibré différentiable" ''Colloq. de Topol., CBRM, Bruxelles'' (1950) pp. 29–55</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993) {{MR|1202431}} {{ZBL|1084.53001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Modugno, M., "Torsion and Ricci tensor for non-linear connections" ''Diff. Geom. Appl.'' , '''1''' (1991) pp. 177–192 {{MR|1244443}} {{ZBL|}} </TD></TR></table>