Ehresmann connection

The genesis of the general concept of connection on an arbitrary fibred manifold , , was inspired by a paper by Ch. Ehresmann, [a1], where he analyzed the classical approaches to connections from the global point of view (cf. also Connections on a manifold; Fibre space; Manifold). The main idea is that at each point one prescribes an -dimensional linear subspace of the tangent space of which is complementary to the tangent space of the fibre passing through . These spaces are called the horizontal spaces of . Hence is an -dimensional distribution on .

There are three main ways to interpret an Ehresmann connection :

1) As the lifting mapping , transforming every vector into the unique vector satisfying , . So, every vector field on is lifted into a vector field on . The parallel transport on along a curve on is determined by the integral curves of the lifts of the tangent vectors of .

2) As the connection form , transforming every vector of into its first component with respect to the direct sum decomposition . Since the vertical tangent bundle is a subbundle of , the connection form is a special tangent-valued one-form on .

3) is identified with an element of the first jet prolongation of . Then is interpreted as a section .

If is a vector bundle and is a linear morphism, then is called a linear connection. (From this viewpoint, an Ehresmannn connection is also said to be a non-linear connection.) A classical connection on a manifold corresponds to a linear connection on the tangent bundle . If is a principal fibre bundle with structure group , and is -invariant, then 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 is its curvature. This is a section , whose definition varies according to the above three cases.

1) This is the obstruction for lifting the bracket of vector fields , on .

2) is one half of the Frölicher–Nijenhuis bracket of the tangent-valued one-form with itself.

3) Consider the jet prolongation . Then characterizes the deviation of from the second jet prolongation of , which is a subspace of .

The curvature of vanishes if and only if the distribution is a foliation.

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

which is one of the basic properties of the Frölicher–Nijenhuis bracket. For a classical connection on , this relation coincides with the second Bianchi identity.

For every section , one defines its absolute differential as the projection of the tangent mapping in the direction of the horizontal spaces. Iterated absolute differentiation is based on the fact that every Ehresmann connection on induces canonically an Ehresmann connection on , [a2].

If a tangent-valued one-form on is given, then the Frölicher–Nijenhuis bracket is called the -torsion of . This leads to a far-reaching generalization of the concept of torsion of a classical connection, [a3]. Even in this case, the basic properties of the Frölicher–Nijenhuis bracket yield a relation

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 [a2].

References

[a1]	C. Ehresmann, "Les connections infinitésimales dans un espace fibré différentiable" Colloq. de Topol., CBRM, Bruxelles (1950) pp. 29–55
[a2]	I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993) MR1202431 Zbl 1084.53001
[a3]	Modugno, M., "Torsion and Ricci tensor for non-linear connections" Diff. Geom. Appl. , 1 (1991) pp. 177–192 MR1244443