Ehresmann connection

The genesis of the general concept of connection on an arbitrary fibred manifold $p : Y \rightarrow M$, $m = \operatorname { dim } M$, 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 $y \in Y$ one prescribes an $m$-dimensional linear subspace $\Gamma ( y )$ of the tangent space $T _ { y } Y$ of $Y$ which is complementary to the tangent space $V _ { y } Y$ of the fibre passing through $y$. These spaces are called the horizontal spaces of $\Gamma$. Hence $\Gamma$ is an $m$-dimensional distribution on $Y$.

There are three main ways to interpret an Ehresmann connection $\Gamma$:

1) As the lifting mapping $Y \times_M TM \rightarrow T Y$, transforming every vector $A \in T _ { x } M$ into the unique vector $A _ { y } \in \Gamma ( y )$ satisfying $T p ( A _ { y } ) = A$, $x = p ( y )$. So, every vector field $X$ on $M$ is lifted into a vector field $\Gamma X$ on $Y$. The parallel transport on $Y$ along a curve $\gamma$ on $M$ is determined by the integral curves of the lifts of the tangent vectors of $\gamma$.

2) As the connection form $T Y \rightarrow V Y$, transforming every vector of $T _ { y } Y$ into its first component with respect to the direct sum decomposition $T _ { y } Y = V _ { y } Y + \Gamma ( y )$. Since the vertical tangent bundle $V Y$ is a subbundle of $T Y$, the connection form is a special tangent-valued one-form on $Y$.

3) $\Gamma ( y )$ is identified with an element of the first jet prolongation $J ^ { 1 } Y$ of $Y$. Then $\Gamma$ is interpreted as a section $Y \rightarrow J ^ { 1 } Y$.

If $Y$ is a vector bundle and $\Gamma : Y \rightarrow J ^ { 1 } Y$ is a linear morphism, then $\Gamma$ 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 $M$ corresponds to a linear connection on the tangent bundle $T M$. If $Y$ is a principal fibre bundle with structure group $G$, and $\Gamma$ is $G$-invariant, then $\Gamma$ 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 $\Gamma$ is its curvature. This is a section $C \Gamma : Y \rightarrow V Y \otimes \wedge ^ { 2 } T ^ { * } M$, whose definition varies according to the above three cases.

1) This is the obstruction $[ \Gamma X _ { 1 } , \Gamma X _ { 2 } ] - \Gamma ( [ X _ { 1 } , X _ { 2 } ] )$ for lifting the bracket of vector fields $X _ { 1 }$, $X _ { 2 }$ on $M$.

2) $C \Gamma$ is one half of the Frölicher–Nijenhuis bracket $[ \Gamma , \Gamma ]$ of the tangent-valued one-form $\Gamma$ with itself.

3) Consider the jet prolongation $J ^ { 1 } \Gamma : J ^ { 1 } Y \rightarrow J ^ { 1 } ( J ^ { 1 } Y \rightarrow M )$. Then $C \Gamma$ characterizes the deviation of $J ^ { 1 } \Gamma ( \Gamma ( Y ) )$ from the second jet prolongation $J ^ { 2 } Y$ of $Y$, which is a subspace of $J ^ { 1 } ( J ^ { 1 } Y \rightarrow M )$.

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

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

\begin{equation*} [ \Gamma , [ \Gamma , \Gamma ] ] = 0, \end{equation*}

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

For every section $s : M \rightarrow Y$, one defines its absolute differential $\nabla _ { \Gamma } s : T M \rightarrow V Y$ as the projection of the tangent mapping $Ts : T M \rightarrow T Y$ in the direction of the horizontal spaces. Iterated absolute differentiation is based on the fact that every Ehresmann connection on $Y$ induces canonically an Ehresmann connection on $V Y \rightarrow M$, [a2].

If a tangent-valued one-form $Q$ on $Y$ is given, then the Frölicher–Nijenhuis bracket $[ Q , \Gamma ]$ is called the $Q$-torsion of $\Gamma$. 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

\begin{equation*} [ Q , [ \Gamma , \Gamma ] ] = 2 [ [ Q , \Gamma ] , \Gamma ] \end{equation*}

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