Ehresmann connection

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.

A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of a vector bundle in a covariant manner.

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 (Ehresmann 1950).

This has the immediate benefit of being definable on a much broader class of structures than mere vector bundles.

Furthermore, many of the features of the covariant derivative still remain: parallel transport, curvature, and holonomy.

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 with respect to the group action.

, called the horizontal bundle of the connection, which is complementary to V, in the sense that it defines a direct sum decomposition

In more sophisticated terms, such an assignment of horizontal spaces satisfying these properties corresponds precisely to a smooth section of the jet bundle J1E → E. Equivalently, let Φ be the projection onto the vertical bundle V along H (so that H = ker Φ).

Thus Φ is a vector bundle homomorphism from TE to itself with the following properties (of projections in general): Conversely, if Φ is a vector bundle endomorphism of TE satisfying these two properties, then H = ker Φ is the horizontal subbundle of an Ehresmann connection.

Finally, note that Φ, 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.

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.

[2][3] These horizontal lifts are a direct analogue of parallel transport for other versions of the connection formalism.

This has important consequences for the differential geometry of fiber bundles: the space of sections of H is not a Lie subalgebra 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.

Thus R ∈ Ω2(E,TE) is the two-form on E with values in TE defined by or, in other terms, where X = XH + XV 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 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: where again [-,-] is the Frölicher-Nijenhuis bracket of Φ ∈ Ω1(E,TE) and R ∈ Ω2(E,TE).

An Ehresmann connection allows curves to have unique horizontal lifts locally.

For a complete Ehresmann connection, a curve can be horizontally lifted over its entire domain.

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 of the connection.

[4] 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[3] if it is invariant with respect to the G action on E in the sense that The one-parameter subgroups of G act vertically on E. The differential of this action allows one to identify the subspace

Thus reinterpreted, the connection form ω satisfies the following two properties: Conversely, it can be shown that such a g-valued 1-form on a principal bundle generates a horizontal distribution satisfying the aforementioned properties.

The fact that the Ehresmann connection is linear implies that in addition it verifies for every function

, and therefore is a covariant derivative of s. Conversely a covariant derivative ∇ on a vector bundle defines a linear Ehresmann connection by defining He, for e ∈ E with x=π(e), to be the image dsx(TxM) where s is a section of E with s(x) = e and ∇Xs = 0 for all X ∈ TxM.

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 : Fx → Fx′ is given by a G-transformation of the fibers (over sufficiently nearby points x and x′ in M joined by a curve).

[5] Given a principal connection on P, one obtains a G-connection on the associated fiber bundle E = P ×G F via pullback.

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-dimensional[6] Lie group acting effectively on F, there must exist a finite configuration of points (y1,...,ym) within F such that the G-orbit R = {(gy1,...,gym) | g ∈ G} 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.