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 bundles, which are required to be equivariant in the principal Lie group action.
Introduction
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. It also allows one to formulate a notion of a parallel section of a bundle in the direction of a vector: a section s is parallel along a vector X if . 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 of each tangent space 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 lies in a horizontal space. Here we are regarding s as a function from the base M to the fiber bundle E, so that is then the pushforward of tangent vectors. The horizontal spaces together form a vector subbundle of .
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.
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 – 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 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. 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 as well.
Formal definition
Let be a smooth fiber bundle. Let
be the vertical bundle consisting of the vectors "tangent to the fibers" of E, i.e. the fiber of V at is . This subbundle of 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" while a product would have two.)
Definition via horizontal subspaces
An Ehresmann connection on E is a smooth subbundle H of , called the horizontal bundle of the connection, which is complementary to V, in the sense that it defines a direct sum decomposition . In more detail, the horizontal bundle has the following properties. For each point , is a vector subspace of the tangent space to E at e, called the horizontal subspace of the connection at e. depends smoothly on e. For each , . Any tangent vector in TeE (for any e∈E) is the sum of a horizontal and vertical component, so that TeE = He + Ve.
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.
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 of the Ehresmann connection. Thus is a vector bundle homomorphism from TE to itself with the following properties (of projections in general): 2 = ; is the identity on V =Im .