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. 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. 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.