The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles

TL;DR

This paper reviews the development of connection theories from Cartan's classical concepts to Ehresmann's fibre bundle approach, highlighting their mathematical structures and applications in physics.

Contribution

It provides a comprehensive review of Charles Ehresmann's work on connections, linking Cartan's classical ideas to modern fibre bundle formulations and applications.

Findings

Connections can be viewed as special cases of fibre bundle connections.

Cartan connections relate to homogeneous spaces and principal bundles.

Applications include modeling particle motion in gravitational fields.

Abstract

Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Elie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkowskian connections which can be considered as special types of affine connections, the group of affine transformations of the affine tangent space being replaced by a suitable subgroup; and more generally, conformal and projective connections, associated to a group which is no more a subgroup of the affine group. Around 1950, Charles Ehresmann introduced connections on a fibre bundle and, when the bundle has a Lie group as structure group, connection forms on the associated principal bundle, with values in the Lie algebra of the structure group. He called Cartan connections the various types of connections on a manifold previously introduced by E. Cartan, and explained how they can be considered as special cases of connections on a fibre bundle with a Lie group G as structure group: the standard fibre of the bundle is then an homogeneous space G/G' ; its dimension is equal to that of the base manifold; a Cartan connection determines an isomorphism of the vector bundle tangent to the the base manifold onto the vector bundle of vertical vectors tangent to the fibres of the bundle along a global section. These works are reviewed and some applications of the theory of connections are sketched.