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

### Volume 76 / 2007

#### Abstract

Around 1923, Élie Cartan introduced affine connections on
manifolds and defined the 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, Élie 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 É. 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.