Charles Ehresmann's concepts in differential geometry

Abstract

We outline some of the tools C. Ehresmann introduced in Differential Geometry (fiber bundles, connections, jets, groupoids, pseudogroups). We emphasize two aspects of C. Ehresmann’s works: use of Cartan notations for the theory of connections and semi-holonomic jets. Introduction. As R. Hermann puts it “Ehresmann’s work was very prophetic and provided a general framework for describing Geometry and the mathematics which have a geometric component”. One of the motivations of C. Ehresmann to build up the foundations of Differential Geometry was to understand Elie Cartan’s work from a global point of view. He was also influenced by S. Lie and E. Vessiot. Besides his written work, C. Ehresmann had a great influence in the development of Geometry. His lectures, his seminars (where well known mathematicians of many countries, as well as young mathematicians like Thom, Reeb, Kuiper, Wu Wen Tsun, . . . explained their results), his private conversations (where generously he gave new ideas) had played a significant part in this development. The written part of Ehresmann’s work in Differential Geometry and Algebraic Topology is published in the first volume (parts 1 and 2) of his “Œuvres completes” [6]. This volume also contains a report of C. Ehresmann on his own work (written in 1955) and comments from W. Van Est (homogeneous spaces and Lie groups), M. Zisman (fiber bundles), G. Reeb (foliations), R. Thom (jets), P. Libermann (connections and Lie pseudogroups), J. Pradines (groupoids), R. Hermann (applications to Physics and control 2000 Mathematics Subject Classification: Primary 53C05; Secondary 58A20, 58H05.