A brief history of contact geometry and topology
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Submission date: 08. Jan. 2001
published in: Expositiones mathematicae, 19 (2001) 1, p. 25-53
DOI number (of the published article): 10.1016/S0723-0869(01)80014-1
The roots of Contact Geometry can be traced back to 1872, when Sophus Lie introduced the notion of contact transformation (Berührungstransformation) as a geometric tool to study systems of differential equations. The subject has manifold connections with other fields of pure mathematics, and a significant place in applied areas such as mechanics, optics, thermodynamics, or control theory. Nonetheless, contact geometry has for a long time been receiving less attention than its twin sister, symplectic geometry.
Contact Topology is of more recent origin. Topological methods have begun to play an important rôle in contact geometry from around the early 1970s, but global topological results remained isolated until the mid 1980s. Since then, 3-dimensional contact geometry and topology have experienced a time of immense and fruitful activity, and some important steps have been taken towards an understanding of higher-dimensional contact topology.
The principal aim of this survey is to trace some classical sources of contact geometry and topology, and to highlight a few results in the development of these fields that are fundamental for the current view of the subject. Given my own interests, the emphasis is on the more topological aspects. Concerning the historical exposition, a great deal of information can also be found in a paper by R. Lutz ("Quelques remarques historiques et prospectives sur la géométrie de contact"), and I have not tried to avoid a certain overlap with that survey.
The development of contact geometry between the years 1960 and 1985 is chronicled in detail by Lutz, so I am deliberately more selective about that period. By contrast, I focus more strongly on connections of contact geometry with other areas of mathematics and physics, and on some research interests of the past decade.