CONTINUOUS GROUPS OF TRANSFORMATIONS: ELIE CARTAN'S STRUCTURAL APPROACH

The thesis deals with E. Cartan's contributions to the theory of continuous groups of transformations from the early 1890's up to the early 1910's. The analysis is focused on both finite and infinite continuous groups. First, Cartan's doctoral dissertation, in which he provided a rigorous classification of what nowadays we would call simple complex Lie algebras, is taken in. A detailed survey of the theory of infinite continuous groups as developed by Sophus Lie, Friedrich Engel, Paolo Medolaghi and Ernst Vessiot follows. The second part of the dissertation concentrates upon Cartan's contributions to the subject. The analysis of the relevant works is preceded by a historical study of the genesis of Cartan's integration theory of general Pfaffian systems (nowadays known as Cartan-Kaehler theory), in which, for the first time, due attention to the work of the German mathematician Eduard Ritter von Weber is paid. Special emphasis is put on the structural aspects of Cartan's highly innovative approach. At the same time, the role played by group theory in the development of 20th century differential geometry is underlined in respect to Cartan's method of moving frames.