The Early Geometrical Works of Sophus Lie and Felix Klein

Abstract

This chapter discusses the geometrical works of Sophus Lie and Felix Klein. Sophus Lie and Felix Klein co-authored two notes that were published in the Comptes rendus of the Paris Academy, and they began work on a paper devoted to the same subject, namely, the theory of W-curves and their associated transformations. When the names Lie and Klein are mentioned together, one thinks of their mutual contributions to the development of group theory, the former through the theory of Lie groups and the latter through the application of group theory to geometry as expounded in Klein's “Erlanger Programm.” During this period, they developed a wealth of interesting ideas, techniques, and results. This work was of decisive importance for the emergence of Lie's theory of transformation groups. Majority of writers referring to the “Erlanger Programm” derive their knowledge of it from secondary sources. Most of these accounts trivialize its content by restricting their discussion to examples in which familiar transformations groups act on real spaces.