Felix Klein and Sophus Lie on quartic surfaces in line geometry

Although rarely appreciated, the collaboration that brought Felix Klein and Sophus Lie together in 1869 had mainly to do with their common interests in the new field of line geometry. As mathematicians, Klein and Lie identified with the latest currents in geometry. Not long before, Klein’s mentor Julius Plücker launched the study of first- and second-degree line complexes, which provided much inspiration for Klein and Lie, though both were busy exploring a broad range of problems and theories. Klein used invariant theory and other algebraic methods to study the properties of line complexes, whereas Lie set his eyes on those aspects related to analysis and differential equations. Much later, historians and mathematicians came to treat the collaboration between Klein and Lie as a famous early chapter in the history of transformation groups, a development often identified with Klein’s “Erlangen Program” from 1872. The present detailed account of their joint work and mutual interests provides a very different picture of their early research, which had relatively little to do with group theory. This essay shows how the geometrical interests of Klein and Lie reflected contemporary trends by focusing on the central importance of quartic surfaces in line geometry.