Volume 285, 2001
An overview of Lie's line-sphere correspondence.
We give an overview of Lie's sphere geometry, and discuss Lie's
discovery of the line-sphere correspondence. The article's aims are twofold.
First we discuss the role played by this result in Lie's mathematical evolu-
tion. Second, we indicate how the line-sphere correspondence can be used to
describe a class of surfaces with integrable asymptotic lines.
Sophus Like was one of the leading figures of late 19th century mathematics.
Today Lie is primarily remembered for the groups and algebras that bear his name,
and for his contributions to symmetry analysis of differential equations. However,
the full scope of Lie's accomplishments is considerably wider. To get a better per-
spective on Lie's research interests one must understand that Lie was first and
foremost a geometer. Geometric topics formed the core of his early research and
were the path that led him to subsequent discoveries. To quote Felix Klein (a close
friend and collaborator)
To fully understand the mathematical genius of Sophus Lie, one
must not turn to the books recently published by him in collaboration
with Dr. Engel, but to his earlier memoirs, written during the
first years of his scientific career. There Lie shows himself the true
geometer that he is, while in his later publications, finding that he
was but imperfectly understood by the mathematicians accustomed
to the analytical point of view, he adopted a very general analytical
form of treatment that is not always easy to follow.
This view is confirmed by a number of recent scholarly works [13, 6].
An important milestone in Lie's mathematical evolution was his research into
sphere-based geometry, as well as the discovery of a contact transformation that
relates sphere geometry to conventional projective geometry (in which lines are the
fundamental elements). These results were the highlight of Lie's doctoral disser-
tation and brought him immediate recognition. The discoveries also exercised a
powerful influence on F. Klein, and provided essential inspiration for the latter's
celebrated Erlanger Programm.
1991 Mathematics Subject Classification. 01-02 53A05.
2001 American Mathematical Society