2.10. Historical interlude: Sophus Lie 27

Lie and Klein traveled to Paris together, and there Lie produced

the famous contact transformation, which mapped straight lines into

spheres. An application of this expertise to the Earth sphere, however,

did not serve him well. After the outbreak of the Franco-Prussian war,

Lie could not find a better way to return to Norway than by first

hiking to Italy. With his peculiar hiking habits, such as taking off his

clothes in the rain and putting them into his backpack, he was not

able to flee very far. The French quickly apprehended him and found

papers filled with mysterious symbols. Lie’s efforts to explain the

meaning of his mathematical notation did not dispel the authorities’

suspicion that he was a German spy. A short stay in prison afforded

him some quiet time to complete his studies, and upon return to

Norway, Lie successfully defended his doctoral dissertation. Unable

to find a job in Norway, Lie resolved to go to Sweden, but Norwegian

patriots intervened, and the Norwegian National Assembly voted by

a large majority to establish a personal extraordinary professorship

for Lie at the University of Christiania. Although the salary offered

was less than extraordinary, he stayed.

Lie’s research on sphere mapping and his lively exchanges with

Klein led both of them to think of more general connections between

group theory and geometry. In 1872 Klein presented his famous Er-

langen Program, in which he suggested unifying specific geometries

under a general framework of projective geometry and using group

theory to organize all geometric knowledge. Lie and Klein clearly ar-

ticulated the notion of a transformation group, the continuous analog

of a permutation group, with promising applications to geometry and

differential equations, but they lacked a general theory of the sub-

ject. The Erlangen Program implied one aspect of this project — the

group classification problem — but Lie had no intention of attacking

this bastion at the time. As he later wrote to Klein, “[I]n your essay

the problem of determining all groups is not posited, probably on the

grounds that at the time such a problem seemed to you absurd or

impossible, as it did to me” [22, pp. 41–42].

By the end of 1873, Lie’s pessimism gave way to a much brighter

outlook. After dipping into the theory of first order differential equa-

tions, developed by Jacobi and his followers, and making considerable