Once Euclidean geometry was no longer considered the only possibility for describing physical space, the question arose how the structure of space, or of spacetime, could be characterized in terms of symmetry structures on manifolds. The question was posed and discussed by Riemann (1854), taken up anew by Helmholtz at the end of the 1860s with ingenious conceptual insight but mathematically quite vague, even from the standpoint of contemporary mathematics. Helmholtz’s argument was refined by Lie (1886) and, independently, by Killing (1885). It acquired a prominent place in the broader discourse on mathematics and reality at the turn to the 20th century. In these considerations the possibility of “freely moving” rigid bodies as measuring devices was crucial.
Erhard Scholze, The problem of space in the light of relativity: the views of H. Weyl and E. Cartan