  More precisely, Weyl generalized the so-called  Riemann-Helmholtz-Lie problem of space in two ways: First,  he allowed for indefinite metrics in order to encompass the  general theory of relativity. Secondly, he considered metrics with  variable gauge \(\lambda(x(p))\) together with an  associated length connection, in order to obtain a  purely infinitesimal geometry. Thus each member of a general class of geometries under consideration is locally determined  relative to a choice of variable gauge by two structural fields  (Strukturfelder): (1) a possibly indefinite Finsler metric   field[55]  \(F_{p}(dx)\), and (2) a length connection that is determined by a 1-form field \(\theta_{i}dx^{i}\). Weyl’s task was to prove: 