Geodesics and distance in classical physics
  We formulate geodesics on a manifold in terms of a parallel transfer of a particle state vector transformed by local Lorentz and Yang-Mills symmetry groups. This formulation leads to an introduction of a canonical one-form the eigenvalues of which define distance on a manifold. We suggest an action based on the canonical distance form and apply it to describe classical particles with spin. Arguments are presented in favour of scaling distance in the space-time with a scalar field. 
A. N. Grigorenko