Differentiable loops on the real line

The paper is devoted to the study of differentiable loops L on the real line such that the group G topologically generated by the left translations is locally compact and hence it is isomorphic to the universal covering group of PSL₂(ℝ). Using the methods developed in [3] we introduce a class of natural parametrizations of the loop manifold L corresponding to the Iwasava decompositions of G and find explicit expressions for the loop multiplication with respect to the given parametrizations. We characterize the differentiable curves ℝ → G consisting of the left translations of a loop L in the biinvariant Lorentzian geometry of G.