Loops, their cores and symmetric spaces

Abstract

This paper is devoted to the relations among affine symmetric spaces, smooth Bol and Moufang loops, smooth left distributive quasigroups and differentiable 3-nets. The results are used to prove the analyticity of smooth Moufang loops and left distributive quasigroups with involutive left translations as well as to show the Lie nature of transformation groups naturally related to some classes of smooth binary systems and 3-nets. In the last section we establish power series expansion for local loops with weak associativity conditions and apply the methods of the previous sections in order to describe geodesic loops having euclidean lines either as their geodesic lines or as geodesic lines of their core.