A frame based approach to computing symmetries with non-trivial isotropy groups

A frame approach to determining the most general solution admitting a desired symmetry group has been examined previously in Riemannian and teleparallel geometries with some success. In teleparallel geometries, one must determine the general form of the frame and spin connection to generate a general solution admitting the desired symmetry group. Current approaches often rely on the use of the proper frame, where the spin connection is zero. However this leads to particular theoretical and practical problems. In this paper we introduce an entirely general approach to determining the most general Riemann-Cartan geometries which admit a given symmetry group and apply these results to teleparallel geometries. To illustrate the approach we determine the most general geometries, with the minimal number of arbitrary functions, for particular choices of symmetry groups with dimension one, three, six and seven. In addition, we rigorously show how the teleparallel analogues of the Robertson-Walker, de Sitter and Einstein static spacetimes can be determined.