Teleparallel geometries not characterized by their scalar polynomial torsion invariants

Abstract

A teleparallel geometry is an n-dimensional manifold equipped with a frame basis and an independent spin connection. For such a geometry, the curvature tensor vanishes and the torsion tensor is non-zero. A straightforward approach to characterizing teleparallel geometries is to compute scalar polynomial invariants constructed from the torsion tensor and its covariant derivatives. An open question has been whether the set of all scalar polynomial torsion invariants, $\mathcal{I}_T$ uniquely characterize a given teleparallel geometry. In this paper we show that the answer is no and construct the most general class of teleparallel geometries in four dimensions which cannot be characterized by $\mathcal{I}_T$. As a corollary we determine all teleparallel geometries which have vanishing scalar polynomial torsion invariants.