A note about the non-commutative three-sphere with torsion and non-metricity

Abstract

The Levi-Civita connection usually adopted on the non-commutative three-sphere is generalized to a connection possessing torsion and a non-metricity of the Weyl kind. This leads to geometrical properties that are qualitatively different from the ordinary ones and, in particular, provides an example of non-commutative “teleparallelism”. The comparison is made with the analogous case in classical (i.e. commutative) differential geometry. A general framework is developed, from which the unique connection having a given metric, torsion and non-metricity can be determined.