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.

中文翻译：

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关于具有挠率和非度量性的非交换三球面的注解

通常在非交换三球面上采用的Levi-Civita联络被推广到Weyl类的具有挠度和非度量性的联络。这导致几何性质与普通性质不同，特别是提供了一个非交换“远平行”的例子。与经典（即交换）微分几何中的类似情况进行了比较。开发了一个通用框架，从中可以确定具有给定度量、扭转和非度量的唯一连接。