This paper is a follow-up on the noncommutative differential geometry on infinitesimal spaces [19]. In the present work, we extend the algebraic convergence from [19] to the geometric setting. On the one hand, we reformulate the definition of finite dimensional compatible Dirac operators using Clifford algebras. This definition also leads to a new construction of a Laplace operator. On the other hand, after a brief introduction of the Von Mises-Fisher distribution on manifolds, we show that when the Dirac operators are interpreted as stochastic matrices, the sequence $(D_n)_{n\in \mathbb{N}}$ converges in average to the usual Dirac operator on a spin manifold. The same conclusion can be drawn for the Laplace operator.

翻译：本文是对无穷小空间上非交换微分几何[19]的后续研究。在现有工作中，我们将[19]中的代数收敛性推广到几何框架。一方面，我们利用Clifford代数重新表述了有限维相容Dirac算子的定义，该定义还导出了Laplace算子的新构造。另一方面，在简要介绍流形上的冯·米塞斯-费舍尔分布后，我们证明了当Dirac算子被解释为随机矩阵时，序列$(D_n)_{n\in \mathbb{N}}$在平均意义下收敛于自旋流形上的标准Dirac算子，且Laplace算子也得出相同结论。