离散圆与环面的黎曼几何
Riemannian Geometry of a Discretized Circle and Torus
摘要：
我们推广了有限群上的黎曼几何的结果，并给出了有限循环群上极小非交换微分的所有线性联络的完全分类。我们解决了挠自由和度量相容性条件一般，并表明有几类解决方案，其中只有特殊的是兼容的度量，给出了一个Hilbert $C^\ast$ -模结构的空间上的一种形式。我们计算这些度量的曲率和标量曲率，并找到它们的连续极限。
英文摘要：
We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and metric compatibility condition in general and show that there are several classes of solutions, out of which only special ones are compatible with a metric that gives a Hilbert $C^\ast$-module structure on the space of the one-forms. We compute curvature and scalar curvature for these metrics and find their continuous limits.
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