双共变群格的黎曼几何
Riemannian Geometry of Bicovariant Group Lattices
摘要：
离散群的群格（Cayley有向图）与群上的微分演算自然对应。在这样的微分几何结构可以介绍以下一般食谱的非交换微分几何。尽管函数和（广义）微分形式之间的非交换性，对于在这项工作中考虑的“双协变”群格的子类，可以将中心几何对象如度量、挠率和曲率理解为具有（左）协变性质的“张量”。这确保了张量分量（相对于1-形式空间的基）在基的变化下以熟悉的齐次方式变换。度量和线性联络有一个自然的相容条件。由此产生的（伪）黎曼几何探讨在这项工作中。它表明，组件的度量确实能够正确地描述离散的几何形状，如长度和角度的属性。一个简单的几何理解，特别是扭转和曲率的实现。这种形式与格点规范理论有许多共同之处。例如，黎曼曲率是由向量围绕一个元格（对应于一个双角形、一个三角形或一个四边形）的平行输运决定的。
英文摘要：
Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite of the non-commutativity between functions and (generalized) differential forms, for the subclass of ``bicovariant'' group lattices considered in this work it is possible to understand central geometric objects like metric, torsion and curvature as ``tensors'' with (left) covariance properties. This ensures that tensor components (with respect to a basis of the space of 1-forms) transform in the familiar homogeneous way under a change of basis. There is a natural compatibility condition for a metric and a linear connection. The resulting (pseudo-) Riemannian geometry is explored in this work. It is demonstrated that the components of the metric are indeed able to properly describe properties of discrete geometries like lengths and angles. A simple geometric understanding in particular of torsion and curvature is achieved. The formalism has much in common with lattice gauge theory. For example, the Riemannian curvature is determined by parallel transport of vectors around a plaquette (which corresponds to a biangle, a triangle or a quadrangle).
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