Discrete Riemannian geometry

摘要: Within a framework of noncommutative geometry, we develop an analog (pseudo-) Riemannian geometry on finite and discrete sets. On set, there is counterpart the continuum metric tensor with simple geometric interpretation. The latter based correspondence between first order differential calculi digraphs (the vertices are given by elements set). Arrows originating from vertex span its (co)tangent space. If to measure length angles at some point, it has be taken as element left-linear product space 1-forms itself, not (nonlocal) over algebra functions, considered previously several authors. It turns out that linear connections can always extended this left product, so compatibility defined in same way geometry. In particular, case universal calculus Euclidean polyhedra recovered conditions vanishing torsion. our rather general (which also comprises structures which far away geometry), is, general, nothing like Ricci or curvature scalar. Because nonlocality products (over functions) forms, corresponding components (with respect module basis) turn nonlocal objects. But one make use parallel transport associated connection “localize” such objects, certain cases distinguished achieve this. leads covariant allow contraction tensor. Several examples worked illustrate procedure. Furthermore, hypercubic lattice propose new analogue (vacuum) Einstein equations.