Discrete Riemannian geometry

Citation
A. Dimakis et F. Muller-hoissen, Discrete Riemannian geometry, J MATH PHYS, 40(3), 1999, pp. 1518-1548
Citations number
36
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Physics
Journal title
JOURNAL OF MATHEMATICAL PHYSICS
ISSN journal
0022-2488 → ACNP
Volume
40
Issue
3
Year of publication
1999
Pages
1518 - 1548
Database
ISI
SICI code
0022-2488(199903)40:3<1518:DRG>2.0.ZU;2-T
Abstract
Within a framework of noncommutative geometry, we develop an analog of (pse udo-) Riemannian geometry on finite and discrete sets. On a finite set, the re is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set). Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some p oint, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (nonlocal) tensor product over the algebra of functions, as considered previously by s everal authors. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined i n the same way as in continuum Riemannian geometry. In particular, in the c ase of the universal differential calculus on a finite set, the Euclidean g eometry of polyhedra is recovered from conditions of metric compatibility a nd vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is, in general, nothing like a Ricci tensor or a curvature scalar. Because of the nonlocality of tensor products (over the algebra of functions) of f orms, corresponding components (with respect to some module basis) turn out to be rather nonlocal objects. But one can make use of the parallel transp ort associated with a connection to "localize'' such objects, and in certai n cases there is a distinguished way to achieve this. In particular, this l eads to covariant components of the curvature tensor which allow a contract ion to a Ricci tensor. Several examples are worked out to illustrate the pr ocedure. Furthermore, in the case of a differential calculus associated wit h a hypercubic lattice we propose a new discrete analogue of the (vacuum) E instein equations. (C) 1999 American Institute of Physics. [S0022-2488(99)0 0303-5].