Authors

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].